1 copy 2019 IOP Publishing Ltd Printed in the UK

Reports on Progress in Physics

P de Groot

Printed in the UK

056101

RPPHAG

copy 2019 IOP Publishing Ltd

82

Rep Prog Phys

ROP

1010881361-6633ab092d

5

Reports on Progress in Physics

Contents

1 Introduction 1 2 Distance and displacement 3 21 Measuring linear motions with a laser 3 22 Multi-dimensional optical encoders 5 23 Absolute distance and position measurement 6 3 Testing of optical components 8 31 The laser Fizeau interferometer 8 32 Vibration sensitivity and environmental

robustness 9 33 Lateral resolution and the instrument

transfer function 11 4 Interference microscopy for surface structure

analysis 14 41 Principles and design of 3D interference

microscopes 14 42 Signal modeling for interference microscopy 15 43 The pursuit of low measurement noise 16 5 Industrial surfaces and holographic interferometry 18 51 Geometrically desensitized interferometry for

rough surfaces 18

52 Infrared wavelengths 20 53 Holographic interferometry of temperature

gradients and sonic waves in fluids 20 6 Metrology for consumer electronics 22 61 Flying height of read-write heads of rigid

disk drives 22 62 Form and relational metrology of microlenses 24 63 Interferometric scatterometry 25 7 Summary and future work 26Acknowledgments 27References 27

1 Introduction

It is difficult to imagine the modern world without dimensional measurements to attach numbers to what we see Distances shapes and topographical maps using units from Angstroms to kilometers require measurement methods and instruments Metrology is fundamental to manufacturing for interchange-ability quality control matching components yield improve-ment and process development [1] The interchange of goods

A review of selected topics in interferometric optical metrology

Peter J de Groot

Zygo Corporation Laurel Brook Road Middlefield CT 06455 United States of America

E-mail peterdzygocom

Received 19 June 2018 revised 20 December 2018Accepted for publication 21 February 2019Published 23 April 2019

Recommended Professor Masud Mansuripur

AbstractThis review gathers together 15 special topics in modern interferometric metrology representing a sampling of historical current and future developments The selected topics cover a wide range of applications including distance and displacement measurement the testing of optical components interference microscopy for surface structure analysis form and dimensional measurements of industrial parts and recent applications in semiconductor manufacturing and consumer electronics Techniques range from laser Fizeau systems to dynamic ellipsometry using polarized heterodyne interferometry

Keywords interferometry metrology optics interferometer laser Fizeau microscopy

Review

IOP

2019

1361-6633

1361-663319056101+32$3300

httpsdoiorg1010881361-6633ab092dRep Prog Phys 82 (2019) 056101 (32pp)

Review

2

requires a common language of dimensions surface forms and textures supported by confident metrology for verifica-tion The scope of application ranges from art preservation to basic research

Metrology and optics have always been closely related Vision aided by angle measuring instruments and telescopes measures the diameters of distant stars while microscopes enable the eye to quantitatively evaluate metallurgical surface grain structure and the interior details of living cells [2] The meter was first determined by optically surveying the land-scape so as to determine the Earthrsquos circumference [3] then in 1875 by a visual comparisons of marks on the international prototype meter with transfer standards [4] Metrology sys-tems that require vision to function today include traditional optical comparators [5] alignment telescopes for finishing guideways on precision stages [5] and surface finish inspec-tion equipment based on a visual detection of scratches digs and defects [6]

Frequently-cited benefits of optical metrology include non-contact sensing parallel detection the ability to mag-nify redirect and de-magnify our line of sight using lenses and mirrors [7] These benefits are already sufficient to pro-mote the use of optical methods based on vision ray mapping and triangulation Yet there is moremdashsomething astonishingly powerful in the light-wave toolboxmdashwhich makes optics a preeminent physical principle for dimensional measurement the wavelength

In 1665 Hooke noted that the colors seen in white-light interference patterns are a sensitive measure of thickness between reflecting surfaces [8] In the second book of Opticks Newton reports of the experimental observation of circular interference fringes produced by two prisms with slightly con-vex surface in contact [9] Although Newton doubted the wave theory of light as an explanation he nonetheless convincingly demonstrated the metrology potential of what he was observ-ing By the 19th century Fizeau made use of Newtonrsquos rings to determine the thermal expansion of crystals using interfer-ence patterns from a sodium lamp which has a narrow enough optical spectrum to generate high-contrast fringes of a single color [10]

A similar setup to that of Newton and Fizeau is the light box traditionally used for rapid visual inspection of the form of partially-transparent optical surfaces [11] The fringes or light bands in figure 1 illustrate the power of this simple meas-urement method which reveals contours of an optical surface having a concave central area on the scale of 260 nm of sur-face height per fringe In the traditional optics shop a range of test plates of differing radii allow for the evaluation of lenses and mirrors with high accuracy [6]

Today the measurement principle illustrated by the inter-ference fringes in figure 1 has been advanced by the availabil-ity of lasers and computers Automated instruments such as the system shown in figure 2 rely on advanced cameras opto-mechanics and computers to provide sub-nm surface height measurements over millions of surface height positions in a few seconds

Interferometric metrology has also played an impor-tant role in fundamental research into the physical universe

The 1887 MichelsonndashMorley experiment using the interfer-ometer of figure 3 was a watershed test of the electromagnetic theory at the time using the extraordinary sensitivity of inter-ferometry in an attempt to confirm the existence of a fixed frame of reference for wave propagation [12] The instrument was configured to detect the difference in the speed of light between two orthogonal directions of propagation as one would expect from the relative motion of the earth through the stationary luminiferous aether Even with longer path lengths to increase sensitivity no such difference has been detected [13] This famous negative result contributed to the special theory of relativity as a way to reconcile Maxwellrsquos equa-tions with moving frames of reference

It is pleasing that basic configuration of the Michelson interferometer continues to play a role in understanding the

Figure 1 Surface metrology using the interference fringes obtained with filtered incandescent light and two glass components in near contact

Figure 2 Modern laser Fizeau interferometer for the testing of optical components using automated analysis of fringes similar to those shown in figure 1 Photo courtesy of Zygo Corporation

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3

physical universe but now on a much grander scale The laser interferometer gravitational-wave observatory (LIGO) also uses orthogonal beam directions but with path lengths meas-ured in kilometers rather than centimeters [14] Recirculating FabryndashPerot interferometer cavities increase the effective dis-tance and highly-refined phase measurements sense motions of the test masses on a scale thousands of times smaller than that of a proton Figure 4 shows a component of one of the 40 kg test masses made of fused silica 34 cm in diameter under inspection for flatness and quality using Fizeau interferometry

On a less ambitious scale than gravitational wave detec-tion the two-beam Michelson type interferometer has been adapted to a wide range of displacement measuring tasks criti-cal to precision machining [15] and photolithography stage positioning With the invention of the laser it became practi-cal to configure interferometers for monitoring motions over dynamic ranges far greater than the few mm offered by the mercury and sodium lamps that came before [16] Augmenting these advances in free-space displacement measurements are fiber sensors which not only measure part positions but temperature pressure strain and other parameters of interest using changes in optical path length [17 18]

Interferometry also plays a key role in the analysis of fine-scale surface topography texture and microsystems [19] Early in the 20th century microscopes were adapted to this purpose using miniature interferometers placed close to the sample surface in place of the normal imaging objective lens [20 21] Many interference objectives use a configuration similar to a figure 5 with illumination and imaging optics for creating areal fringe patterns indicative of surface texture [22] Interferometric microscopy is a common technique today for quantifying the functional aspects of surfaces such as lubrica-tion adhesion friction leaks corrosion and wear [23 24]

In this review I gather together a sampling of techniques and applications relevant to recent developments in modern interferometric metrology The selection of topics is neither comprehensive nor inclusive rather it represents a sampling of challenges and solutions viewed from the perspective of an instrument maker The review is organized as a linear sequence however each individual topic can be treated as a separate article with some reference to previously-defined equations and symbols

2 Distance and displacement

21 Measuring linear motions with a laser

A HeNe laser running at 633 nm has individual spectral modes each with a nearly infinite coherence length and a light beam that for all practical purposes is equivalent to a collimated point source Simple two-beam interference follows from dividing the source light into two paths in the manner of a Michelson interferometer with one path to a measurement mirror and the other to a reference mirror The recombined the beams result in an oscillating intensity that is highly sensitive to displacements of the measurement mirror [25] Laser-based displacement measuring interferometry or DMI has been a foundational metrology technique for automated sub-micron stage control since the 1970s [16] It also serves as a starting point for the understanding of the operating principles of a wide variety of instruments based on interferometry

DMI owes its precision to the 360deg phase shift for every half wavelength of test object movement The fact that we

Figure 4 One of the test masses for the advanced LIGO project under evaluation using a laser Fizeau interferometer similar to figure 2 Courtesy CaltechMITLIGO Laboratory

Figure 5 Watson-type interference objective from the 1960s together with an example interference fringe pattern generated by an average surface roughness height of 30 nm

Figure 3 The essential components of the Michelson interferometer

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have access to the wavelength as a traceable distance metric is not to be taken for grantedmdashthe frequency of oscillation of visible light is approximately 6 times 1014 Hz far too high for direct detection using a simple square-law detection of inter-ference fringes This is why we need two beams together with the principle of coherent superposition to detect just the dif-ference in phase change between the two light paths

A straightforward mathematical development clarifies the relatively fundamentals of two-beam single-wavelength interferometry [25 26] We begin with an oscillating electric field of a source beam of amplitude E0

E (t z) = E0 exp [2πi (νt minus nzλ )] (1)

where ν is the frequency of oscillation λ is the vacuum wave-length z is the geometrical path length and n is the index of refraction The source light divides into two beams in the interferometer of figure 3 to follow different paths and there-fore have different phase offsets related to the propagation term nzλ

E1 (t z1) = r1E0 exp (2πiνt) exp (minus2πinz1λ ) (2)

E2 (t z2) = r2E0 exp (2πiνt) exp (minus2πinz2λ ) (3)

Where z1 z2 are the path lengths traversed by beams from the point that they are separated by the beam splitter to the point that they are recombined and r1 r2 are their relative strengths with respect to the original complex amplitude |E0| When the two beams superimpose on a square-law detector the time average of the resulting intensity is

I (z1 z2) = |E0|2uml|exp (2πiνt)|2

part

|r1 exp (minus2πinz1λ ) + r2 exp (minus2πinz2λ )|2 (4)The frequency term νt averaged over time becomes a constant

uml|exp (2πiνt)|2

part= 1 (5)

and the final expression is

I (L) = I1 + I2 +radic

I1I2 cos [φ (L)] (6)

where

I1 = |r1E0|2 (7)

I2 = |r2E0|2 (8)

φ (L) = 2πnLλ (9)

for a path difference

L = z1 minus z2 (10)

The principle of DMI is to detect changes in the distance L by evaluation of the interference phase φ (L) via its effect on the time-averaged intensity I(L) The beauty of this technique is that equation (6) does not include the frequency ν of the light oscillation which is averaged away leaving only the constructive or destructive interference From this interfer-ence signal we may determine the difference in path length L for the two beams

Extracting the phase information φ (L) presents an inter-esting challenge in signal processing Even in the ideal case there are three unknowns related to signal strength aver-age signal and phase Phase determination using automated means nearly always involves three or more discrete signal samples to solve for φ (L) independent of other factors In the homodyne detection method polarizing optics and multiple detectors measure the intensity for several static phase shifts imposed by polarizing optics [27ndash29]

A widespread phase estimation technique is based on het-erodyne methods wherein the interference intensity is con-tinuously shifted in phase with time by imparting a frequency difference between the reference and measurement beams [16 30 31] The light source is modulated to provide an offset frequency ∆ν = ν2 minus ν1 between reference and measurement beams which results in a continuous time-dependent phase shift

Figure 6 Stabilized HeNe laser with an acousto-optic modulator for heterodyne interferometry Lower left a 20 mW laser tube on a test bench Upper right view of an AOM assembly for generating a 20 MHz beat signal

Figure 7 Plane-mirror interferometer for stage motion metrology The double pass with a retroreflector compensates for tip and tilt of the object mirror The design relies on polarized light and heterodyne detectionmdashcommon features in modern systems

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I (Lα) = I1 + I2 +radic

I1I2 cos [φ (L) + φ0 +∆νt] (11)

The output beam intensity oscillates at a rate of a few kHz to hundreds of MHz depending on the system design and the maximum object velocity A single detector captures this signal for each measurement direction or axis and firmware applies a sliding-window Fourier analysis or some other tech-nique to determine the phase [32]

Whether the phase detection is by homodyne or heterodyne techniques polarization plays an important role in nearly all DMI systems Heterodyne laser sources usually emit simultaneously in two orthogonal polarizations encoding the frequency shift ∆ν between the polarization states The interferometer optics include polarizing elements to separate the measurement and reference beams and direct them to the reference and object mirrors This free-space delivery system can feed many interferometer optical assemblies monitoring multiple displacement directions or axes of motion in com-plex machines The motions include linear motions angles and straightness [33]

Polarized heterodyne laser sources may rely on the Zeeman effect wherein a strong magnetic field along the axis of the tube splits the laser emission into orthogonally circularly polarized modes with a frequency difference of 1 to 4 MHz [35 36] Higher-performance systems use acousto-optic mod-ulators with split frequencies of 20 to 80 MHz to overcome the Doppler shift in high-speed stage systems (figure 6) while preserving the characteristic of generating copropagating collinear measurement and reference beams with orthogonal polarizations [34 37ndash40]

Modern displacement interferometers are widely used as sensors for stage motion monitoring and control In such applications the simple geometry of the Michelson is not quite good enough because of the sensitivity of the geometry to tilt-ing of the mirrors In one-axis systems this issue is overcome by using retroreflective optics such as corner-cubes in place of measurement and reference mirrors For x-y stages the problem becomes more interesting because the interferometer must accommodate motions orthogonal to the line of site The interferometer optical design of figure 7 overcomes this issue by a double pass to a plane mirror plane mirror rather than a single pass to a retroreflector [31 41] The design relies on a quarter-wave plate (QWP) to convert linear polarization to circular and then back to an orthogonal linear polarization after reflection from the object mirror The polarizing beam splitter (PBS) and QWP enable the double pass with the retro-reflection that compensates for the tip and tilt of the object mirror The object mirror is free to move orthogonal to the line of sight without disturbing the beam paths [42] An additional benefit of the double pass is a finer measurement resolution by magnifying the optical path change with displacement by 4times Although shown as 2D in the figures to clarify their func-tion actual interferometers are 3D and arrange beam paths for compactness and light efficiency Commercial interferometers for advanced stage positioning use optics of BK7 or crystal-line quartz vacuum-grade low-volatility adhesives stainless steel housings and thermal compensation [43]

The performance of DMI systems has progressed signifi-cantly driven in large part by the demands of the photolithog-raphy industry which depends on sub-nm positioning at stage motions of 4 m sminus1 The high precision of these measurements relates to the need to register successive exposures during the creation of transistors with critical dimensions currently approaching 7 nm [44] Double-exposure techniques allocate only 05 nm in the uncertainty budget for stage metrology [45]

22 Multi-dimensional optical encoders

The evolution of stage positioning requirements in photoli-thography to the impressive numbers in the previous para-graph is an interesting story In the early 1990s the stage metrology demands were on the order of a few tens of nm and were adequately addressed with free-space heterodyne interferometry Tightening tolerances were met with improve-ments in optical designs and active techniques for suppress-ing cyclic errors related to polarization mixing and other imperfections [33 46ndash48] In the latter half of the 1990s it became clear that operating in open air for the primary stage motion metrology as required for Michelson-type interferom-eters introduced seemingly intractable problems related to air turbulence errors The first line of defense was to com-pletely redesign airflow within the photolithography system but this clearly had its limits There were serious attempts to compensate for air-index using dispersion interferometrymdashactually measuring the air density along the line of site in real time by comparing optical path length measurements at two or more wavelengths [26 49 50] and to replace multi-pass interferometers with single-beam designs with dynamic beam pointing [51] These advances were continuously overtaken by rapidly-tightening requirements

Ultimately the solution to the air turbulence problem has been to use optical encoders in combination with 2D gratings (figure 9) because encoders have much shorter beam paths and are therefore less sensitive to air turbulence [52 53] This transition to encoders was not easymdashthere are significant chal-lenges with regard to grating manufacture Abbe offset errors and with the encoders themselves which initially were not up to the task This in part was related to the more traditional role

Figure 8 Optical encoder for simultaneous measurement of y z displacements

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of optical encoders for machine tool positioning feedback as opposed to sub-nm high-speed photolithography

Of the several possible approaches to high-precision encoder-based positioning one of the more straightforward strategies involves placing a grating in the path of a conven-tional Michelson-type interferometer that employs retroreflec-tors as targets as shown in figure 8 [54] The signal detection once again uses the heterodyne technique with polarization to encode the reference and measurement paths In the geometry of figure 8 there are two measurements of interference phase φ+φminus to enable simultaneous determination of the vertical z and lateral y displacements in the figure [55ndash57] The calcul-ations are

x =(φ+ minus φminus)

8πΛ (12)

z =(φ+ + φminus)

8πλ

1 +raquo

1 minus (λΛ )2 (13)

where Λ is the grating pitch The encoder design of figure 8 is to first order insensitive to tip and tilt of the diffraction gratingmdashan important practical requirement However out-of-plane motion along the z axis has the effect of laterally shearing the measurement beam with respect to the reference beam which reduces signal strength Newer designs recircu-late the beams in such a way that the double pass passively compensates for this effect and include means for reduc-ing accidental mixing of beam paths using polarized angled beams [56 58 59]

23 Absolute distance and position measurement

So far we have considered high-precision displacement mea-surements meaning that the DMI system tracks changes in position without necessarily determining the starting point In a displacement measurement if the measurement beam is blocked and then reestablished there is no memory of any change in position that may have occurred when the object was not being tracked An alternative class of position-sensing

interferometer measures the so-called absolute distance as opposed to a relative displacement These systems are also of interest where the target surface may have a rough surface tex-ture that complicates fringe tracking especially when using a single-point probe for surface profiling There is abundant literature on this topic representing a wide range of solutions [60 61]

Most interferometric systems for absolute distance meas-urement employ multiple or swept-wavelength sources The essential principles were known from the earliest interferom-eters for length standards [62] and has been in continuous use for measuring gage blocks for the past century [63] The principle of measurement follows from the dependence of the interferometric phase on wavelength given in equation (9) which we can rewrite as

φ (L) = knL (14)

where the angular wavenumber is given by

k = 2πλ (15)

The classical method of excess fractions relies on match-ing phase values for a sequence of discrete wavelengths to a specific distance using tables or a special slide rule The wavelengths may be selected by a prism or be generated by multiple light sources or a single light source that is tunable or selectable in wavelength

Another algorithmic approach to multiple-wavelength interferometry relies on the definition of equivalent or synth-etic wavelengths For a pair of optical wavelengths λ1 gt λ2 the synthetic wavelength is

Λ = λ1λ2(λ1 minus λ2) (16)

for a corresponding synthetic phase

Φ (L) = 2πnLΛ (17)

given by the difference in interference phase measured at each of the two wavelengths λ1λ2 Figure 10 illustrates a synthetic wavelength interferometer using two single-mode tunable diode lasers locked to a common FabryndashPerot etalon that provides high relative wavelength stability for large vari-ations in optical path length [64 65] Synthetic wavelength

Figure 9 A 2D grating for encoder-based stage positioning

Figure 10 Synthetic wavelength interferometer using two lasers locked to a common resonance frequency of a confocal etalon

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interferometry has also been applied to the profiling of unpol-ished or technical surfaces using laterally-scanning probes [66 67]

The essential principle of multiple wavelength interferom-etry is that the rate of change of phase φ with wavenumber k is proportional to the optical path difference nL as is directly shown by taking the derivative of equation (15)

dφdk

= nL (18)

An alternative to a light source with a sequence of discrete wavelengths is a frequency-swept light source so as to cal-culate this rate of change as a continuous derivative The heterodyne frequency ∆ν of the interference signal during wavelength tuning for a tunable optical frequency ν is

∆ν =dνdt

nLc

(19)

It is then straightforward to solve for the optical path differ-ence nL This method is sometimes referred to as frequency-modulated continuous wave ranging Laser diodes are logical choices for tunable sources [68 69] High precision on the order of one part in 109 of the measured distance is feasible with advances sources and sufficient care [70]

A complication when using tunable laser diodes is that these devices have a tendency to be highly sensitive to min-ute amounts of backscattered light requiring optical isolators and special care to avoid feedback [71] Light reflected light directed into the laser cavity produces wavelength and inten-sity modulations corresponding to the phase of the returned light just as in an interferometer although in this case the reference beam is essentially internal to the laser [72 73] Figure 11 shows that the geometry for a self-mixing or opti-cal feedback interferometer for distance measurement can be very simple All that is required is a path for reflected light to enter the lasermdasha condition that is almost unavoidable in many casesmdashand a detector for observing the modulations in the laser output in response to the phase of this reflected light The technique has high sensitivity to small amounts of

returned light and is best suited for scattering surfaces that couple less than 001 of the source light back into the cavity

Simple FabryndashPerot type laser diodes are easily tuned by injection current modulation which changes the cavity length by rapid heating and cooling of the active area of the diode [74] Alternatively an external cavity using mirrors or grat-ings tunes the laser by controlled mechanical motions [75] The resulting signal has a distinctive sawtooth shape as a result of the laser system phase lock to the external reflec-tion but the frequency of this signal corresponds to that of any other type of interferometric swept-wavelength laser ranging system Self-mixing has been investigated for a wide range of sensing applications including displacement velocimetry and absolute distance measurement [76]

The interferometer configurations so far have been shown with free-space light propagation throughout whereas in practice today it is common to use fiber optics to control light paths For the plane-mirror DMI and the encoder figures 7 and 8 respectively it is convenient to transmit the source light by a polarization-preserving single-mode fiber subsystem and to capture the output light in a multi-mode fiber for detec-tion some distance away [77 78] Fiber-based interferometer systems have advantages in flexibility and heat management when placing interferometer optics close to the application area such as a precision stage in a thermally-sensitive area

The final example of an absolute distance interferometer is multiplexed fiber-optic sensors designed for small size and multi-axis position sensing [79] Fiber sensors are widely used for measurements of strain temperature and pressure using interferometric methods [79 80] Figure 12 illustrates an example system intended for up to 64 sensors external to a control box that contains all of the complex hardware required to illuminate the sensors provide phase modulation and heter-odyne detection [81] Spectrally-broadband light source light follows two paths with phase modulation being applied along one path and a phase delay along the other The modulated and delayed copies of the source illumination are combined and delivered via an optical fiber to the miniature fiber-optic Fizeau interferometers The short coherence length of the source means that only obtained between pairs of reflections that have travelled the same optical path length The light reflected from the sensors travels back through fibers to an array of detectors that pick up the resulting beat signals Data

Figure 11 Optical-feedback laser distance measuring system The target object in most cases has a rough surface texture scattering a small fraction of the source light back into the laser

Figure 12 Fiber-based positioning system with multiple remote sensors

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rates are up to 208 kHz per channel with extraordinarily low noise levels of 0003 nm (

radicHz)minus1 [18] Recent applications

include the in situ measurement of deformable mirrors in syn-chrotrons [82]

3 Testing of optical components

31 The laser Fizeau interferometer

The light box that provided us with the interference fringe pattern in figure 1 was for most of the history of interferom-etry representative of the state of the art in interferometric surface form testing Generalizations of the idea including proper imaging instruments with physically-separated refer-ence and measurement surfaces such as the TwymanndashGreen interferometer invented a hundred years ago [83] had lim-ited range of application because of the short coherence length of the available light sources and the skill required to interpret the resulting patterns These challenges meant that optical shops often preferred geometric ray tests such as the Foucault knife-edge or shadow method to evaluate sur-face form using zonal masks for aspheric surfaces such as parabolas [84 85]

A breakthrough in the use of interferometry for optical testing was the laser which enabled the modern laser Fizeau geometry with unequal paths [86] High coherence means that the instrument is easily reconfigured to accommodate flats concave spheres convex spheres lenses prisms and even complete optical assemblies [11 87] In combination with refractive reflective or holographic null correctors the Fizeau geometry also measures aspherical and free-form optics [88 89] In addition to its flexibility the Fizeau geometry is attractive in that there are no additional beam shaping optics between the reference and the object which significantly simplifies the instrument calibrationmdashin most cases it is suf-ficient to accurately characterize the reference surface [90] These advances led to commercial instruments such as the one

shown in figures 13 and 14 that moved interferometers from the physics lab to the optical shop floor [91]

The need for high accuracy surface measurement requires a more sophisticated approach than the visual interpretation of fringe patterns This requires automated fringe analysis using electronic cameras and heterodyne techniques [92] Phase shifting interferometry or PSI relies on controlled phase shifts ξ (t) usually imparted by the mechanical motion of the refer-ence mirror [93] Sometimes referred to as temporal PSI to distinguish the method from the spatial interpretation of indi-vidual fringe pattern images PSI analyzes the interference signals from each pixel individually calculating an interfer-ence phase φ (x y) proportional to each point h (x y) in the surface topography where the height h is orthogonal to the nominal surface shape

Surface topography measurements using PSI assume two-beam interference similar to what we have already seen for displacement measurement in equation (6)

I (x y t) = I1 + I2 +radic

I1I2 cos [φ (x y) + ξ (t)] (20)

Figure 13 A commercial implementation of a laser Fizeau system from the early 1970s This was a visual system with an integrated film camera for recording interference fringe images Photo courtesy of Zygo Corporation

Figure 14 Laser Fizeau interferometer with a phase shifted reference setup for a concave spherical object surface

Figure 15 Upper graph intensity signal and digital sampling in linear phase shifting Lower graph intensity signal with a sinusoidal phase shift

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φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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10

frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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11

mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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12

wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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13

for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

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[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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3

physical universe but now on a much grander scale The laser interferometer gravitational-wave observatory (LIGO) also uses orthogonal beam directions but with path lengths meas-ured in kilometers rather than centimeters [14] Recirculating FabryndashPerot interferometer cavities increase the effective dis-tance and highly-refined phase measurements sense motions of the test masses on a scale thousands of times smaller than that of a proton Figure 4 shows a component of one of the 40 kg test masses made of fused silica 34 cm in diameter under inspection for flatness and quality using Fizeau interferometry

On a less ambitious scale than gravitational wave detec-tion the two-beam Michelson type interferometer has been adapted to a wide range of displacement measuring tasks criti-cal to precision machining [15] and photolithography stage positioning With the invention of the laser it became practi-cal to configure interferometers for monitoring motions over dynamic ranges far greater than the few mm offered by the mercury and sodium lamps that came before [16] Augmenting these advances in free-space displacement measurements are fiber sensors which not only measure part positions but temperature pressure strain and other parameters of interest using changes in optical path length [17 18]

Interferometry also plays a key role in the analysis of fine-scale surface topography texture and microsystems [19] Early in the 20th century microscopes were adapted to this purpose using miniature interferometers placed close to the sample surface in place of the normal imaging objective lens [20 21] Many interference objectives use a configuration similar to a figure 5 with illumination and imaging optics for creating areal fringe patterns indicative of surface texture [22] Interferometric microscopy is a common technique today for quantifying the functional aspects of surfaces such as lubrica-tion adhesion friction leaks corrosion and wear [23 24]

In this review I gather together a sampling of techniques and applications relevant to recent developments in modern interferometric metrology The selection of topics is neither comprehensive nor inclusive rather it represents a sampling of challenges and solutions viewed from the perspective of an instrument maker The review is organized as a linear sequence however each individual topic can be treated as a separate article with some reference to previously-defined equations and symbols

2 Distance and displacement

21 Measuring linear motions with a laser

A HeNe laser running at 633 nm has individual spectral modes each with a nearly infinite coherence length and a light beam that for all practical purposes is equivalent to a collimated point source Simple two-beam interference follows from dividing the source light into two paths in the manner of a Michelson interferometer with one path to a measurement mirror and the other to a reference mirror The recombined the beams result in an oscillating intensity that is highly sensitive to displacements of the measurement mirror [25] Laser-based displacement measuring interferometry or DMI has been a foundational metrology technique for automated sub-micron stage control since the 1970s [16] It also serves as a starting point for the understanding of the operating principles of a wide variety of instruments based on interferometry

DMI owes its precision to the 360deg phase shift for every half wavelength of test object movement The fact that we

Figure 4 One of the test masses for the advanced LIGO project under evaluation using a laser Fizeau interferometer similar to figure 2 Courtesy CaltechMITLIGO Laboratory

Figure 5 Watson-type interference objective from the 1960s together with an example interference fringe pattern generated by an average surface roughness height of 30 nm

Figure 3 The essential components of the Michelson interferometer

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4

have access to the wavelength as a traceable distance metric is not to be taken for grantedmdashthe frequency of oscillation of visible light is approximately 6 times 1014 Hz far too high for direct detection using a simple square-law detection of inter-ference fringes This is why we need two beams together with the principle of coherent superposition to detect just the dif-ference in phase change between the two light paths

A straightforward mathematical development clarifies the relatively fundamentals of two-beam single-wavelength interferometry [25 26] We begin with an oscillating electric field of a source beam of amplitude E0

E (t z) = E0 exp [2πi (νt minus nzλ )] (1)

where ν is the frequency of oscillation λ is the vacuum wave-length z is the geometrical path length and n is the index of refraction The source light divides into two beams in the interferometer of figure 3 to follow different paths and there-fore have different phase offsets related to the propagation term nzλ

E1 (t z1) = r1E0 exp (2πiνt) exp (minus2πinz1λ ) (2)

E2 (t z2) = r2E0 exp (2πiνt) exp (minus2πinz2λ ) (3)

Where z1 z2 are the path lengths traversed by beams from the point that they are separated by the beam splitter to the point that they are recombined and r1 r2 are their relative strengths with respect to the original complex amplitude |E0| When the two beams superimpose on a square-law detector the time average of the resulting intensity is

I (z1 z2) = |E0|2uml|exp (2πiνt)|2

part

|r1 exp (minus2πinz1λ ) + r2 exp (minus2πinz2λ )|2 (4)The frequency term νt averaged over time becomes a constant

uml|exp (2πiνt)|2

part= 1 (5)

and the final expression is

I (L) = I1 + I2 +radic

I1I2 cos [φ (L)] (6)

where

I1 = |r1E0|2 (7)

I2 = |r2E0|2 (8)

φ (L) = 2πnLλ (9)

for a path difference

L = z1 minus z2 (10)

The principle of DMI is to detect changes in the distance L by evaluation of the interference phase φ (L) via its effect on the time-averaged intensity I(L) The beauty of this technique is that equation (6) does not include the frequency ν of the light oscillation which is averaged away leaving only the constructive or destructive interference From this interfer-ence signal we may determine the difference in path length L for the two beams

Extracting the phase information φ (L) presents an inter-esting challenge in signal processing Even in the ideal case there are three unknowns related to signal strength aver-age signal and phase Phase determination using automated means nearly always involves three or more discrete signal samples to solve for φ (L) independent of other factors In the homodyne detection method polarizing optics and multiple detectors measure the intensity for several static phase shifts imposed by polarizing optics [27ndash29]

A widespread phase estimation technique is based on het-erodyne methods wherein the interference intensity is con-tinuously shifted in phase with time by imparting a frequency difference between the reference and measurement beams [16 30 31] The light source is modulated to provide an offset frequency ∆ν = ν2 minus ν1 between reference and measurement beams which results in a continuous time-dependent phase shift

Figure 6 Stabilized HeNe laser with an acousto-optic modulator for heterodyne interferometry Lower left a 20 mW laser tube on a test bench Upper right view of an AOM assembly for generating a 20 MHz beat signal

Figure 7 Plane-mirror interferometer for stage motion metrology The double pass with a retroreflector compensates for tip and tilt of the object mirror The design relies on polarized light and heterodyne detectionmdashcommon features in modern systems

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5

I (Lα) = I1 + I2 +radic

I1I2 cos [φ (L) + φ0 +∆νt] (11)

The output beam intensity oscillates at a rate of a few kHz to hundreds of MHz depending on the system design and the maximum object velocity A single detector captures this signal for each measurement direction or axis and firmware applies a sliding-window Fourier analysis or some other tech-nique to determine the phase [32]

Whether the phase detection is by homodyne or heterodyne techniques polarization plays an important role in nearly all DMI systems Heterodyne laser sources usually emit simultaneously in two orthogonal polarizations encoding the frequency shift ∆ν between the polarization states The interferometer optics include polarizing elements to separate the measurement and reference beams and direct them to the reference and object mirrors This free-space delivery system can feed many interferometer optical assemblies monitoring multiple displacement directions or axes of motion in com-plex machines The motions include linear motions angles and straightness [33]

Polarized heterodyne laser sources may rely on the Zeeman effect wherein a strong magnetic field along the axis of the tube splits the laser emission into orthogonally circularly polarized modes with a frequency difference of 1 to 4 MHz [35 36] Higher-performance systems use acousto-optic mod-ulators with split frequencies of 20 to 80 MHz to overcome the Doppler shift in high-speed stage systems (figure 6) while preserving the characteristic of generating copropagating collinear measurement and reference beams with orthogonal polarizations [34 37ndash40]

Modern displacement interferometers are widely used as sensors for stage motion monitoring and control In such applications the simple geometry of the Michelson is not quite good enough because of the sensitivity of the geometry to tilt-ing of the mirrors In one-axis systems this issue is overcome by using retroreflective optics such as corner-cubes in place of measurement and reference mirrors For x-y stages the problem becomes more interesting because the interferometer must accommodate motions orthogonal to the line of site The interferometer optical design of figure 7 overcomes this issue by a double pass to a plane mirror plane mirror rather than a single pass to a retroreflector [31 41] The design relies on a quarter-wave plate (QWP) to convert linear polarization to circular and then back to an orthogonal linear polarization after reflection from the object mirror The polarizing beam splitter (PBS) and QWP enable the double pass with the retro-reflection that compensates for the tip and tilt of the object mirror The object mirror is free to move orthogonal to the line of sight without disturbing the beam paths [42] An additional benefit of the double pass is a finer measurement resolution by magnifying the optical path change with displacement by 4times Although shown as 2D in the figures to clarify their func-tion actual interferometers are 3D and arrange beam paths for compactness and light efficiency Commercial interferometers for advanced stage positioning use optics of BK7 or crystal-line quartz vacuum-grade low-volatility adhesives stainless steel housings and thermal compensation [43]

The performance of DMI systems has progressed signifi-cantly driven in large part by the demands of the photolithog-raphy industry which depends on sub-nm positioning at stage motions of 4 m sminus1 The high precision of these measurements relates to the need to register successive exposures during the creation of transistors with critical dimensions currently approaching 7 nm [44] Double-exposure techniques allocate only 05 nm in the uncertainty budget for stage metrology [45]

22 Multi-dimensional optical encoders

The evolution of stage positioning requirements in photoli-thography to the impressive numbers in the previous para-graph is an interesting story In the early 1990s the stage metrology demands were on the order of a few tens of nm and were adequately addressed with free-space heterodyne interferometry Tightening tolerances were met with improve-ments in optical designs and active techniques for suppress-ing cyclic errors related to polarization mixing and other imperfections [33 46ndash48] In the latter half of the 1990s it became clear that operating in open air for the primary stage motion metrology as required for Michelson-type interferom-eters introduced seemingly intractable problems related to air turbulence errors The first line of defense was to com-pletely redesign airflow within the photolithography system but this clearly had its limits There were serious attempts to compensate for air-index using dispersion interferometrymdashactually measuring the air density along the line of site in real time by comparing optical path length measurements at two or more wavelengths [26 49 50] and to replace multi-pass interferometers with single-beam designs with dynamic beam pointing [51] These advances were continuously overtaken by rapidly-tightening requirements

Ultimately the solution to the air turbulence problem has been to use optical encoders in combination with 2D gratings (figure 9) because encoders have much shorter beam paths and are therefore less sensitive to air turbulence [52 53] This transition to encoders was not easymdashthere are significant chal-lenges with regard to grating manufacture Abbe offset errors and with the encoders themselves which initially were not up to the task This in part was related to the more traditional role

Figure 8 Optical encoder for simultaneous measurement of y z displacements

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6

of optical encoders for machine tool positioning feedback as opposed to sub-nm high-speed photolithography

Of the several possible approaches to high-precision encoder-based positioning one of the more straightforward strategies involves placing a grating in the path of a conven-tional Michelson-type interferometer that employs retroreflec-tors as targets as shown in figure 8 [54] The signal detection once again uses the heterodyne technique with polarization to encode the reference and measurement paths In the geometry of figure 8 there are two measurements of interference phase φ+φminus to enable simultaneous determination of the vertical z and lateral y displacements in the figure [55ndash57] The calcul-ations are

x =(φ+ minus φminus)

8πΛ (12)

z =(φ+ + φminus)

8πλ

1 +raquo

1 minus (λΛ )2 (13)

where Λ is the grating pitch The encoder design of figure 8 is to first order insensitive to tip and tilt of the diffraction gratingmdashan important practical requirement However out-of-plane motion along the z axis has the effect of laterally shearing the measurement beam with respect to the reference beam which reduces signal strength Newer designs recircu-late the beams in such a way that the double pass passively compensates for this effect and include means for reduc-ing accidental mixing of beam paths using polarized angled beams [56 58 59]

23 Absolute distance and position measurement

So far we have considered high-precision displacement mea-surements meaning that the DMI system tracks changes in position without necessarily determining the starting point In a displacement measurement if the measurement beam is blocked and then reestablished there is no memory of any change in position that may have occurred when the object was not being tracked An alternative class of position-sensing

interferometer measures the so-called absolute distance as opposed to a relative displacement These systems are also of interest where the target surface may have a rough surface tex-ture that complicates fringe tracking especially when using a single-point probe for surface profiling There is abundant literature on this topic representing a wide range of solutions [60 61]

Most interferometric systems for absolute distance meas-urement employ multiple or swept-wavelength sources The essential principles were known from the earliest interferom-eters for length standards [62] and has been in continuous use for measuring gage blocks for the past century [63] The principle of measurement follows from the dependence of the interferometric phase on wavelength given in equation (9) which we can rewrite as

φ (L) = knL (14)

where the angular wavenumber is given by

k = 2πλ (15)

The classical method of excess fractions relies on match-ing phase values for a sequence of discrete wavelengths to a specific distance using tables or a special slide rule The wavelengths may be selected by a prism or be generated by multiple light sources or a single light source that is tunable or selectable in wavelength

Another algorithmic approach to multiple-wavelength interferometry relies on the definition of equivalent or synth-etic wavelengths For a pair of optical wavelengths λ1 gt λ2 the synthetic wavelength is

Λ = λ1λ2(λ1 minus λ2) (16)

for a corresponding synthetic phase

Φ (L) = 2πnLΛ (17)

given by the difference in interference phase measured at each of the two wavelengths λ1λ2 Figure 10 illustrates a synthetic wavelength interferometer using two single-mode tunable diode lasers locked to a common FabryndashPerot etalon that provides high relative wavelength stability for large vari-ations in optical path length [64 65] Synthetic wavelength

Figure 9 A 2D grating for encoder-based stage positioning

Figure 10 Synthetic wavelength interferometer using two lasers locked to a common resonance frequency of a confocal etalon

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7

interferometry has also been applied to the profiling of unpol-ished or technical surfaces using laterally-scanning probes [66 67]

The essential principle of multiple wavelength interferom-etry is that the rate of change of phase φ with wavenumber k is proportional to the optical path difference nL as is directly shown by taking the derivative of equation (15)

dφdk

= nL (18)

An alternative to a light source with a sequence of discrete wavelengths is a frequency-swept light source so as to cal-culate this rate of change as a continuous derivative The heterodyne frequency ∆ν of the interference signal during wavelength tuning for a tunable optical frequency ν is

∆ν =dνdt

nLc

(19)

It is then straightforward to solve for the optical path differ-ence nL This method is sometimes referred to as frequency-modulated continuous wave ranging Laser diodes are logical choices for tunable sources [68 69] High precision on the order of one part in 109 of the measured distance is feasible with advances sources and sufficient care [70]

A complication when using tunable laser diodes is that these devices have a tendency to be highly sensitive to min-ute amounts of backscattered light requiring optical isolators and special care to avoid feedback [71] Light reflected light directed into the laser cavity produces wavelength and inten-sity modulations corresponding to the phase of the returned light just as in an interferometer although in this case the reference beam is essentially internal to the laser [72 73] Figure 11 shows that the geometry for a self-mixing or opti-cal feedback interferometer for distance measurement can be very simple All that is required is a path for reflected light to enter the lasermdasha condition that is almost unavoidable in many casesmdashand a detector for observing the modulations in the laser output in response to the phase of this reflected light The technique has high sensitivity to small amounts of

returned light and is best suited for scattering surfaces that couple less than 001 of the source light back into the cavity

Simple FabryndashPerot type laser diodes are easily tuned by injection current modulation which changes the cavity length by rapid heating and cooling of the active area of the diode [74] Alternatively an external cavity using mirrors or grat-ings tunes the laser by controlled mechanical motions [75] The resulting signal has a distinctive sawtooth shape as a result of the laser system phase lock to the external reflec-tion but the frequency of this signal corresponds to that of any other type of interferometric swept-wavelength laser ranging system Self-mixing has been investigated for a wide range of sensing applications including displacement velocimetry and absolute distance measurement [76]

The interferometer configurations so far have been shown with free-space light propagation throughout whereas in practice today it is common to use fiber optics to control light paths For the plane-mirror DMI and the encoder figures 7 and 8 respectively it is convenient to transmit the source light by a polarization-preserving single-mode fiber subsystem and to capture the output light in a multi-mode fiber for detec-tion some distance away [77 78] Fiber-based interferometer systems have advantages in flexibility and heat management when placing interferometer optics close to the application area such as a precision stage in a thermally-sensitive area

The final example of an absolute distance interferometer is multiplexed fiber-optic sensors designed for small size and multi-axis position sensing [79] Fiber sensors are widely used for measurements of strain temperature and pressure using interferometric methods [79 80] Figure 12 illustrates an example system intended for up to 64 sensors external to a control box that contains all of the complex hardware required to illuminate the sensors provide phase modulation and heter-odyne detection [81] Spectrally-broadband light source light follows two paths with phase modulation being applied along one path and a phase delay along the other The modulated and delayed copies of the source illumination are combined and delivered via an optical fiber to the miniature fiber-optic Fizeau interferometers The short coherence length of the source means that only obtained between pairs of reflections that have travelled the same optical path length The light reflected from the sensors travels back through fibers to an array of detectors that pick up the resulting beat signals Data

Figure 11 Optical-feedback laser distance measuring system The target object in most cases has a rough surface texture scattering a small fraction of the source light back into the laser

Figure 12 Fiber-based positioning system with multiple remote sensors

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rates are up to 208 kHz per channel with extraordinarily low noise levels of 0003 nm (

radicHz)minus1 [18] Recent applications

include the in situ measurement of deformable mirrors in syn-chrotrons [82]

3 Testing of optical components

31 The laser Fizeau interferometer

The light box that provided us with the interference fringe pattern in figure 1 was for most of the history of interferom-etry representative of the state of the art in interferometric surface form testing Generalizations of the idea including proper imaging instruments with physically-separated refer-ence and measurement surfaces such as the TwymanndashGreen interferometer invented a hundred years ago [83] had lim-ited range of application because of the short coherence length of the available light sources and the skill required to interpret the resulting patterns These challenges meant that optical shops often preferred geometric ray tests such as the Foucault knife-edge or shadow method to evaluate sur-face form using zonal masks for aspheric surfaces such as parabolas [84 85]

A breakthrough in the use of interferometry for optical testing was the laser which enabled the modern laser Fizeau geometry with unequal paths [86] High coherence means that the instrument is easily reconfigured to accommodate flats concave spheres convex spheres lenses prisms and even complete optical assemblies [11 87] In combination with refractive reflective or holographic null correctors the Fizeau geometry also measures aspherical and free-form optics [88 89] In addition to its flexibility the Fizeau geometry is attractive in that there are no additional beam shaping optics between the reference and the object which significantly simplifies the instrument calibrationmdashin most cases it is suf-ficient to accurately characterize the reference surface [90] These advances led to commercial instruments such as the one

shown in figures 13 and 14 that moved interferometers from the physics lab to the optical shop floor [91]

The need for high accuracy surface measurement requires a more sophisticated approach than the visual interpretation of fringe patterns This requires automated fringe analysis using electronic cameras and heterodyne techniques [92] Phase shifting interferometry or PSI relies on controlled phase shifts ξ (t) usually imparted by the mechanical motion of the refer-ence mirror [93] Sometimes referred to as temporal PSI to distinguish the method from the spatial interpretation of indi-vidual fringe pattern images PSI analyzes the interference signals from each pixel individually calculating an interfer-ence phase φ (x y) proportional to each point h (x y) in the surface topography where the height h is orthogonal to the nominal surface shape

Surface topography measurements using PSI assume two-beam interference similar to what we have already seen for displacement measurement in equation (6)

I (x y t) = I1 + I2 +radic

I1I2 cos [φ (x y) + ξ (t)] (20)

Figure 13 A commercial implementation of a laser Fizeau system from the early 1970s This was a visual system with an integrated film camera for recording interference fringe images Photo courtesy of Zygo Corporation

Figure 14 Laser Fizeau interferometer with a phase shifted reference setup for a concave spherical object surface

Figure 15 Upper graph intensity signal and digital sampling in linear phase shifting Lower graph intensity signal with a sinusoidal phase shift

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9

φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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12

wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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13

for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

Rep Prog Phys 82 (2019) 056101

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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4

have access to the wavelength as a traceable distance metric is not to be taken for grantedmdashthe frequency of oscillation of visible light is approximately 6 times 1014 Hz far too high for direct detection using a simple square-law detection of inter-ference fringes This is why we need two beams together with the principle of coherent superposition to detect just the dif-ference in phase change between the two light paths

A straightforward mathematical development clarifies the relatively fundamentals of two-beam single-wavelength interferometry [25 26] We begin with an oscillating electric field of a source beam of amplitude E0

E (t z) = E0 exp [2πi (νt minus nzλ )] (1)

where ν is the frequency of oscillation λ is the vacuum wave-length z is the geometrical path length and n is the index of refraction The source light divides into two beams in the interferometer of figure 3 to follow different paths and there-fore have different phase offsets related to the propagation term nzλ

E1 (t z1) = r1E0 exp (2πiνt) exp (minus2πinz1λ ) (2)

E2 (t z2) = r2E0 exp (2πiνt) exp (minus2πinz2λ ) (3)

Where z1 z2 are the path lengths traversed by beams from the point that they are separated by the beam splitter to the point that they are recombined and r1 r2 are their relative strengths with respect to the original complex amplitude |E0| When the two beams superimpose on a square-law detector the time average of the resulting intensity is

I (z1 z2) = |E0|2uml|exp (2πiνt)|2

part

|r1 exp (minus2πinz1λ ) + r2 exp (minus2πinz2λ )|2 (4)The frequency term νt averaged over time becomes a constant

uml|exp (2πiνt)|2

part= 1 (5)

and the final expression is

I (L) = I1 + I2 +radic

I1I2 cos [φ (L)] (6)

where

I1 = |r1E0|2 (7)

I2 = |r2E0|2 (8)

φ (L) = 2πnLλ (9)

for a path difference

L = z1 minus z2 (10)

The principle of DMI is to detect changes in the distance L by evaluation of the interference phase φ (L) via its effect on the time-averaged intensity I(L) The beauty of this technique is that equation (6) does not include the frequency ν of the light oscillation which is averaged away leaving only the constructive or destructive interference From this interfer-ence signal we may determine the difference in path length L for the two beams

Extracting the phase information φ (L) presents an inter-esting challenge in signal processing Even in the ideal case there are three unknowns related to signal strength aver-age signal and phase Phase determination using automated means nearly always involves three or more discrete signal samples to solve for φ (L) independent of other factors In the homodyne detection method polarizing optics and multiple detectors measure the intensity for several static phase shifts imposed by polarizing optics [27ndash29]

A widespread phase estimation technique is based on het-erodyne methods wherein the interference intensity is con-tinuously shifted in phase with time by imparting a frequency difference between the reference and measurement beams [16 30 31] The light source is modulated to provide an offset frequency ∆ν = ν2 minus ν1 between reference and measurement beams which results in a continuous time-dependent phase shift

Figure 6 Stabilized HeNe laser with an acousto-optic modulator for heterodyne interferometry Lower left a 20 mW laser tube on a test bench Upper right view of an AOM assembly for generating a 20 MHz beat signal

Figure 7 Plane-mirror interferometer for stage motion metrology The double pass with a retroreflector compensates for tip and tilt of the object mirror The design relies on polarized light and heterodyne detectionmdashcommon features in modern systems

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5

I (Lα) = I1 + I2 +radic

I1I2 cos [φ (L) + φ0 +∆νt] (11)

The output beam intensity oscillates at a rate of a few kHz to hundreds of MHz depending on the system design and the maximum object velocity A single detector captures this signal for each measurement direction or axis and firmware applies a sliding-window Fourier analysis or some other tech-nique to determine the phase [32]

Whether the phase detection is by homodyne or heterodyne techniques polarization plays an important role in nearly all DMI systems Heterodyne laser sources usually emit simultaneously in two orthogonal polarizations encoding the frequency shift ∆ν between the polarization states The interferometer optics include polarizing elements to separate the measurement and reference beams and direct them to the reference and object mirrors This free-space delivery system can feed many interferometer optical assemblies monitoring multiple displacement directions or axes of motion in com-plex machines The motions include linear motions angles and straightness [33]

Polarized heterodyne laser sources may rely on the Zeeman effect wherein a strong magnetic field along the axis of the tube splits the laser emission into orthogonally circularly polarized modes with a frequency difference of 1 to 4 MHz [35 36] Higher-performance systems use acousto-optic mod-ulators with split frequencies of 20 to 80 MHz to overcome the Doppler shift in high-speed stage systems (figure 6) while preserving the characteristic of generating copropagating collinear measurement and reference beams with orthogonal polarizations [34 37ndash40]

Modern displacement interferometers are widely used as sensors for stage motion monitoring and control In such applications the simple geometry of the Michelson is not quite good enough because of the sensitivity of the geometry to tilt-ing of the mirrors In one-axis systems this issue is overcome by using retroreflective optics such as corner-cubes in place of measurement and reference mirrors For x-y stages the problem becomes more interesting because the interferometer must accommodate motions orthogonal to the line of site The interferometer optical design of figure 7 overcomes this issue by a double pass to a plane mirror plane mirror rather than a single pass to a retroreflector [31 41] The design relies on a quarter-wave plate (QWP) to convert linear polarization to circular and then back to an orthogonal linear polarization after reflection from the object mirror The polarizing beam splitter (PBS) and QWP enable the double pass with the retro-reflection that compensates for the tip and tilt of the object mirror The object mirror is free to move orthogonal to the line of sight without disturbing the beam paths [42] An additional benefit of the double pass is a finer measurement resolution by magnifying the optical path change with displacement by 4times Although shown as 2D in the figures to clarify their func-tion actual interferometers are 3D and arrange beam paths for compactness and light efficiency Commercial interferometers for advanced stage positioning use optics of BK7 or crystal-line quartz vacuum-grade low-volatility adhesives stainless steel housings and thermal compensation [43]

The performance of DMI systems has progressed signifi-cantly driven in large part by the demands of the photolithog-raphy industry which depends on sub-nm positioning at stage motions of 4 m sminus1 The high precision of these measurements relates to the need to register successive exposures during the creation of transistors with critical dimensions currently approaching 7 nm [44] Double-exposure techniques allocate only 05 nm in the uncertainty budget for stage metrology [45]

22 Multi-dimensional optical encoders

The evolution of stage positioning requirements in photoli-thography to the impressive numbers in the previous para-graph is an interesting story In the early 1990s the stage metrology demands were on the order of a few tens of nm and were adequately addressed with free-space heterodyne interferometry Tightening tolerances were met with improve-ments in optical designs and active techniques for suppress-ing cyclic errors related to polarization mixing and other imperfections [33 46ndash48] In the latter half of the 1990s it became clear that operating in open air for the primary stage motion metrology as required for Michelson-type interferom-eters introduced seemingly intractable problems related to air turbulence errors The first line of defense was to com-pletely redesign airflow within the photolithography system but this clearly had its limits There were serious attempts to compensate for air-index using dispersion interferometrymdashactually measuring the air density along the line of site in real time by comparing optical path length measurements at two or more wavelengths [26 49 50] and to replace multi-pass interferometers with single-beam designs with dynamic beam pointing [51] These advances were continuously overtaken by rapidly-tightening requirements

Ultimately the solution to the air turbulence problem has been to use optical encoders in combination with 2D gratings (figure 9) because encoders have much shorter beam paths and are therefore less sensitive to air turbulence [52 53] This transition to encoders was not easymdashthere are significant chal-lenges with regard to grating manufacture Abbe offset errors and with the encoders themselves which initially were not up to the task This in part was related to the more traditional role

Figure 8 Optical encoder for simultaneous measurement of y z displacements

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of optical encoders for machine tool positioning feedback as opposed to sub-nm high-speed photolithography

Of the several possible approaches to high-precision encoder-based positioning one of the more straightforward strategies involves placing a grating in the path of a conven-tional Michelson-type interferometer that employs retroreflec-tors as targets as shown in figure 8 [54] The signal detection once again uses the heterodyne technique with polarization to encode the reference and measurement paths In the geometry of figure 8 there are two measurements of interference phase φ+φminus to enable simultaneous determination of the vertical z and lateral y displacements in the figure [55ndash57] The calcul-ations are

x =(φ+ minus φminus)

8πΛ (12)

z =(φ+ + φminus)

8πλ

1 +raquo

1 minus (λΛ )2 (13)

where Λ is the grating pitch The encoder design of figure 8 is to first order insensitive to tip and tilt of the diffraction gratingmdashan important practical requirement However out-of-plane motion along the z axis has the effect of laterally shearing the measurement beam with respect to the reference beam which reduces signal strength Newer designs recircu-late the beams in such a way that the double pass passively compensates for this effect and include means for reduc-ing accidental mixing of beam paths using polarized angled beams [56 58 59]

23 Absolute distance and position measurement

So far we have considered high-precision displacement mea-surements meaning that the DMI system tracks changes in position without necessarily determining the starting point In a displacement measurement if the measurement beam is blocked and then reestablished there is no memory of any change in position that may have occurred when the object was not being tracked An alternative class of position-sensing

interferometer measures the so-called absolute distance as opposed to a relative displacement These systems are also of interest where the target surface may have a rough surface tex-ture that complicates fringe tracking especially when using a single-point probe for surface profiling There is abundant literature on this topic representing a wide range of solutions [60 61]

Most interferometric systems for absolute distance meas-urement employ multiple or swept-wavelength sources The essential principles were known from the earliest interferom-eters for length standards [62] and has been in continuous use for measuring gage blocks for the past century [63] The principle of measurement follows from the dependence of the interferometric phase on wavelength given in equation (9) which we can rewrite as

φ (L) = knL (14)

where the angular wavenumber is given by

k = 2πλ (15)

The classical method of excess fractions relies on match-ing phase values for a sequence of discrete wavelengths to a specific distance using tables or a special slide rule The wavelengths may be selected by a prism or be generated by multiple light sources or a single light source that is tunable or selectable in wavelength

Another algorithmic approach to multiple-wavelength interferometry relies on the definition of equivalent or synth-etic wavelengths For a pair of optical wavelengths λ1 gt λ2 the synthetic wavelength is

Λ = λ1λ2(λ1 minus λ2) (16)

for a corresponding synthetic phase

Φ (L) = 2πnLΛ (17)

given by the difference in interference phase measured at each of the two wavelengths λ1λ2 Figure 10 illustrates a synthetic wavelength interferometer using two single-mode tunable diode lasers locked to a common FabryndashPerot etalon that provides high relative wavelength stability for large vari-ations in optical path length [64 65] Synthetic wavelength

Figure 9 A 2D grating for encoder-based stage positioning

Figure 10 Synthetic wavelength interferometer using two lasers locked to a common resonance frequency of a confocal etalon

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7

interferometry has also been applied to the profiling of unpol-ished or technical surfaces using laterally-scanning probes [66 67]

The essential principle of multiple wavelength interferom-etry is that the rate of change of phase φ with wavenumber k is proportional to the optical path difference nL as is directly shown by taking the derivative of equation (15)

dφdk

= nL (18)

An alternative to a light source with a sequence of discrete wavelengths is a frequency-swept light source so as to cal-culate this rate of change as a continuous derivative The heterodyne frequency ∆ν of the interference signal during wavelength tuning for a tunable optical frequency ν is

∆ν =dνdt

nLc

(19)

It is then straightforward to solve for the optical path differ-ence nL This method is sometimes referred to as frequency-modulated continuous wave ranging Laser diodes are logical choices for tunable sources [68 69] High precision on the order of one part in 109 of the measured distance is feasible with advances sources and sufficient care [70]

A complication when using tunable laser diodes is that these devices have a tendency to be highly sensitive to min-ute amounts of backscattered light requiring optical isolators and special care to avoid feedback [71] Light reflected light directed into the laser cavity produces wavelength and inten-sity modulations corresponding to the phase of the returned light just as in an interferometer although in this case the reference beam is essentially internal to the laser [72 73] Figure 11 shows that the geometry for a self-mixing or opti-cal feedback interferometer for distance measurement can be very simple All that is required is a path for reflected light to enter the lasermdasha condition that is almost unavoidable in many casesmdashand a detector for observing the modulations in the laser output in response to the phase of this reflected light The technique has high sensitivity to small amounts of

returned light and is best suited for scattering surfaces that couple less than 001 of the source light back into the cavity

Simple FabryndashPerot type laser diodes are easily tuned by injection current modulation which changes the cavity length by rapid heating and cooling of the active area of the diode [74] Alternatively an external cavity using mirrors or grat-ings tunes the laser by controlled mechanical motions [75] The resulting signal has a distinctive sawtooth shape as a result of the laser system phase lock to the external reflec-tion but the frequency of this signal corresponds to that of any other type of interferometric swept-wavelength laser ranging system Self-mixing has been investigated for a wide range of sensing applications including displacement velocimetry and absolute distance measurement [76]

The interferometer configurations so far have been shown with free-space light propagation throughout whereas in practice today it is common to use fiber optics to control light paths For the plane-mirror DMI and the encoder figures 7 and 8 respectively it is convenient to transmit the source light by a polarization-preserving single-mode fiber subsystem and to capture the output light in a multi-mode fiber for detec-tion some distance away [77 78] Fiber-based interferometer systems have advantages in flexibility and heat management when placing interferometer optics close to the application area such as a precision stage in a thermally-sensitive area

The final example of an absolute distance interferometer is multiplexed fiber-optic sensors designed for small size and multi-axis position sensing [79] Fiber sensors are widely used for measurements of strain temperature and pressure using interferometric methods [79 80] Figure 12 illustrates an example system intended for up to 64 sensors external to a control box that contains all of the complex hardware required to illuminate the sensors provide phase modulation and heter-odyne detection [81] Spectrally-broadband light source light follows two paths with phase modulation being applied along one path and a phase delay along the other The modulated and delayed copies of the source illumination are combined and delivered via an optical fiber to the miniature fiber-optic Fizeau interferometers The short coherence length of the source means that only obtained between pairs of reflections that have travelled the same optical path length The light reflected from the sensors travels back through fibers to an array of detectors that pick up the resulting beat signals Data

Figure 11 Optical-feedback laser distance measuring system The target object in most cases has a rough surface texture scattering a small fraction of the source light back into the laser

Figure 12 Fiber-based positioning system with multiple remote sensors

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rates are up to 208 kHz per channel with extraordinarily low noise levels of 0003 nm (

radicHz)minus1 [18] Recent applications

include the in situ measurement of deformable mirrors in syn-chrotrons [82]

3 Testing of optical components

31 The laser Fizeau interferometer

The light box that provided us with the interference fringe pattern in figure 1 was for most of the history of interferom-etry representative of the state of the art in interferometric surface form testing Generalizations of the idea including proper imaging instruments with physically-separated refer-ence and measurement surfaces such as the TwymanndashGreen interferometer invented a hundred years ago [83] had lim-ited range of application because of the short coherence length of the available light sources and the skill required to interpret the resulting patterns These challenges meant that optical shops often preferred geometric ray tests such as the Foucault knife-edge or shadow method to evaluate sur-face form using zonal masks for aspheric surfaces such as parabolas [84 85]

A breakthrough in the use of interferometry for optical testing was the laser which enabled the modern laser Fizeau geometry with unequal paths [86] High coherence means that the instrument is easily reconfigured to accommodate flats concave spheres convex spheres lenses prisms and even complete optical assemblies [11 87] In combination with refractive reflective or holographic null correctors the Fizeau geometry also measures aspherical and free-form optics [88 89] In addition to its flexibility the Fizeau geometry is attractive in that there are no additional beam shaping optics between the reference and the object which significantly simplifies the instrument calibrationmdashin most cases it is suf-ficient to accurately characterize the reference surface [90] These advances led to commercial instruments such as the one

shown in figures 13 and 14 that moved interferometers from the physics lab to the optical shop floor [91]

The need for high accuracy surface measurement requires a more sophisticated approach than the visual interpretation of fringe patterns This requires automated fringe analysis using electronic cameras and heterodyne techniques [92] Phase shifting interferometry or PSI relies on controlled phase shifts ξ (t) usually imparted by the mechanical motion of the refer-ence mirror [93] Sometimes referred to as temporal PSI to distinguish the method from the spatial interpretation of indi-vidual fringe pattern images PSI analyzes the interference signals from each pixel individually calculating an interfer-ence phase φ (x y) proportional to each point h (x y) in the surface topography where the height h is orthogonal to the nominal surface shape

Surface topography measurements using PSI assume two-beam interference similar to what we have already seen for displacement measurement in equation (6)

I (x y t) = I1 + I2 +radic

I1I2 cos [φ (x y) + ξ (t)] (20)

Figure 13 A commercial implementation of a laser Fizeau system from the early 1970s This was a visual system with an integrated film camera for recording interference fringe images Photo courtesy of Zygo Corporation

Figure 14 Laser Fizeau interferometer with a phase shifted reference setup for a concave spherical object surface

Figure 15 Upper graph intensity signal and digital sampling in linear phase shifting Lower graph intensity signal with a sinusoidal phase shift

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9

φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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23

β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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5

I (Lα) = I1 + I2 +radic

I1I2 cos [φ (L) + φ0 +∆νt] (11)

The output beam intensity oscillates at a rate of a few kHz to hundreds of MHz depending on the system design and the maximum object velocity A single detector captures this signal for each measurement direction or axis and firmware applies a sliding-window Fourier analysis or some other tech-nique to determine the phase [32]

Whether the phase detection is by homodyne or heterodyne techniques polarization plays an important role in nearly all DMI systems Heterodyne laser sources usually emit simultaneously in two orthogonal polarizations encoding the frequency shift ∆ν between the polarization states The interferometer optics include polarizing elements to separate the measurement and reference beams and direct them to the reference and object mirrors This free-space delivery system can feed many interferometer optical assemblies monitoring multiple displacement directions or axes of motion in com-plex machines The motions include linear motions angles and straightness [33]

Polarized heterodyne laser sources may rely on the Zeeman effect wherein a strong magnetic field along the axis of the tube splits the laser emission into orthogonally circularly polarized modes with a frequency difference of 1 to 4 MHz [35 36] Higher-performance systems use acousto-optic mod-ulators with split frequencies of 20 to 80 MHz to overcome the Doppler shift in high-speed stage systems (figure 6) while preserving the characteristic of generating copropagating collinear measurement and reference beams with orthogonal polarizations [34 37ndash40]

Modern displacement interferometers are widely used as sensors for stage motion monitoring and control In such applications the simple geometry of the Michelson is not quite good enough because of the sensitivity of the geometry to tilt-ing of the mirrors In one-axis systems this issue is overcome by using retroreflective optics such as corner-cubes in place of measurement and reference mirrors For x-y stages the problem becomes more interesting because the interferometer must accommodate motions orthogonal to the line of site The interferometer optical design of figure 7 overcomes this issue by a double pass to a plane mirror plane mirror rather than a single pass to a retroreflector [31 41] The design relies on a quarter-wave plate (QWP) to convert linear polarization to circular and then back to an orthogonal linear polarization after reflection from the object mirror The polarizing beam splitter (PBS) and QWP enable the double pass with the retro-reflection that compensates for the tip and tilt of the object mirror The object mirror is free to move orthogonal to the line of sight without disturbing the beam paths [42] An additional benefit of the double pass is a finer measurement resolution by magnifying the optical path change with displacement by 4times Although shown as 2D in the figures to clarify their func-tion actual interferometers are 3D and arrange beam paths for compactness and light efficiency Commercial interferometers for advanced stage positioning use optics of BK7 or crystal-line quartz vacuum-grade low-volatility adhesives stainless steel housings and thermal compensation [43]

The performance of DMI systems has progressed signifi-cantly driven in large part by the demands of the photolithog-raphy industry which depends on sub-nm positioning at stage motions of 4 m sminus1 The high precision of these measurements relates to the need to register successive exposures during the creation of transistors with critical dimensions currently approaching 7 nm [44] Double-exposure techniques allocate only 05 nm in the uncertainty budget for stage metrology [45]

22 Multi-dimensional optical encoders

The evolution of stage positioning requirements in photoli-thography to the impressive numbers in the previous para-graph is an interesting story In the early 1990s the stage metrology demands were on the order of a few tens of nm and were adequately addressed with free-space heterodyne interferometry Tightening tolerances were met with improve-ments in optical designs and active techniques for suppress-ing cyclic errors related to polarization mixing and other imperfections [33 46ndash48] In the latter half of the 1990s it became clear that operating in open air for the primary stage motion metrology as required for Michelson-type interferom-eters introduced seemingly intractable problems related to air turbulence errors The first line of defense was to com-pletely redesign airflow within the photolithography system but this clearly had its limits There were serious attempts to compensate for air-index using dispersion interferometrymdashactually measuring the air density along the line of site in real time by comparing optical path length measurements at two or more wavelengths [26 49 50] and to replace multi-pass interferometers with single-beam designs with dynamic beam pointing [51] These advances were continuously overtaken by rapidly-tightening requirements

Ultimately the solution to the air turbulence problem has been to use optical encoders in combination with 2D gratings (figure 9) because encoders have much shorter beam paths and are therefore less sensitive to air turbulence [52 53] This transition to encoders was not easymdashthere are significant chal-lenges with regard to grating manufacture Abbe offset errors and with the encoders themselves which initially were not up to the task This in part was related to the more traditional role

Figure 8 Optical encoder for simultaneous measurement of y z displacements

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6

of optical encoders for machine tool positioning feedback as opposed to sub-nm high-speed photolithography

Of the several possible approaches to high-precision encoder-based positioning one of the more straightforward strategies involves placing a grating in the path of a conven-tional Michelson-type interferometer that employs retroreflec-tors as targets as shown in figure 8 [54] The signal detection once again uses the heterodyne technique with polarization to encode the reference and measurement paths In the geometry of figure 8 there are two measurements of interference phase φ+φminus to enable simultaneous determination of the vertical z and lateral y displacements in the figure [55ndash57] The calcul-ations are

x =(φ+ minus φminus)

8πΛ (12)

z =(φ+ + φminus)

8πλ

1 +raquo

1 minus (λΛ )2 (13)

where Λ is the grating pitch The encoder design of figure 8 is to first order insensitive to tip and tilt of the diffraction gratingmdashan important practical requirement However out-of-plane motion along the z axis has the effect of laterally shearing the measurement beam with respect to the reference beam which reduces signal strength Newer designs recircu-late the beams in such a way that the double pass passively compensates for this effect and include means for reduc-ing accidental mixing of beam paths using polarized angled beams [56 58 59]

23 Absolute distance and position measurement

So far we have considered high-precision displacement mea-surements meaning that the DMI system tracks changes in position without necessarily determining the starting point In a displacement measurement if the measurement beam is blocked and then reestablished there is no memory of any change in position that may have occurred when the object was not being tracked An alternative class of position-sensing

interferometer measures the so-called absolute distance as opposed to a relative displacement These systems are also of interest where the target surface may have a rough surface tex-ture that complicates fringe tracking especially when using a single-point probe for surface profiling There is abundant literature on this topic representing a wide range of solutions [60 61]

Most interferometric systems for absolute distance meas-urement employ multiple or swept-wavelength sources The essential principles were known from the earliest interferom-eters for length standards [62] and has been in continuous use for measuring gage blocks for the past century [63] The principle of measurement follows from the dependence of the interferometric phase on wavelength given in equation (9) which we can rewrite as

φ (L) = knL (14)

where the angular wavenumber is given by

k = 2πλ (15)

The classical method of excess fractions relies on match-ing phase values for a sequence of discrete wavelengths to a specific distance using tables or a special slide rule The wavelengths may be selected by a prism or be generated by multiple light sources or a single light source that is tunable or selectable in wavelength

Another algorithmic approach to multiple-wavelength interferometry relies on the definition of equivalent or synth-etic wavelengths For a pair of optical wavelengths λ1 gt λ2 the synthetic wavelength is

Λ = λ1λ2(λ1 minus λ2) (16)

for a corresponding synthetic phase

Φ (L) = 2πnLΛ (17)

given by the difference in interference phase measured at each of the two wavelengths λ1λ2 Figure 10 illustrates a synthetic wavelength interferometer using two single-mode tunable diode lasers locked to a common FabryndashPerot etalon that provides high relative wavelength stability for large vari-ations in optical path length [64 65] Synthetic wavelength

Figure 9 A 2D grating for encoder-based stage positioning

Figure 10 Synthetic wavelength interferometer using two lasers locked to a common resonance frequency of a confocal etalon

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7

interferometry has also been applied to the profiling of unpol-ished or technical surfaces using laterally-scanning probes [66 67]

The essential principle of multiple wavelength interferom-etry is that the rate of change of phase φ with wavenumber k is proportional to the optical path difference nL as is directly shown by taking the derivative of equation (15)

dφdk

= nL (18)

An alternative to a light source with a sequence of discrete wavelengths is a frequency-swept light source so as to cal-culate this rate of change as a continuous derivative The heterodyne frequency ∆ν of the interference signal during wavelength tuning for a tunable optical frequency ν is

∆ν =dνdt

nLc

(19)

It is then straightforward to solve for the optical path differ-ence nL This method is sometimes referred to as frequency-modulated continuous wave ranging Laser diodes are logical choices for tunable sources [68 69] High precision on the order of one part in 109 of the measured distance is feasible with advances sources and sufficient care [70]

A complication when using tunable laser diodes is that these devices have a tendency to be highly sensitive to min-ute amounts of backscattered light requiring optical isolators and special care to avoid feedback [71] Light reflected light directed into the laser cavity produces wavelength and inten-sity modulations corresponding to the phase of the returned light just as in an interferometer although in this case the reference beam is essentially internal to the laser [72 73] Figure 11 shows that the geometry for a self-mixing or opti-cal feedback interferometer for distance measurement can be very simple All that is required is a path for reflected light to enter the lasermdasha condition that is almost unavoidable in many casesmdashand a detector for observing the modulations in the laser output in response to the phase of this reflected light The technique has high sensitivity to small amounts of

returned light and is best suited for scattering surfaces that couple less than 001 of the source light back into the cavity

Simple FabryndashPerot type laser diodes are easily tuned by injection current modulation which changes the cavity length by rapid heating and cooling of the active area of the diode [74] Alternatively an external cavity using mirrors or grat-ings tunes the laser by controlled mechanical motions [75] The resulting signal has a distinctive sawtooth shape as a result of the laser system phase lock to the external reflec-tion but the frequency of this signal corresponds to that of any other type of interferometric swept-wavelength laser ranging system Self-mixing has been investigated for a wide range of sensing applications including displacement velocimetry and absolute distance measurement [76]

The interferometer configurations so far have been shown with free-space light propagation throughout whereas in practice today it is common to use fiber optics to control light paths For the plane-mirror DMI and the encoder figures 7 and 8 respectively it is convenient to transmit the source light by a polarization-preserving single-mode fiber subsystem and to capture the output light in a multi-mode fiber for detec-tion some distance away [77 78] Fiber-based interferometer systems have advantages in flexibility and heat management when placing interferometer optics close to the application area such as a precision stage in a thermally-sensitive area

The final example of an absolute distance interferometer is multiplexed fiber-optic sensors designed for small size and multi-axis position sensing [79] Fiber sensors are widely used for measurements of strain temperature and pressure using interferometric methods [79 80] Figure 12 illustrates an example system intended for up to 64 sensors external to a control box that contains all of the complex hardware required to illuminate the sensors provide phase modulation and heter-odyne detection [81] Spectrally-broadband light source light follows two paths with phase modulation being applied along one path and a phase delay along the other The modulated and delayed copies of the source illumination are combined and delivered via an optical fiber to the miniature fiber-optic Fizeau interferometers The short coherence length of the source means that only obtained between pairs of reflections that have travelled the same optical path length The light reflected from the sensors travels back through fibers to an array of detectors that pick up the resulting beat signals Data

Figure 11 Optical-feedback laser distance measuring system The target object in most cases has a rough surface texture scattering a small fraction of the source light back into the laser

Figure 12 Fiber-based positioning system with multiple remote sensors

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rates are up to 208 kHz per channel with extraordinarily low noise levels of 0003 nm (

radicHz)minus1 [18] Recent applications

include the in situ measurement of deformable mirrors in syn-chrotrons [82]

3 Testing of optical components

31 The laser Fizeau interferometer

The light box that provided us with the interference fringe pattern in figure 1 was for most of the history of interferom-etry representative of the state of the art in interferometric surface form testing Generalizations of the idea including proper imaging instruments with physically-separated refer-ence and measurement surfaces such as the TwymanndashGreen interferometer invented a hundred years ago [83] had lim-ited range of application because of the short coherence length of the available light sources and the skill required to interpret the resulting patterns These challenges meant that optical shops often preferred geometric ray tests such as the Foucault knife-edge or shadow method to evaluate sur-face form using zonal masks for aspheric surfaces such as parabolas [84 85]

A breakthrough in the use of interferometry for optical testing was the laser which enabled the modern laser Fizeau geometry with unequal paths [86] High coherence means that the instrument is easily reconfigured to accommodate flats concave spheres convex spheres lenses prisms and even complete optical assemblies [11 87] In combination with refractive reflective or holographic null correctors the Fizeau geometry also measures aspherical and free-form optics [88 89] In addition to its flexibility the Fizeau geometry is attractive in that there are no additional beam shaping optics between the reference and the object which significantly simplifies the instrument calibrationmdashin most cases it is suf-ficient to accurately characterize the reference surface [90] These advances led to commercial instruments such as the one

shown in figures 13 and 14 that moved interferometers from the physics lab to the optical shop floor [91]

The need for high accuracy surface measurement requires a more sophisticated approach than the visual interpretation of fringe patterns This requires automated fringe analysis using electronic cameras and heterodyne techniques [92] Phase shifting interferometry or PSI relies on controlled phase shifts ξ (t) usually imparted by the mechanical motion of the refer-ence mirror [93] Sometimes referred to as temporal PSI to distinguish the method from the spatial interpretation of indi-vidual fringe pattern images PSI analyzes the interference signals from each pixel individually calculating an interfer-ence phase φ (x y) proportional to each point h (x y) in the surface topography where the height h is orthogonal to the nominal surface shape

Surface topography measurements using PSI assume two-beam interference similar to what we have already seen for displacement measurement in equation (6)

I (x y t) = I1 + I2 +radic

I1I2 cos [φ (x y) + ξ (t)] (20)

Figure 13 A commercial implementation of a laser Fizeau system from the early 1970s This was a visual system with an integrated film camera for recording interference fringe images Photo courtesy of Zygo Corporation

Figure 14 Laser Fizeau interferometer with a phase shifted reference setup for a concave spherical object surface

Figure 15 Upper graph intensity signal and digital sampling in linear phase shifting Lower graph intensity signal with a sinusoidal phase shift

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φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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12

wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

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tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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6

of optical encoders for machine tool positioning feedback as opposed to sub-nm high-speed photolithography

Of the several possible approaches to high-precision encoder-based positioning one of the more straightforward strategies involves placing a grating in the path of a conven-tional Michelson-type interferometer that employs retroreflec-tors as targets as shown in figure 8 [54] The signal detection once again uses the heterodyne technique with polarization to encode the reference and measurement paths In the geometry of figure 8 there are two measurements of interference phase φ+φminus to enable simultaneous determination of the vertical z and lateral y displacements in the figure [55ndash57] The calcul-ations are

x =(φ+ minus φminus)

8πΛ (12)

z =(φ+ + φminus)

8πλ

1 +raquo

1 minus (λΛ )2 (13)

where Λ is the grating pitch The encoder design of figure 8 is to first order insensitive to tip and tilt of the diffraction gratingmdashan important practical requirement However out-of-plane motion along the z axis has the effect of laterally shearing the measurement beam with respect to the reference beam which reduces signal strength Newer designs recircu-late the beams in such a way that the double pass passively compensates for this effect and include means for reduc-ing accidental mixing of beam paths using polarized angled beams [56 58 59]

23 Absolute distance and position measurement

So far we have considered high-precision displacement mea-surements meaning that the DMI system tracks changes in position without necessarily determining the starting point In a displacement measurement if the measurement beam is blocked and then reestablished there is no memory of any change in position that may have occurred when the object was not being tracked An alternative class of position-sensing

interferometer measures the so-called absolute distance as opposed to a relative displacement These systems are also of interest where the target surface may have a rough surface tex-ture that complicates fringe tracking especially when using a single-point probe for surface profiling There is abundant literature on this topic representing a wide range of solutions [60 61]

Most interferometric systems for absolute distance meas-urement employ multiple or swept-wavelength sources The essential principles were known from the earliest interferom-eters for length standards [62] and has been in continuous use for measuring gage blocks for the past century [63] The principle of measurement follows from the dependence of the interferometric phase on wavelength given in equation (9) which we can rewrite as

φ (L) = knL (14)

where the angular wavenumber is given by

k = 2πλ (15)

The classical method of excess fractions relies on match-ing phase values for a sequence of discrete wavelengths to a specific distance using tables or a special slide rule The wavelengths may be selected by a prism or be generated by multiple light sources or a single light source that is tunable or selectable in wavelength

Another algorithmic approach to multiple-wavelength interferometry relies on the definition of equivalent or synth-etic wavelengths For a pair of optical wavelengths λ1 gt λ2 the synthetic wavelength is

Λ = λ1λ2(λ1 minus λ2) (16)

for a corresponding synthetic phase

Φ (L) = 2πnLΛ (17)

given by the difference in interference phase measured at each of the two wavelengths λ1λ2 Figure 10 illustrates a synthetic wavelength interferometer using two single-mode tunable diode lasers locked to a common FabryndashPerot etalon that provides high relative wavelength stability for large vari-ations in optical path length [64 65] Synthetic wavelength

Figure 9 A 2D grating for encoder-based stage positioning

Figure 10 Synthetic wavelength interferometer using two lasers locked to a common resonance frequency of a confocal etalon

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7

interferometry has also been applied to the profiling of unpol-ished or technical surfaces using laterally-scanning probes [66 67]

The essential principle of multiple wavelength interferom-etry is that the rate of change of phase φ with wavenumber k is proportional to the optical path difference nL as is directly shown by taking the derivative of equation (15)

dφdk

= nL (18)

An alternative to a light source with a sequence of discrete wavelengths is a frequency-swept light source so as to cal-culate this rate of change as a continuous derivative The heterodyne frequency ∆ν of the interference signal during wavelength tuning for a tunable optical frequency ν is

∆ν =dνdt

nLc

(19)

It is then straightforward to solve for the optical path differ-ence nL This method is sometimes referred to as frequency-modulated continuous wave ranging Laser diodes are logical choices for tunable sources [68 69] High precision on the order of one part in 109 of the measured distance is feasible with advances sources and sufficient care [70]

A complication when using tunable laser diodes is that these devices have a tendency to be highly sensitive to min-ute amounts of backscattered light requiring optical isolators and special care to avoid feedback [71] Light reflected light directed into the laser cavity produces wavelength and inten-sity modulations corresponding to the phase of the returned light just as in an interferometer although in this case the reference beam is essentially internal to the laser [72 73] Figure 11 shows that the geometry for a self-mixing or opti-cal feedback interferometer for distance measurement can be very simple All that is required is a path for reflected light to enter the lasermdasha condition that is almost unavoidable in many casesmdashand a detector for observing the modulations in the laser output in response to the phase of this reflected light The technique has high sensitivity to small amounts of

returned light and is best suited for scattering surfaces that couple less than 001 of the source light back into the cavity

Simple FabryndashPerot type laser diodes are easily tuned by injection current modulation which changes the cavity length by rapid heating and cooling of the active area of the diode [74] Alternatively an external cavity using mirrors or grat-ings tunes the laser by controlled mechanical motions [75] The resulting signal has a distinctive sawtooth shape as a result of the laser system phase lock to the external reflec-tion but the frequency of this signal corresponds to that of any other type of interferometric swept-wavelength laser ranging system Self-mixing has been investigated for a wide range of sensing applications including displacement velocimetry and absolute distance measurement [76]

The interferometer configurations so far have been shown with free-space light propagation throughout whereas in practice today it is common to use fiber optics to control light paths For the plane-mirror DMI and the encoder figures 7 and 8 respectively it is convenient to transmit the source light by a polarization-preserving single-mode fiber subsystem and to capture the output light in a multi-mode fiber for detec-tion some distance away [77 78] Fiber-based interferometer systems have advantages in flexibility and heat management when placing interferometer optics close to the application area such as a precision stage in a thermally-sensitive area

The final example of an absolute distance interferometer is multiplexed fiber-optic sensors designed for small size and multi-axis position sensing [79] Fiber sensors are widely used for measurements of strain temperature and pressure using interferometric methods [79 80] Figure 12 illustrates an example system intended for up to 64 sensors external to a control box that contains all of the complex hardware required to illuminate the sensors provide phase modulation and heter-odyne detection [81] Spectrally-broadband light source light follows two paths with phase modulation being applied along one path and a phase delay along the other The modulated and delayed copies of the source illumination are combined and delivered via an optical fiber to the miniature fiber-optic Fizeau interferometers The short coherence length of the source means that only obtained between pairs of reflections that have travelled the same optical path length The light reflected from the sensors travels back through fibers to an array of detectors that pick up the resulting beat signals Data

Figure 11 Optical-feedback laser distance measuring system The target object in most cases has a rough surface texture scattering a small fraction of the source light back into the laser

Figure 12 Fiber-based positioning system with multiple remote sensors

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8

rates are up to 208 kHz per channel with extraordinarily low noise levels of 0003 nm (

radicHz)minus1 [18] Recent applications

include the in situ measurement of deformable mirrors in syn-chrotrons [82]

3 Testing of optical components

31 The laser Fizeau interferometer

The light box that provided us with the interference fringe pattern in figure 1 was for most of the history of interferom-etry representative of the state of the art in interferometric surface form testing Generalizations of the idea including proper imaging instruments with physically-separated refer-ence and measurement surfaces such as the TwymanndashGreen interferometer invented a hundred years ago [83] had lim-ited range of application because of the short coherence length of the available light sources and the skill required to interpret the resulting patterns These challenges meant that optical shops often preferred geometric ray tests such as the Foucault knife-edge or shadow method to evaluate sur-face form using zonal masks for aspheric surfaces such as parabolas [84 85]

A breakthrough in the use of interferometry for optical testing was the laser which enabled the modern laser Fizeau geometry with unequal paths [86] High coherence means that the instrument is easily reconfigured to accommodate flats concave spheres convex spheres lenses prisms and even complete optical assemblies [11 87] In combination with refractive reflective or holographic null correctors the Fizeau geometry also measures aspherical and free-form optics [88 89] In addition to its flexibility the Fizeau geometry is attractive in that there are no additional beam shaping optics between the reference and the object which significantly simplifies the instrument calibrationmdashin most cases it is suf-ficient to accurately characterize the reference surface [90] These advances led to commercial instruments such as the one

shown in figures 13 and 14 that moved interferometers from the physics lab to the optical shop floor [91]

The need for high accuracy surface measurement requires a more sophisticated approach than the visual interpretation of fringe patterns This requires automated fringe analysis using electronic cameras and heterodyne techniques [92] Phase shifting interferometry or PSI relies on controlled phase shifts ξ (t) usually imparted by the mechanical motion of the refer-ence mirror [93] Sometimes referred to as temporal PSI to distinguish the method from the spatial interpretation of indi-vidual fringe pattern images PSI analyzes the interference signals from each pixel individually calculating an interfer-ence phase φ (x y) proportional to each point h (x y) in the surface topography where the height h is orthogonal to the nominal surface shape

Surface topography measurements using PSI assume two-beam interference similar to what we have already seen for displacement measurement in equation (6)

I (x y t) = I1 + I2 +radic

I1I2 cos [φ (x y) + ξ (t)] (20)

Figure 13 A commercial implementation of a laser Fizeau system from the early 1970s This was a visual system with an integrated film camera for recording interference fringe images Photo courtesy of Zygo Corporation

Figure 14 Laser Fizeau interferometer with a phase shifted reference setup for a concave spherical object surface

Figure 15 Upper graph intensity signal and digital sampling in linear phase shifting Lower graph intensity signal with a sinusoidal phase shift

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9

φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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12

wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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13

for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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7

interferometry has also been applied to the profiling of unpol-ished or technical surfaces using laterally-scanning probes [66 67]

The essential principle of multiple wavelength interferom-etry is that the rate of change of phase φ with wavenumber k is proportional to the optical path difference nL as is directly shown by taking the derivative of equation (15)

dφdk

= nL (18)

An alternative to a light source with a sequence of discrete wavelengths is a frequency-swept light source so as to cal-culate this rate of change as a continuous derivative The heterodyne frequency ∆ν of the interference signal during wavelength tuning for a tunable optical frequency ν is

∆ν =dνdt

nLc

(19)

It is then straightforward to solve for the optical path differ-ence nL This method is sometimes referred to as frequency-modulated continuous wave ranging Laser diodes are logical choices for tunable sources [68 69] High precision on the order of one part in 109 of the measured distance is feasible with advances sources and sufficient care [70]

A complication when using tunable laser diodes is that these devices have a tendency to be highly sensitive to min-ute amounts of backscattered light requiring optical isolators and special care to avoid feedback [71] Light reflected light directed into the laser cavity produces wavelength and inten-sity modulations corresponding to the phase of the returned light just as in an interferometer although in this case the reference beam is essentially internal to the laser [72 73] Figure 11 shows that the geometry for a self-mixing or opti-cal feedback interferometer for distance measurement can be very simple All that is required is a path for reflected light to enter the lasermdasha condition that is almost unavoidable in many casesmdashand a detector for observing the modulations in the laser output in response to the phase of this reflected light The technique has high sensitivity to small amounts of

returned light and is best suited for scattering surfaces that couple less than 001 of the source light back into the cavity

Simple FabryndashPerot type laser diodes are easily tuned by injection current modulation which changes the cavity length by rapid heating and cooling of the active area of the diode [74] Alternatively an external cavity using mirrors or grat-ings tunes the laser by controlled mechanical motions [75] The resulting signal has a distinctive sawtooth shape as a result of the laser system phase lock to the external reflec-tion but the frequency of this signal corresponds to that of any other type of interferometric swept-wavelength laser ranging system Self-mixing has been investigated for a wide range of sensing applications including displacement velocimetry and absolute distance measurement [76]

The interferometer configurations so far have been shown with free-space light propagation throughout whereas in practice today it is common to use fiber optics to control light paths For the plane-mirror DMI and the encoder figures 7 and 8 respectively it is convenient to transmit the source light by a polarization-preserving single-mode fiber subsystem and to capture the output light in a multi-mode fiber for detec-tion some distance away [77 78] Fiber-based interferometer systems have advantages in flexibility and heat management when placing interferometer optics close to the application area such as a precision stage in a thermally-sensitive area

The final example of an absolute distance interferometer is multiplexed fiber-optic sensors designed for small size and multi-axis position sensing [79] Fiber sensors are widely used for measurements of strain temperature and pressure using interferometric methods [79 80] Figure 12 illustrates an example system intended for up to 64 sensors external to a control box that contains all of the complex hardware required to illuminate the sensors provide phase modulation and heter-odyne detection [81] Spectrally-broadband light source light follows two paths with phase modulation being applied along one path and a phase delay along the other The modulated and delayed copies of the source illumination are combined and delivered via an optical fiber to the miniature fiber-optic Fizeau interferometers The short coherence length of the source means that only obtained between pairs of reflections that have travelled the same optical path length The light reflected from the sensors travels back through fibers to an array of detectors that pick up the resulting beat signals Data

Figure 11 Optical-feedback laser distance measuring system The target object in most cases has a rough surface texture scattering a small fraction of the source light back into the laser

Figure 12 Fiber-based positioning system with multiple remote sensors

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rates are up to 208 kHz per channel with extraordinarily low noise levels of 0003 nm (

radicHz)minus1 [18] Recent applications

include the in situ measurement of deformable mirrors in syn-chrotrons [82]

3 Testing of optical components

31 The laser Fizeau interferometer

The light box that provided us with the interference fringe pattern in figure 1 was for most of the history of interferom-etry representative of the state of the art in interferometric surface form testing Generalizations of the idea including proper imaging instruments with physically-separated refer-ence and measurement surfaces such as the TwymanndashGreen interferometer invented a hundred years ago [83] had lim-ited range of application because of the short coherence length of the available light sources and the skill required to interpret the resulting patterns These challenges meant that optical shops often preferred geometric ray tests such as the Foucault knife-edge or shadow method to evaluate sur-face form using zonal masks for aspheric surfaces such as parabolas [84 85]

A breakthrough in the use of interferometry for optical testing was the laser which enabled the modern laser Fizeau geometry with unequal paths [86] High coherence means that the instrument is easily reconfigured to accommodate flats concave spheres convex spheres lenses prisms and even complete optical assemblies [11 87] In combination with refractive reflective or holographic null correctors the Fizeau geometry also measures aspherical and free-form optics [88 89] In addition to its flexibility the Fizeau geometry is attractive in that there are no additional beam shaping optics between the reference and the object which significantly simplifies the instrument calibrationmdashin most cases it is suf-ficient to accurately characterize the reference surface [90] These advances led to commercial instruments such as the one

shown in figures 13 and 14 that moved interferometers from the physics lab to the optical shop floor [91]

The need for high accuracy surface measurement requires a more sophisticated approach than the visual interpretation of fringe patterns This requires automated fringe analysis using electronic cameras and heterodyne techniques [92] Phase shifting interferometry or PSI relies on controlled phase shifts ξ (t) usually imparted by the mechanical motion of the refer-ence mirror [93] Sometimes referred to as temporal PSI to distinguish the method from the spatial interpretation of indi-vidual fringe pattern images PSI analyzes the interference signals from each pixel individually calculating an interfer-ence phase φ (x y) proportional to each point h (x y) in the surface topography where the height h is orthogonal to the nominal surface shape

Surface topography measurements using PSI assume two-beam interference similar to what we have already seen for displacement measurement in equation (6)

I (x y t) = I1 + I2 +radic

I1I2 cos [φ (x y) + ξ (t)] (20)

Figure 13 A commercial implementation of a laser Fizeau system from the early 1970s This was a visual system with an integrated film camera for recording interference fringe images Photo courtesy of Zygo Corporation

Figure 14 Laser Fizeau interferometer with a phase shifted reference setup for a concave spherical object surface

Figure 15 Upper graph intensity signal and digital sampling in linear phase shifting Lower graph intensity signal with a sinusoidal phase shift

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φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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11

mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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12

wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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13

for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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8

rates are up to 208 kHz per channel with extraordinarily low noise levels of 0003 nm (

radicHz)minus1 [18] Recent applications

include the in situ measurement of deformable mirrors in syn-chrotrons [82]

3 Testing of optical components

31 The laser Fizeau interferometer

The light box that provided us with the interference fringe pattern in figure 1 was for most of the history of interferom-etry representative of the state of the art in interferometric surface form testing Generalizations of the idea including proper imaging instruments with physically-separated refer-ence and measurement surfaces such as the TwymanndashGreen interferometer invented a hundred years ago [83] had lim-ited range of application because of the short coherence length of the available light sources and the skill required to interpret the resulting patterns These challenges meant that optical shops often preferred geometric ray tests such as the Foucault knife-edge or shadow method to evaluate sur-face form using zonal masks for aspheric surfaces such as parabolas [84 85]

A breakthrough in the use of interferometry for optical testing was the laser which enabled the modern laser Fizeau geometry with unequal paths [86] High coherence means that the instrument is easily reconfigured to accommodate flats concave spheres convex spheres lenses prisms and even complete optical assemblies [11 87] In combination with refractive reflective or holographic null correctors the Fizeau geometry also measures aspherical and free-form optics [88 89] In addition to its flexibility the Fizeau geometry is attractive in that there are no additional beam shaping optics between the reference and the object which significantly simplifies the instrument calibrationmdashin most cases it is suf-ficient to accurately characterize the reference surface [90] These advances led to commercial instruments such as the one

shown in figures 13 and 14 that moved interferometers from the physics lab to the optical shop floor [91]

The need for high accuracy surface measurement requires a more sophisticated approach than the visual interpretation of fringe patterns This requires automated fringe analysis using electronic cameras and heterodyne techniques [92] Phase shifting interferometry or PSI relies on controlled phase shifts ξ (t) usually imparted by the mechanical motion of the refer-ence mirror [93] Sometimes referred to as temporal PSI to distinguish the method from the spatial interpretation of indi-vidual fringe pattern images PSI analyzes the interference signals from each pixel individually calculating an interfer-ence phase φ (x y) proportional to each point h (x y) in the surface topography where the height h is orthogonal to the nominal surface shape

Surface topography measurements using PSI assume two-beam interference similar to what we have already seen for displacement measurement in equation (6)

I (x y t) = I1 + I2 +radic

I1I2 cos [φ (x y) + ξ (t)] (20)

Figure 13 A commercial implementation of a laser Fizeau system from the early 1970s This was a visual system with an integrated film camera for recording interference fringe images Photo courtesy of Zygo Corporation

Figure 14 Laser Fizeau interferometer with a phase shifted reference setup for a concave spherical object surface

Figure 15 Upper graph intensity signal and digital sampling in linear phase shifting Lower graph intensity signal with a sinusoidal phase shift

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9

φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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11

mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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12

wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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13

for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

Rep Prog Phys 82 (2019) 056101

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interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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9

φ (x y) =4πλ

h (x y)

(21)

To analyze such a signal one approach is to expand the intensity

I = I1 + I2 +radic

I1I2 [cos (φ) cos (ξ)minus sin (φ) sin (ξ)] (22)

where the x y t is implicit for brevity In linear PSI the phase shift ξ varies continuously and proportionally with time over a fixed range at a constant rate ω

ξ = ωt (23)

The result is oscillatory sine and cosine modulations in equa-tion (22) Fitting sine and cosine functions to the interference signal gives the sin (φ) and cos (φ) terms respectively pro-viding quadrature signals that allow an accurate determination of φ and hence the surface height h at that pixel An analytical expression for the method is

tan (φ) = N (φ)D (φ) (24)

N (φ) = minusacute πminusπ

I (φ ξ) sin (ξ)dξD (φ) =

acute πminusπ

I (φ ξ) cos (ξ)dξ

(25)

This autocorrelation or synchronous detection technique is equivalent to Fourier analysis at a single frequency deter-mined by the rate of movement of the reference mirror [92]

In practice a sequence of camera frames samples the interference signal as shown in the upper graph of figure 15 Assuming for example an acquisition of four sample intensi-ties I0123 corresponding to four phase shifts ξ0123 at evenly-spaced intervals ω ∆t = π2 the integrations are

N = I0 + I1 minus I2 + I3

D = minusI0 + I1 + I2 minus I3 (26)

Discrete sampling requires full normalization the orthogo-nality of N D and the suppression of the constant offset I1 + I2 Design rules for PSI algorithms cover a wide range of mathematical methods that include the use of Fourier the-ory [94ndash96] and characteristic polynomials [97] to optimize performance Review articles provide comprehensive lists of these algorithms together with the relative performances in the presence of error sources [98]

An alternative to linear PSI with a phase shift ξ that is directly proportional to time is a phase shift which is itself sinusoidal as in

ξ = u cos (ωt) (27)

The resulting intensity pattern is a cosine of a cosine which generates a signal comprised of harmonics of the original phase modulation ξ The lower graph of figure 15 shows an example signal which changes shape with the interference phase φ as opposed to position on the time axis as is the case with linear phase shifting Sinusoidal PSI has the attraction that the phase shift is continuous and does not require inter-ruption to reverse and restart the shift as is the case with linear PSI It is a single-frequency modulation that can be easier to implement with mechanical phase shift mechanisms [99 100]

The phase estimation for sinusoidal PSI again involves the arctangent of ratio calculated from a discrete sampling of the intensity signal Rewriting equation (24) we have

tan (φ) = Nprime (φ)Dprime (φ) (28)

but now the numerator and denominator represent the relative strengths of the odd and even harmonics of the fundamental signal frequency ω respectively For discrete sampling using j = 0 1 M minus 1 for M camera frames the general form of the calcul ation is similar to linear PSI

Nprime (φ) =Mminus1sumj=0

ajIj

Dprime (φ) =Mminus1sumj=0

bjIj

(29)

where the coefficients aj bj define the sinusoidal phase shift algorithm A simple 4-frame algorithm attributed to Sasaki has coefficient vectors

a =Auml

1 1 minus1 minus1auml

b =Auml

1 minus1 1 minus1auml (30)

where the data samples are at intervals ω∆t = π2 and the sinusoidal phase shift amplitude is u = 245 radians [99]

As is the case for linear phase shifting strategies have been developed for designing sinusoidal PSI algorithms with high resistance to camera nonlinearity phase shifting errors vibra-tion and intensity noise [101] Greater performance generally requires more camera frames High-performance algorithms for applications in Fizeau interferometry and interference microscopy use up to 20 camera frames a sampling interval ω ∆t = π10 The 20 camera frames are acquired over one cycle of the sinusoidal phase modulation For a 200 Hz cam-era this is ten phase shift cycles and 10 complete measure-ments per second [102 103]

When the light source has a tunable wavelength it is pos-sible to phase shift without a mechanical device using a wavelength modulation in an unequal-path geometry [104] Wavelength tuning is particularly interesting when the size of the test part exceeds what can be accommodated by mechani-cal phase shifting methods [105] This approach has the additional benefit of providing a means to separate parallel surfaces of an optical component or system since the amount of phase shift depends on the distance to the surface [106] The interference signal can be analyzed in frequency space to identify each surface individually [107]

32 Vibration sensitivity and environmental robustness

Phase-shifting interferometric techniques depend on mechan-ical stability during the time-based data acquisition As a con-sequence most optical testing laboratories are equipped with vibration isolation systems which may include an air table at a minimum and specialized facilities for the most precise mea-surements [108] When these isolation measures are imperfect there can be measurement errors that manifest themselves as ripple patterns in the measured topography with a spatial

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frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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10

frequency equal to twice that of the interference fringes in the field of view [109]

There has been extensive work to determine the sensitiv-ity of PSI algorithms to specific vibrational frequencies and amplitudes Small phase measurement errors can be treated as linear perturbations that allow for closed-form expressions of vibration sensitivity [110ndash112] A linear approximation first defines an arbitrary vibration n (t) complex frequency spec-trum N (ν)

n (t) = Re

ˆ infin

minusinfinN (ν) exp (iνt) dt (31)

and then calculates the effect of each vibrational frequency ν on the phase measurement The phase error for each frequency can be determined by multiplying the vibrational noise spec-trum N (ν) by a transfer function P (νφ) that depends both on the frequency ν of the vibration and the interference phase φ The net phase error ∆φ can be calculated by summing up the individual phase errors corresponding to the various fre-quencies The results of this multiplication integrated over all frequencies yields the net measurement error

∆ϕ (φ) = Re

ˆ infin

0N(ν)P(ν θ)dν (32)

The transfer function P (νφ) remains the same for all forms of vibration and is characteristic of the interferometer and the PSI algorithm The transfer function P (νφ) is calculated in closed-form solution for a variety of common PSI algorithms in [112]

The upper graph of figure 16 shows the response of an interferometer to single vibrational tones for a common linear

PSI algorithms that uses five camera frames to discretely sample the phase-shifted interference signal [109 112] The vibrational frequency is normalized to the rate of phase shift in the interferometer in terms of fringes per unit time The phase errors were averaged over all vibrational phase angles α A number of important conclusions may be drawn from the vibration-sensitivity spectrum Firstly there is a prominent peak near a normalized vibrational frequency of ν = 2 corre-sponding to two unintended vibrational oscillations during a single ξ = 2π cycle of controlled phase shifting Another gen-eral observation is that the sensitivity to vibration decreases with vibrational frequency thanks primarily to the smoothing effect of the camera integration The lower graph of figure 16 shows the results for sinusoidal phase shifting with a 12-frame algorithm [113] The response curve is more complex given the interaction of vibrational motions with the harmonics of the intensity pattern The general conclusions are nonetheless similar to those for linear PSI in that there are peak sensi-tivities that can be identified using the transfer function and that the sensitivity declines at higher frequencies thanks to the camera frame integration time

There are two primary practical uses of the vibrational error transfer function The first is to aid in algorithm design usually with an emphasis on reducing low-frequency sensi-tivity by increasing the number of camera frames in the data acquisition [95] The other common use is as a guide to sup-pression or avoidance of specific vibrational frequencies that are known from this analysis to be particularly bothersome

There is a temptation to develop further PSI algorithms to further incrementally improve vibration performance However with advancing computing power PSI algorithms based fixed design frequency with an inflexible data reduction algorithm are becoming less common in optical testing inter-ferometers Iterative least-squares methods are now capable of unraveling the contributions from intentional and accidental

Figure 16 Vibrational sensitivity transfer functions showing the predicted RMS measurement error at 633 nm wavelength for a 1 nm synchronous vibration Upper graph linear phase shift and a 5-frame algorithm Lower graph sinusoidal phase shift with a 12-frame algorithm

Figure 17 Spatial phase shifting technique using carrier fringes to enable measurements of surface topography with a single camera frame

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mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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mechanical motions [114ndash117] These new methods are sig-nificantly more robust in the presence of vibration than tradi-tional PSI although they remain dependent on a controlled phase shift over time to function

As a further alternative to traditional time-based PSI methods instantaneous interferometers impart phase shifts onto the interference signal that are measurable at the same instant in time or at least within a time window of a single shuttered camera frame on the order of 10 micros to 100 micros The usual approach is to replace the time-based phase shift with a spatial one A classical technique relies on a large amount of static tilt between the reference and object surfaces which results in a large number of carrier fringes as shown upper left hand image of figure 17 The carrier fringes introduce the equivalent to a continuous phase shift ξ orthogonal to the fringe direction which may be analyzed by convolution or Fourier methods to extract a surface profile with a single cam-era frame of data [118] This method not only provides near immunity to vibrational errors but also the option of captur-ing dynamically-changing events and viewing the resulting topography changes in real time [119] An interesting aspect of the spatial carrier fringe method is its similarity to off-axis digital holography The holographic recording technique is essentially identical with the only tangible difference being that holographic methods usually employ digital propagation for depth perception or focus correction when reconstructing the measurement wavefront [120 121]

The use of polarization at the pixel level provides another approach to spatial phase shifting proved that the measure-ment and reference paths can be distinguished by orthogonal polarizations similar to what is done for homodyne DMI [118 122 123]

33 Lateral resolution and the instrument transfer function

Of practical importance in optics manufacturing is the toler-ancing of surface topography deviations from the intended design as a function of the separation of features or surface spatial frequency There are a number of ways of doing this including structure functions [124] Zernike polynomials and Fourier analysis [125]

The Fourier approach is one of the more established meth-ods in spite of recognized limitations related to boundary conditions The Fourier-based power spectral density (PSD) quantifies the surface topography as a function of spatial fre-quency [126] Mathematically the PSD P (fx fy) is calculated from the surface profile spatial frequency distribution h (fx fy) obtained from the Fourier transform F of the measured sur-face topography h (x y) over the full surface as follows

P (fx fy) =∣∣∣h (fx fy)

∣∣∣2

(33)

h (fx fy) = F h (x y) (34)

F h (x y) =

umlh (x y) exp [2πi (fxx + fyy)] dxdy (35)

The spatial frequencies fx fy along the x y directions respec-tively are the inverse of the spatial periods of the individual sinusoidal components of the surface height map Often the normalized PSD is over a line profile or is integrated annu-larly so that it may be plotted in terms of a single frequency coordinate f with frequency ranges corresponding to form waviness and roughness An especially useful feature of the PSD is that integration over a range of frequencies provides the variance or square of the RMS deviation

Of particular interest to the optics manufacturer who is tar-geting a PSD specification is the range of spatial frequencies that can be captured and accurately measured interferometri-cally [92] A relevant performance parameter is the instru-ment transfer function (ITF) The ITF catalogs the response of an interferometer to the Fourier components of the surface topography An ITF function T ( f ) can be determined empiri-cally by comparing the measured PSD Pprime ( f ) to the known or calibrated PSD P ( f ) of a surface feature having a broad distribution of spatial frequencies

T ( f ) =raquo

Pprime ( f )P ( f ) (36)

The function T ( f ) is a linear filter for the mutually-independent Fourier components

radicP ( f ) The ITF is sometimes referred

to as the system transfer function [127] or the height trans-fer function [128] The ITF is a more complete description of an instrument than a simple statement of the Rayleigh or Sparrow resolution limit [129]

The experimental determination of the ITF may use a sharp surface step height or edge feature which has a broad-spec-trum PSD [130] An example test specimen for Fizeau inter-ferometry is the patterned 100 mm diameter super-polished disk shown in figure 18 The specimen has two step features one horizontal and the other vertical and includes additional features lying in the quadrants defined to aid in focusing the instrument These features are fabricated using conventional lithography methods and should be small with respect to the

Figure 18 Upper left surface topography of a 100 mm diameter test object used for ITF evaluation Lower right interference microscope image of one of the 30 nm step features

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wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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12

wavelength for reasons that will become apparent shortly The traces are differentiated to produce a line spread function and then Fourier transformed to produce a quantitative result for the local ITF [131]

Optimizing the ITF requires an instrument model for deter-mining the limiting factors to resolution and measurement fidelity Laser interferometer systems for optical testing fall well within the scope of classical Fourier optics as a modeling tool and it is relatively straightforward to predict the instru-ment response given a surface topography the imaging prop-erties of the optical system and the measurement principle As we shall see diffraction in combination with the spatial fil-tering properties places some limits on when we can approxi-mate the instrument response with a linear ITF

The system model is as in figure 19 where the pupil is at the back focal plane of the objective lens L1 the wavefront spatial frequencies are filtered at the pupil and the ocular lens L2 images the object onto the detector The pupil plane wave-front is in this model the Fourier Transform of the object plane wavefront according to Fraunhofer diffraction The model is silent regarding how we add a reference wave and measure interference phase rather we assume that the detected surface heights are based on evaluation of the phase of the imaged electric field without error

In the model we assume a unit-magnitude plane wave normally incident on a surface height distribution h (x y) The surface height variations are coded into a complex phase φ (x y) for the reflected wavefront E (x y) for each individual image point in the same way as linear displacement measure-ments (see section 21)

E (x y) = exp [iφ (x y)] (37)

φ (x y) =4πλ

h (x y) (38)

This simple relationship assumes that the local topography variations are well within the depth of focus of the optical imaging system In this way we tie the wavefront phase to the object surface as linearly scaled versions of each other

The next step is to image the wavefront through the sys-tem to the detector The amplitude transfer function (ATF) sometimes referred to as the coherent optical transfer function (CTF) is a convenient way to express the contribution of the optical design to the instrument response when the illumina-tion is a laser or monochromatic point The ATF is the Fourier transform of the coherent point spread function and is a shift-invariant linear filter for complex amplitudes source and may incorporate modest optical aberrations resulting from imper-fections in the instrument design [132ndash134] To use the filter we start with an input electric field expressed as a complex amplitude E (x y) propagate to the far-field E (fx fy) using a Fourier transform apply the ATF ψ (fx fy) to determine the fil-tered spectrum Eψ (fx fy) then transform to an imaged wave-front Eψ (x y) by an inverse Fourier transform

E (fx fy) = F E (x y) (39)

Eψ (fx fy) = ψ (fx fy) E (fx fy) (40)

Eψ (x y) = Fminus1para

Eψ (fx fy)copy

(41)

This formalism is similar to the ITF but relates only to the optical systems and wavefronts as opposed to object surfaces The spatial frequencies fx fy scale with the wavelength λ for the diffraction angles αxαy through the familiar scaler dif-fraction grating formula which for the x direction reads

sin (αx) = λfx (42)

For optical systems that obey the Abbe sine condition the spa-tial frequencies fx fy map linearly to the pupil plane

The final step is to reconstruct the surface topography as a measured 3D image by again assuming a linear relation-ship between the phase φψ (x y) for the imaged wavefront Eψ (x y) as follows

φψ (x y) = arg [Eψ (x y)] (43)

hψ (x y) =λ

4πφψ (x y) (44)

This last step is accepted practice in interferometry notwith-standing the implicit assumption that the linear filtering of the electric field is equivalent to linearly filtering the complex phase of the electric fieldmdashan assumption that cannot be true in general based on the mathematical behavior of complex numbers Consequently the effect of a 3D variation in height

Figure 19 Simplified generic model of an optical system for measuring 3D surface topography

Figure 20 Illustration of the difference between diffraction from a 1D sinusoidal reflectivity pattern (upper graph) and a sinusoidal surface topography variation (lower graph)

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13

for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

Rep Prog Phys 82 (2019) 056101

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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13

for a uniformly-reflecting smooth surface is quite different from conventional imaging of pure reflectivity objects

To understand the effect of 3D structures on the reflected wavefront we consider a sinusoidal object of peak height b and pitch 1ν such that the height map is

h (x y) = b cos (2πνx) (45)

Applying the JacobindashAnger expansion for h (x y) we have the renormalized expression

E (x y) = 1 + a1 cos (2πνx) +infinsum

p=2

i pap cos (2πpνx) (46)

and the coefficients ap for p gt 0 are

ap = 2Jp

Aring4πbλ

atildeJ0

Aring4πbλ

atilde (47)

The first two terms of the complex amplitude E (x y) in equation (46) look like an intensity reflectivity pattern at the frequency ν of the sinusoidal profile However these two terms are followed by a sum of contributions at higher spatial frequencies pν that are not actually present in the surface structure h (x y) The resulting diffraction pattern from the sinusoidal topography consists of the first-order diffractions at fx = ν plus a sequence of higher-order dif-fracted beams These higher orders (p gt 1) are absent for objects that have only a reflectivity variation as opposed to a surface height variation Figure 20 illustrates the differ-ence between diffraction from an intensity-reflectivity pat-tern and diffraction from a variation in surface height The larger the surface height variation the greater the number of diffracted orders

The presence of a sequence of high-order diffracted beams from 3D surface structures together with the use of complex phase as a means of evaluating surface heights means that it is possible for the instrument response to be nonlinear even though the optical system itself can be meaningfully mod-eled by a linear ATF [135 136] It is worth emphasizing that this conclusion is not a limitation unique to interferometry rather it is fundamental to optical imaging of 3D surface structures

Fortunately the linear response for interferometric topog-raphy measurements is a reasonable approximation for a wide range of practical situations [129 135] If local surface height variations are small then only the p = 0 1 diffraction orders are of significance and the 3D imaging has transfer proper-ties similar to that of 2D variations in surface reflectivity The small surface-height limit is most relevant to high spatial frequencies and sharp-edged discrete features with the usual rule of thumb being that the peak-valley variation should be much less than a limit value of λ4 The limit value follows from the observation that neighboring points that are λ4 dif-ferent in height are in phase opposition (180deg) which results in strongly nonlinear behavior if the point spread functions at these neighboring points overlap

Another practical regime is low spatial frequencies for which all of the significant diffracted beams of order p gt 1 are passed through the limiting apertures of the optical system [137] This is the case for overall form deviations with mod-est slopes corresponding to clearly-resolved high-contrast

Figure 21 Comparison of the experimentally-determined ITF for an advanced laser Fizeau interferometer with the results of a theoretical model based on Fourier optics

Figure 22 Schematic diagram of an interference microscope with an equal-path objective for operation with an extended white-light source

Figure 23 Left interference fringes with white-light illumination in an interference microscope Right the same fringes viewed with a black and white camera showing maximum fringe contrast at zero group-velocity path difference

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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14

interference fringes Within these restrictions the interferom-eter response is sufficiently linear that the ITF may be approx-imated by the modulus of the ATF Figure 21 for example shows good agreement between a Fourier optics model that includes aberrations and system limits with the exper imental results of an empirical ITF measurement using the sample shown in figure 18 [131]

4 Interference microscopy for surface structure analysis

41 Principles and design of 3D interference microscopes

One of the fundamental characteristics of interferometric sur-face topography measurement is that the sensitivity to varia-tions in surface height is independent of the field of view This is in contrast to other optical metrology techniques based on triangulation moireacute fringes or focus effects Laser Fizeau interferometers can have working apertures of 1000 mm in diameter while maintaining RMS measurement noise levels well below 1 nm (radicHz)minus1 over millions of data points At the opposite extreme interference microscopes have fields of view ranging from about 10 mm down to 008 mm in stan-dard commercial implementations using turreted interference objectives with comparable noise levels for all magnifications [138ndash140]

The typical optical configuration of a modern surface-topography measuring interference microscope for reflective surfaces comprises an epi Koumlhler illuminator based on an LED light source a digital electronic camera an interference objective and a mechanism for scanning the objective axially as shown in figure 22 Given the spatially and spectrally inco-herent light source the interferometer must have equal group-velocity path lengths for high-contrast fringes as illustrated in figure 23 This is why the partially-transparent reference mirror in figure 22 is positioned at a distance to the beam split-ter that is equal to the distance to the object surface and the reference and beam splitter plates are of equal thickness [141]

Interference microscopes may use phase shifting inter-ferometry (PSI) in a manner similar to a laser Fizeau espe-cially for optical-quality polished surfaces [142 143] A more common measurement technique today for more complex surface textures is coherence scanning interferometry or CSI [140 144] The benefit of CSI is that it enables interferometric

measurements of surface topography structures that do not exhibit clear fringes across the surface such as those shown in figure 23 With a broad spectrum light source and high numer-ical aperture the narrow range of surface heights for which there are high-contrast interference fringes means that it is not always necessary to see continuous interference fringes on a surface to perform useful metrologymdashit is sufficient to detect the presence of an interference effect [144 145] CSI enables highly flexible measurements of areal surface topog-raphy that would otherwise be beyond the reach of interfer-ence microscopy

Data acquisition for CSI consists of mechanically moving the interference objective shown in figure 22 axially (verti-cally in the figure) which simultaneously and synchronously scans the focus and optical path difference The camera col-lects images at the rate of a few images for each full cycle of interference resulting in a scan-dependent signal for each pixel in the field of view (see figure 24) showing interference fringes modulated by the coherence effect

There is a variety of signal processing options for CSI One approach makes use of high-speed digital signal processing (DSP) electronics to demodulate the CSI signal in real time [146] This method enabled the first commercial implementa-tion of interference microscopy for rough surfaces in 1992 A more common approach today is digital post-processing of signals stored in dynamic memory followed by algorithms designed to extract both the position of the peak signal con-trast and the interference phase at the mean wavelength The location of the contrast peak is often referred to as the coher-ence profile data while the interference fringe information provides the phase profile The phase profile has approxi-mately the same noise level as conventional PSI on smooth surfaces whereas the coherence profile is effective even when the interference phase is randomized by surface texture or dis-crete features [147 148]

In CSI it is often the case that the coherence profile pro-vides an initial estimate of the surface topography which is then refined to a much lower noise level using phase data [146 148ndash150] Combining phase and coherence data is not as straightforward as one might think The first issue encounter ed which is actually quite an interesting one is that the fundamental metric of measurement is different for the two methods On the one hand the phase data scales to height based on our knowledge of the wavelength and optical geom-etry The coherence-based topography on the other hand is calculated based on the scan position and has nothing what-soever to do with the wavelength

One way to link the coherence and phase calculations together to the same length scale is to work entirely in the frequency domain by means of a Fourier transform of the CSI signal [147 151] The result is a series of Fourier coeffi-cients of phases ϕ associated with fringe frequencies K scaled according to the known scan step between camera data acqui-sitions These Fourier phases are

ϕ (K) = hK (48)

and the coherence-based estimation hC of surface height h is given by the phase slope

Figure 24 CSI signal for a single camera pixel as a function of the axial scan position of the interference objective

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15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

15

hC = dKdϕ (49)

estimated from a least-squares linear fit to the discrete values ϕ K calculated from the Fourier transform This same linear fit provides a mean phase value at a chosen center frequency K0

φ = hK0 (50)

The phase φ equivalent to a PSI measurement using all of the data frames that have nonzero signal modulation and repre-sents in a least-squares sense the lowest achievable noise level for a given CSI signal To calculate the final height value we apply

h =φ

K0+

2πK0

roundAring

A minus 〈A〉2π

atilde (51)

where the so-called phase gap is

A = θ minus K0hC (52)

and the brackets 〈〉 represent a field average [151]Applications in semiconductor medical and automo-

tive surface structures encompass fields of view from 01 to 40 mm and surface height ranges from less than 1 nm

up to several mm Commercial instruments typically have extensive post-processing capabilities for identifying fea-tures and textures that correlate to functional behavior such as lubrication adhesion friction leaks corrosion and wear [24 142 152ndash157]

Two examples illustrate the range of application of modern interference microscopes Figure 25 is a PSI measurement of a super-polished texture 1 mm square having a height range of plusmn025 nm At the other texture extreme figure 26 is a 3D image of a Tindash6Alndash4V laser powder bed fusion sample show-ing steep slopes and highly irregular features captured using the CSI measurement principle [158 159]

42 Signal modeling for interference microscopy

There is significant interest in understanding how signals are generated in interference microscopy particularly for CSI which has a useful signal structure as illustrated in figure 24 The shape of the interference contrast envelope depends on the illumination spectrum and spatial extent as well as on the surface structure Understanding these contributions enables both instrument optimization and in some cases solving for surface structures using modeling methods

The usual approach to signal modeling is to assume a fully incoherent superposition of interference intensity pat-terns over a range of illumination angles and wavelengths [160ndash162] Starting with the simplest case of an object surface that is perfectly flat and without structure and oriented so that there is no tip or tilt the interference signal as a function of the axial scan position of the interference objective ζ has the form of an inverse Fourier transform

I (ζ) =ˆ infin

minusinfinq (K) exp (minusiKζ) dK (53)

where the K gt 0 part of the frequency spectrum is

q (K) = ρ (K) exp (iKh) (54)

In principle we should expect to have to calculate the inter-ference contributions for all possible source spectrum wave-numbers k = 2πλ and incident angles ψ However the interference fringe frequency K links these two factors as follows

K = 2kβ (55)

where the directional cosine is

β = cos (ψ) (56)

Therefore we can determine β given the two frequencies K k and calculate the height-independent Fourier coefficients from the single integral

ρ (K) =

ˆ infin

K2

K4k2 rOB (K k) rRF (K k) S (k) ΦIN (K k) ΦEX (K k) dk

(57)where the complex reflectivity rRF is for the reference surface rOB is the reflectivity of the object surface and S is the spec-trum of the light source [163] The incident light distribution in the objective pupil is ΦIN and the imaging aperture after

Figure 25 Measurement of a super-polished optical surface using a low-noise interference microscope The measured RMS surface roughness Sq over a 1 times 1 mm field of view is 005 nm

Figure 26 True color surface topography measurement for an additive manufactured part measured with a CSI microscope [33] The measurement represents a 4 times 4 laterally-stitched 3D image for a 50 times objective lens (NA = 055) The height range is 150 microm

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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23

β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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16

reflection from the object surface is ΦEX In the simplest case the imaging aperture follows from the numerical aperture AN of the optical system

ΦEX =K2k

minusraquo

1 minus A2N (58)

and may be identical to the illumination aperture ΦIN The K lt 0 values for q (K) are the complex conjugates of these K gt 0 values The K = 0 coefficient is the sum of the height-independent terms

ρ (0) =ˆ infin

0

ˆ infin

Kprime2

Kprime

4k2

Auml|rOB (Kprime k)|2 + |rRF (Kprime k)|2

auml

S (k) ΦIN (Kprime k) ΦEX (Kprime k) dk dKprime

(59)

where Kprime is a free variable of the integrationThis formalism readily extends to layered media that

is to a surface that has a semi-transparent film but is other-wise still nominally flat and featureless so that only specular reflection need be considered in the model This has proven to be a successful strategy for solving the inverse problem for CSI signals distorted by material properties or the presence of thin transparent films [164 165] To calculate the signal we need only include these wavelength and incident angle effects in the complex reflectivity rOB of the object surface An example calculation shown in figure 27 The methodol-ogy is to match the experimental surface height-independent Fourier components ρ (K) to a library of theoretically-calcu-lated values over a range of surface parameters such as film thickness or other optical properties Figure 28 illustrates one use of this capability to measure the top surface film thick-ness and substrate topography for an oxide film on a silicon

wafer This approach has been successfully applied to films down to 10 nm thick [165]

The situation becomes more complicated if we introduce significant topography structure even when there is no addi-tional surface film present There are fundamental open ques-tions regarding the instrument response over a range of feature types particularly in the presence of high surface slopes or when operating close to the traditional limits of optical resolution

With an abundance of caution an analysis similar to that of the coherent Fourier optics model of section 33 can be extended to the incoherent imaging system of a CSI micro-scope Such modeling shows that with comparable restrictions allows for estimating the ITF for interferometers that use dif-fuse white-light sources by using the incoherent modulation transfer function (MTF) in place of the modulus of the ATF [129 138 166 167] Figure 29 demonstrates this approach by comparing an experimental measurement of the ITF using the step-height method and an experimental determination of the incoherent MTF using an opaque screen [129] Both transfer functions closely match the predicted MTF

An alternative to a conventional Fourier optics modeling of instrument response is to define a 3D point spread function and a corresponding 3D optical transfer function in the frequency domain [168ndash173] This approach is particularly interesting if the surface height variations are significant when compared to the depth of focus of the system [174] Assuming scalar diffraction and weak scattering the interference fringes are consistent with a 3D convolution filtering operation applied to a thin refractive index profile or foil representing the sur-face The filtering operation can be determined empirically by calibration [175] This approach to interferometric imaging allows in many cases for compensation of measurement errors by application of an inverse filter after calibration

43 The pursuit of low measurement noise

Measurement noise is a fundamental metrological character-istic and is an essential component of instrument calibration and specification [176] Noise is distinguished from other sources of error by its random quality as opposed to sys-tematic effects distortions and thermal drift [177] Noise is added to the measured topography and limits our ability to view fine textures (figure 25) and to detect small signals returned from weakly-reflecting strongly-sloped or scatter-ing textures (figure 26)

A straightforward method for quantifying noise is the sur-face topography repeatability (STR) Two measurements are performed one right after the other resulting in two topogra-phy maps h0 (x y) h1 (x y) The difference between these two measurements is a map of random errors indicative of instru-ment noise environmental effects or other instabilities in the system If we assume that the errors are essentially random we identify the STR with measurement noise

NM =1radic2

Atilde1

XY

Xminus1sumx=0

Yminus1sumy=0

[h1 (x y)minus h0 (x y)]2 (60)

Figure 27 Comparison of a calculated and an experimentally-measured interference signal for a visible-wavelength 08 NA CSI instrument for a silicon surface coated with a 067 microm oxide (SiO2) film

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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17

where X Y are the total number of discrete pixel positions x y respectively This is equivalent to the RMS or Sq param-eter applied to the difference map divided by the square root of 2 since both height maps contribute to the observed data randomization There are several ways of estimating the mea-surement noise NM with increasing confidence over many data sets but the simple two-measurement test corresponds to the essential idea [177ndash180] As an illustration of the calcul-ation figure 30 shows a height map h0 (x y) for a SiC optical flat in the upper image the difference h1 (x y)minus h0 (x y) in the lower image The noise is essentially random on the spatial scale of individual pixels and has a calculated noise value of 0007 nm

There are a number of ways to reduce the measurement noise including acquiring data over a longer time to pro-duce more stable results at the cost of reduced measurement speed or low-pass filtering at the cost of lateral resolution [181 182] This can make it confusing to compare noise specification between measuring technologies and implemen-tations given the wide range of acquisition speeds and pixel counts characteristic of different instruments Good practice in dimensional sensor specification therefore is to express noise as a spectral density ηM such that

NM = ηM

raquoTt (61)

where T is the number of independent uncorrelated image points and t is the data acquisition time A proper noise speci-fication should either explicitely state the noise density ηM or equivalently provide both t and T together with the corre-sponding noise NM

As an illustration of the time dependence of NM figure 31 shows the measured STR using a commercial interference microscope operating in sinusoidal PSI mode [101 140 183] The data show the expected trend with a noise value of NM = 003 nm

radict middot Hz The total number of pixels in this

case is 1000 000 with 3 times 3 median filtering also applied

Figure 28 Practical application of model-based films analysis for a CSI microscope

Figure 29 Experimental and theoretical ITF for an interference microscope in the small-surface height regime compared to the conventional imaging MTF for diffuse white-light illumination

Figure 30 Upper image measured surface topography map for a SiC flat for a 26s data acquisition Lower image difference between two successive individual measurements

Figure 31 Data-acquisition time dependence for the measurement noise NM

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

Rep Prog Phys 82 (2019) 056101

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

Rep Prog Phys 82 (2019) 056101

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23

β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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18

In an ideal metrology environment with a smooth part the majority of noise arises from the digital electronic camera which has random noise contributions from pixel to pixel determined in part by the well depth [140 184] In many prac-tical applications however the observed noise level is much higher than can be accounted for by these limited instrument noise contributions

A well-known error source is vibration Instruments based on interferometry are equipped with systems to dampen or isolate vibrations [112 143] The residual errors depend on the vibrational frequency with low-frequency vibrations asso-ciated with form errors and high-frequency vibrations con-tributing to ripple errors at twice the spatial frequency of the interference fringes [112 178 185 186] Figure 32 illustrates these effects using experimental CSI data at 20times magnifica-tion for a flat mirror The intensity image at the upper left of figure 32 shows the orientation of the interference fringes Without vibration the measured RMS surface roughness in the 3D topography is 1 nm The errors caused by environ-mental vibrations shown in the lower left and lower right images for the same surface are 4 nm RMS for low- and high-frequency vibrations respectively although the effects on the measured surface form are clearly different

The magnitude and frequency dependence of vibrational errors in traditional interference microscopy are well known [112 186] Newer instruments employ technologies to reduce these effects and make it easier to use interference microscopy in production environments [187ndash189] but vibration remains an important consideration in instrument design and good practice

5 Industrial surfaces and holographic interferometry

51 Geometrically desensitized interferometry for rough surfaces

An lsquoindustrial partrsquo or a lsquotechnical surfacersquo broadly refers to objects other than the lenses prisms and mirrors that make

up optical systems Very often these objects have machined ground wire-polished or burnished surfaces with average roughness greater than a few tenths of a micron exceeding the customary limits for what we would call an optically-smooth surface For the metrology instrument maker the challenge of industrial part measurement is that the surface form toler-ances may be in the micron range but the surface structure is inconsistent with the measurement strategies described in the previous section [190] This covers a wide range of precision-engineered partsmdashfar more than are captured by the optical components and systems that not so long ago were the unique area of application for interferometric surface metrology

A technique known for over a century is grazing inci-dence interferometry [8] The idea rest on the observation that roughly-textures surfaces appear smoother when viewed at a steep angle of incidence This observation translates quantita-tively into an equivalent wavelength given by

Λ = λcos (α) (62)

where λ is the source wavelength and α is the angle of inci-dence [191] Often this equivalent wavelength can be made long enough to overcome the scattering effects of rough sur-faces and provide interference fringes that can be modulated analyzed and converted into overall form maps in a manner similar to that of conventional interferometers A simple way to do this is the so-called skip test described by Wilson [192] in 1983 for use with a standard Fizeau interferometer which places the angled test object surfaces between the Fizeau ref-erence flat and an auxiliary mirror in the double-pass configu-ration shown in figure 33 This configuration is equally useful for large optically-smooth surfaces because of the elongated measurement area [193 194]

In the early 1970s Birch invented a grazing incidence interferometer using a pair of linear gratings arranged in series [195 196] A collimated beam passes through a linear diffraction grating to the object surface while the first-order diffracted beam followes a path that avoids the object and served as a reference beam A second grating combines the reference beam with the grazing-incidence reflection of the measurement beam to generate the interference effect This clever design is particularly effective for objects that are rather more long than wide such as machine tool ways Birchrsquos idea has been adapted for reflective diffraction gratings [197] and for cylindrical optics [198] Although these systems enable surface form measurements of industrial parts they have the disadvantage of requiring a laser because of unequal path

Figure 32 Experimental demonstration of potential errors in a CSI instrument without inadequate vibration compensation or isolation

Figure 33 Grazing incidence skip test for form measurements over large areas or in the presence of rough surface textures

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19

lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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lengths and the wavefront inversion between the measurement and reference beams

The diffractive-incidence interferometer in figure 34 over-comes the wavefront inversion problem described in the pre-vious paragraph by a configuration in which the sample and a reference mirror symmetrically reflect the first-order dif-fracted waves to a fold mirror and back again to the grating [199] The interference pattern appears on a scattering screen oriented at an oblique incidence to compensate for the image compression that results from the grazing incidence geometry Similar symmetric geometries appear in optical encoders and in the context of achromatic fringe generation for velocimetry and particle counting

The system is sufficiently achromatic to that it does not require a laser since in principle all source wavelengths have the same equivalent wavelength given by the grating period This together with the non-inverting equal-path geometry relaxes the coherence requirements so that one may use a mul-timode laser diode and a rotating diffuser as the light source A low coherence source eliminates speckle noise and other coherent artifacts This system enables metrology of ground-surface parts as shown in figure 35 which has an average sur-face roughness of 1 micron that is too high for conventional visible-wavelength interferometry

Grazing incidence is an effective technique for form meas-urements using a long equivalent wavelength generated simply

by the geometry of the measurement However there are some fundamental disadvantages in imaging fidelity including lat-eral resolution and form-dependent image distortion attribut-able to the steep oblique incidence An additional limitation is the effect of shadowing when viewing structured parts with recessed surfaces

An alternative geometry for geometrically-synthesizing a long equivalent wavelength involves illuminating the part simultaneously with both the measurement and reference beam but with these beams at different angles of incidence α1α2 The result as derived in 1910 by Barus [201] is an equivalent wavelength

Λ =λ

cos (α1)minus cos (α2) (63)

With a suitable choice of angles α1α2 almost any equivalent wavelength is possible Importantly these angles need not be anywhere as large as required for a grazing incidence geometry for example one angle can be 182deg and the other angle zero for normal incidence resulting in a magnification of the source wavelength λ by a factor of 20 The idea has been implemented for form metrology using a single diffraction grating [202] and a hologram [203 204] A disadvantage of single-grating con-figurations is that they require near contact with the part

A more practical arrangement for geometrically-desensi-tized interferometry near normal incidence was invented in the 1990s and uses two gratings in series to create working distance [200] As shown in figure 36 collimated source light from the multimode fiber bundle illuminates a first grating and diffracts into two first-order beams A second grating has a grating frequency twice that of the first and redirects beams so that they combine on the object surface The reflected light from the surface passes back through the gratings and interfere on the camera A characteristic of this dual-grating design is that the incident angles α1α2 on the object surface are estab-lished by diffraction angles with the net result that the design is to first order achromatic [205] The equivalent wavelength is given by the angle of incidence γ of the source light on the first grating and the first grating period N irrespective of the source wavelength

Figure 34 Diffractive grazing incidence interferometer for industrial metrology

Figure 35 Grazing-incidence form measurement for a 60 times 40 mm automotive pump part

Figure 36 Geometrically-desensitized interferometer

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Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

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of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

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20

Λ =1

2N sin (γ) (64)

As a practical example a grating pair with 250 lines mmminus1 for the first grating and 500 lines mmminus1 illuminated at an angle γ equal to 92deg results in an equivalent wavelength of 125 microm

Among the benefits of the longer equivalent wavelength is that the part setup and vibration control requirements are better suited to production metrology applications than many conventional types of interferometry Figure 37 shows a suc-cessful application of the technology for in-line testing of unfinished 96 mm diameter patters for computer rigid disk drives The platter sits a three-point mechanical fixture that locates the sample surface at the optimum metrology plane with the correct tip and tilt for measurement The fixture is adjusted during the initial setup of the instrument A robotic arm inserts and removes 700 platters per hour The system measures and sorts flatness results for both the rough surface aluminum blanks as well as the finished media [200]

52 Infrared wavelengths

So far we have considered systems operating with visi-ble-wavelength light but having a surface-height sensitiv-ity corresponding to a long equivalent wavelength so as to accommodate rough surface textures and large departures from flatness It is logical to ask if we can instead perform interferometry at a longer emission wavelength beyond the visible range in the infrared Such systems have indeed been in use for some time with many of the original applications for the testing of optical components in the grinding phase The fringe contrast contribution C resulting from surface scat-ter varies with RMS surface roughness σ as follows [206]

C = exp(minus8π2σ2λ2 )

(65)

This equation translates to the rule of thumb that a σλ of 10 results in a 50 drop in fringe contrast from the smooth-surface result Carbon-dioxide laser systems operating at 106 microm emission wavelengths enabling 50 contrast at a surface roughness of approximately 1 microm Such systems have been shown to be effective for measuring ground surfaces up to 2 microm RMS roughness [207] Interferometers at this wave-length require specialized pyroelectric or microbolometer cameras and germanium zinc-selenide or sapphire optical materials for lenses and prisms

Infrared wavelengths are of course all around us in the form of radiant heat If instead of a CO2 laser we employ a thermal source in a near-equal path interferometer geometry we can create the low-coherence interference fringes shown in figure 38 for a machined-metal part An advantage of the low coherence is the ability to measure large step height features without continuous fringes between the step features When paired with a DMI system as shown in figure 39 an infrared low-coherence interferometer measures both surface form and step heights between separated nominally-parallel surfaces The same idea can be further extended to simultaneous flat-ness thickness and parallelism of industrial components as shown in figure 40 [208]

53 Holographic interferometry of temperature gradients and sonic waves in fluids

There are many dimensional metrology tasks for which the result of interest is not the surface form itself but rather the change in form in response to a stimulus The broad field of non-destructive testing evaluates the response of test pieces or volumes to stress strain thermal gradients mechanical shock vibration or acoustic impulse [209ndash211] For these applications holographic interferometry and the closely-associated field of speckle metrology are attractive techniques [190 212ndash214] In the applications considered here holo-graphic interferometry is applied to thermal gradients and ultrasonic standing waves in fluids

Figure 37 A geometrically-desensitized interferometer using the geometry of figure 36 for automated inspection of brush-finished aluminum hard disk blanks

Figure 38 Low-coherence interference fringes (a) for a machined metal part (b) using an interferometer illuminated by a mid-infrared thermal source (c)

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Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

Rep Prog Phys 82 (2019) 056101

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

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Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

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21

Classical off-axis film-based holography involves recording the phase and amplitude of a wavefront using two-beam interference in a geometry the results in tilt or carrier fringes similar to those shown in figure 17 for single-frame laser Fizeau interferometry [121 215] Unlike inter-ferometry however the holographic interference pattern or hologram is usually not evaluated directly to quantify the complex wavefront Instead the hologram recreates the measurement wavefront by re-illuminating the hologram the reference beam alone Mathematically the sequence is as follows for a measurement wavefront reflected from a test object

E =radic

I exp (iφ) (66)

and a reference wavefront

E0 =radic

I0 exp (iφ0) (67)

the exposed and developed hologram has a transmission dis-tribution proportional to

IH = |E + E0 exp (2πiTxλ )|2 (68)

IH = (I + I0) + ElowastE0 exp (2πiTxλ ) + Elowast0 E exp (minus2πiTxλ )

(69)

where the phase change across the field caused by the relative tilt between the measurement and reference beams is

2πiTxλ (70)and T is the directional cosine

The interference fringes in the exposed hologram typically require an exceptionally high sampling density that up until recently required holographic film as opposed to a digital cam-era The combination of speckle patterns and dense fringes is often beyond the practical limits of direct interpretation yet it is possible to use the hologram to reproduce the measurement wavefront by the straightforward procedure of re-illuminating the hologram with the original reference beam The resulting complex field is

[E0 exp (2πiTx)] IH =(I + I0) [E0 exp (iTx)]

+ ElowastE0 exp (2iTx) + |E0|2E (71)The various terms in this equation are separable by the dif-fracting effect of the carrier fringes which allows us to isolate the term

Eprime = |E0|2E (72)

which is the original measurement wavefront with an inten-sity distribution proportional to the reference beam intensity profile The reconstructed wavefront may be further focused redirected or viewed from multiple angles in nearly the same way as the original wavefront reflected from the test object

In holographic interferometry the holographic process allows us to compare the measurement wavefront from instant in time to another instant for example after applying a stress a change in temperature or other distorting influence on the object and effectively interfere the measurement wave-front with itself before and after the application of the stress [210 216] We need not concern ourselves with the evaluation of the wavefront itself which can be a complicated mix of relevant and irrelevant information In double-exposure inter-ferometry the irrelevant information is spatially filtered leav-ing only the readily-interpreted difference fringes Figure 41 shows an example of a hologram where the change between exposures is the heating of a resister and the consequent ther-mal gradients in a fluid After the two exposures the measure-ment beam is blocked and the camera views the interference effect between the superimposed measurement beams repre-senting the sample area before and after heating The holo-gram is recorded on Holotest 8E75 film whereas the camera uses conventional film

Another important application of holographic interferom-etry is the areal or even volumetric evaluation of vibrations and other oscillatory motions using time averaging If we add a periodic oscillation of phase amplitude b and frequency ν to a two-beam interference signal we have

Figure 40 Flatness thickness and parallelism for a 32 mm diameter component having a machine-ground surface using a double-sided low-coherence interferometer operating near the 10 microm wavelength

Figure 39 System for surface flatness and step height measurement using a combination of displacement measuring interferometry (DMI) and broadband infrared surface interferometry

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22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

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β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

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24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

Rep Prog Phys 82 (2019) 056101

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25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

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Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

Review

27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

22

I = I1 + I2 +radic

I1I2 cos [φ+ b sin (2πνt)] (73)

If we average this signal over a sufficiently long time the result is an intensity that varies with the amplitude b of the oscillation according to

〈I〉 = J20 (b) (74)

which has an overall maximum (bright fringe) at b = 0 a first dark fringe at 240 a bright fringe at 38 the next dark fringe at 552 and so on This phenomenon provides a means for map-ping the oscillation amplitude as it changes across an object surface in response to an acoustic or vibrational stimulus

Figure 42 shows the results of a time-averaged holographic analysis of standing sonic waves in fluids using the apparatus of figure 41 The results for the index of refraction change are determined from the bright and dark fringes in the hologram and the geometry of the fluid cell [217] The results provide the relative compressibility of three different fluids including liquid nitrogen and Helium II in the presence of local varia-tions in pressure induced by the oscillating sonic wave [218] Time-averaged methods are applicable to a variety of applica-tions from industrial inspection to the analysis of the reso-nances of musical instruments

6 Metrology for consumer electronics

61 Flying height of read-write heads of rigid disk drives

Introduced by IBM in 1956 by the early 1960s the hard disk drive (HDD) became the dominant long-term data storage device for computers An HDD stores and retrieve digital information using a readwrite head or slider suspended close to the magnetic coating of a rapidly rotating (5000 RPM) disk A critical functional parameter is the flying height defined as the distance between the readwrite head and the disk The height should be close enough for dense data storage but suf-ficiently high to avoid striking the disk surface Performance

drivers linked to capacity and readwrite speed have driven the flying height from about 20 microns in the first commercial devices to a few nanometers today [219 220]

Optical flying-height testers use a rotating glass disk in place of the real magnetic disk providing direct visual access to the slider air bearing surface Pioneers in optical flying height testing readily appreciated the usefulness of the inter-ference effect Already in 1921 W Stone measured flying height of a slider through a spinning glass disk using a sodium flame and visual interpretation of the interference fringes resulting from the effects of the thin film of air within the gap [221] Stone was interested in the behavior of thrust bearings in railroad cars but the fundamental ideas have found their way into modern flying-height testers

Modern flying height testers use a laser in place of the sodium flame [222ndash224] while others use spectroscopic analy-sis of reflected white light or two or more selected wavelengths [225] These instruments work at normal incidence to the disk and measure the reflected intensity to estimate flying height An interesting alternative is to treat the air gap as a thin film and to analyze its thickness using a high-speed dynamic ellipsometer This technique provides an interesting example of polarization interferometry for a consumer-electronics application

The ellipsometric idea moves the illumination off normal incidence to create a polarization-dependent complex reflec-tivity at the slider-glass interface Referring to the geom-etry shown in figure 43 the polarization-dependent effective reflectivities of the slider-glass interface are

rps (β) =ςps + ςps exp (iβ)1 + ςpsςps exp (iβ)

(75)

where p s denote the polarization states parallel and perpend-icular to the plane of incidence The amplitude reflectivities ςps are for the glass-air surface on the underside of the disk while the reflectivities ς primeps refer to the slider surface The phase

Figure 42 Time-averaged holographic interferometry of standing sonic waves in fluids Left data showing the relationship between relative pressure in a standing sonic wave and the corresponding change in refractive index Right example hologram of a 094 MHz wave in liquid nitrogen at 77 K

Figure 41 Double exposure holographic interferogram of the temperature field around a 54 Ω 2 mm diameter resister immersed in methyl alcohol at 30 mA current

Rep Prog Phys 82 (2019) 056101

Review

23

β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

Review

24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

Rep Prog Phys 82 (2019) 056101

Review

25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

Rep Prog Phys 82 (2019) 056101

Review

26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

Review

27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

23

β =4πλ

h cos (φ) (76)

depends on the wavelength λ the angle of incidence φ and the flying height h The intensity of the reflected beam is then

I (β) = |zs (β)|2 + |zp (β)|2 (77)

The phase difference between the polarization components of the reflected beam is

θ (β) = arg [rs (β)]minus arg [rp (β)] (78)

The sensitivities of intensity and phase data are complemen-tary in that when the rate of change with flying height is flat for the intensity the phase data are changing rapidly and vice-versa The curve in a parameter plot of phase versus inten-sity is nearly circular and the unambiguous measurement range is one-half micron at a single visible wavelength [226] Figure 44 compares theoretical (solid line) and experimental intensity and phase data for a read-write slider in steady flight at several different locations on the tilted air-bearing surface The arrow indicates the direction of increasing flying height starting from contact

The flying height is calculated by minimizing the differ-ence between experimental and theoretical values of both the intensity and the phase using a merit function

χ2 (β) = [Iexp minus I (β)]2 + [θexp minus θ (β)]2

(79)

and varying β to minimize χ2(β) A four-channel homodyne receiver uses a combination of two Wollaston prisms a wave plate and four photodetectors to generate the required inten-sity signals for simultaneous intensity and phase estimation The experimental results figure 45 show variation in flying height for one position on the slider as a function of spindle speed for a measurement dwell time of 20 micros (50 kHz)

Any optical measurement of flying height necessarily requires knowledge of the optical properties of the slider surface which is comprised of an amalgam of metals The

amplitude reflectivities ς primeps depend on the complex index of refraction of the slider via the Fresnel equations [227] An interesting feature of the ellipsometric method presented here is that the instrument simultaneously measures the full com-plex reflectivity of the slider-glass interface during the flying height measurement [226] The in situ measurement simpli-fies production testing and eliminates uncertainties caused by differences in measurement geometry and material variations across the slider surface

An aspect of the ellipsometric measurement of flying height that requires special attention is potential errors related to the birefringent properties of the rapidly-rotating disk The platters in contemporary HDDs are spun at speeds varying from 4200 RPM in portable devices to 15 000 rpm for high-performance servers These rotation speeds generate signifi-cant stresses in the glass disk surrogate of the flying height

Figure 44 Parameter plot of the ellipsometric phase and reflected intensity over the 05 microm unambiguous range of the dynamic flying height measurement

Figure 45 Dynamic measurement of flying height while varying the speed of the rotating glass disk shown in figure 43

Figure 43 Configuration for the dynamic measurement of the fly height of read-write heads used in rigid disk drives

Rep Prog Phys 82 (2019) 056101

Review

24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

Rep Prog Phys 82 (2019) 056101

Review

25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

Rep Prog Phys 82 (2019) 056101

Review

26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

Review

27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

24

tester resulting in radial and tangential stress birefringence that compensated dynamically In the operation of the instru-ment the dynamic ellipsometer maps the background bire-fringence over the active area of the disk as a first step in the measurement sequence [228] Figure 46 is an example of such a mapping comparing the birefringence predicted by mod-eling the stresses and optical effects and experimental results The overall agreement is reasonable with the largest differ-ence at small radii caused by additional stress induced by the hub clamp

The instrument described here was developed in the late 1990s but it did not survive the technology competition to common usage today One of the disadvantages of the approach is disappointingly simple It is difficult to properly image at high resolution the microscopic read-write heads given the φ = 50 viewing angle through glass Modern sys-tems rely on updated embodiments of the more traditional spectral reflectometry techniques that operate at normal incidence Nonetheless the interferometric ellipsometer of figure 43 remains an interesting example of the applica-tion of interferometry to consumer electronics component inspection

62 Form and relational metrology of microlenses

Miniature cameras for smart phones laptops and automotive sensors have evolved into sophisticated optical assemblies of injection-molded micro optics In-process metrology for con-trolling lens surfaces and alignment features is an important part of improving performance and production yield [229] The surfaces are almost exclusively aspherical for wide fields compact design and high lateral resolution consistent with ever-increasing pixel count for the camera sensors

Contact stylus metrology has been a common choice for microlenses often limited to just two orthogonal cross-sectional profile measurements sometimes supplemented by optical measurements of some of the alignment features on the lenses [230] The need for greater detail full areal measurements and combined surface and feature measurements is driving

development of optical interferometry as a preferred solu-tion However measurements of aspheric form are a signifi-cant challenge for interferometry which is most precise and accurate when used to compare object surface shape to a well-known reference flat or sphere Deviations from the reference shape result in high interference fringe density and sensitiv-ity to optical imperfections and aberrations in the metrology instrument

A strategy for measuring aspheric form is to combine or lsquostitchrsquo a sequence of areal topography maps at high lat-eral resolution into a full surface reconstruction [231ndash233] The stitching approach can also be applied to micro optics [234 235] Figure 47 shows an implentation than combines coherence scanning interferometry (CSI) microscopy with five-axis staging for positioning micro-optics [140 236] The measurement sequence consists of individual CSI imaging scans performed over an array of pre-programmed imaging areas each of which is presented to the instrument with minimal tip and tilt Software adjusts the position and orientation of each map with respect to the others consist-ent with the concept of surface form and texture continuity including the roughness present in each map at micron-scale spatial periods [236 237] The most common measurement recipe relies on a targeted aspheric surface equation but it is also possible to discover the equation with no prior knowl-edge using a surface follower strategy following surface gradients

The instrument is an effective means of discovering issues that can affect image quality [238] Figure 48 illustrates a man-ufacturing problem with the flow and distribution of mat erial during injection molding In addition to the deviation from the design aspheric form the material has failed to completely fill the mold The measured topography includes other details regarding texture diamond turning marks and local surface defects on the lateral scale of a few microns The flexibility of the platform enables the determination of geometrical shape parameters critical to assembly of stacks of aspheres into a final lens which may comprise five or six elements Some

Figure 46 Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle as a function of skew angle and radial position for a disk rotating at 10 000 RPM The range of phase shift between orthogonal polarizations is approximately minus25deg to +40deg

Figure 47 Interference microscopy combined with multi-axis staging for microlens metrology

Rep Prog Phys 82 (2019) 056101

Review

25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

Rep Prog Phys 82 (2019) 056101

Review

26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

Review

27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

25

of these relational parameters are shown in figure 49 Table 1 provides expected performance results with the caution that the actual results depend on the specific lens geometry

63 Interferometric scatterometry

Semiconductor device fabrication is an essential industry for the modern world providing integrated circuits for commer-cial and consumer products Semiconductor wafers are created with billions of individual transistors comprised of surface features that are separated by less than 20 nm far below the resolving power of conventional visible-wavelength optical systems interferometric or otherwise Nonetheless optical methods provide critical non-contact inspection of wafers in production using principles of light scattering on test fea-tures specifically designed to facilitate production parameter controls

The topographical and optical properties of surfaces are literally reflected in the light scattered from the surface at dif-ferent angles polarizations and wavelengths Scatterometry is a technique for determining detailed material and dimensional surface structure information from these optical properties

Common applications include measurements of partially-transparent films and the measurement of specific parameters on test structures that may include feature widths and separa-tions that are far below the diffraction limit of conventional imaging [239] The technical challenge is to collect as much angle- polarization- and wavelength-resolved information as possible in a short length of time

An approach to rapidly collecting data over a wide range of angles is to image the pupil (or back focal plane) of a high-NA microscope objective onto a camera [240] High-NA pupil-plane imaging provides the angle-resolved optical properties of surfaces at a single focused measurement point [241 242] A white light source adds the extra dimension of wavelength using data collection and analysis techniques closely related to coherence scanning interferometry (CSI) [140 243 244] The technique is often referred to as pupil-plane CSI interference scatterometry or white light Fourier scatterometry [245 246]

The instrument configuration in figure 50 includes a field stop to reduce the view to a single 10-micron diameter spot on the object surface and a Bertrand lens to relay the pupil image to the camera The addition of a polarizer enables reflectivity analysis as a function of polarization and angle of incidence

Figure 48 Form deviation map showing a manufacturing error (flat area around the apex) The lens diameter is 24 mm and the height scale in this image is 15 microm

Figure 49 Relational metrology features critical to lens function and alignment

Table 1 Performance specifications of the interferometric microlens metrology system

Specification 1σ reproducibility

Aspheric Surface Form 0025 microm RMSSurface Roughness (Sa) 0005 micromLens Height 017 micromTilt Control Interlock Flatness 002 micromApex Concentricity 025 micromLens Concentricity to Interlock 016 micromInterlock Diameter 005 micromThickness at Center 034 micromTilt Control Interlock Parallelism 10 arcsecTilt Control Interlock Thickness 024 microm

Figure 50 CSI microscope geometry with pupil-plane imaging for interferometric scatterometry

Rep Prog Phys 82 (2019) 056101

Review

26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

Review

27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

26

Figure 52 Results of a visible-wavelength interferometric scatterometry measurement of a semiconductor surface featureData acquisition involves an axial scan of the interference

objective and Fourier analysis separates the wavelengths A model-based interpretation of the multiple-wavelength multi-ple-angle ellipsometric data provides the desired information about surface structure

Figure 51 shows plots of the standard ellipsometric param-eters Δ and Ψ measured as a function of wavelength and angle of incidence for a two-layer film stack of 56 nm Diamond Like Carbon (DLC) on 5 nm alumina over a metal alloy Markers represent experimental data points computed from a single measurement Lines correspond to an optimized model of the film stack Each line maps to a different angle of incidence covering a 20deg to 46deg range

Scatterometry for optically-unresolved structures relies on test gratings created on semiconductor wafers during manufacture [239] These regular structures serve as witness samples for a range of parameter variations including shape height width and the influence of transparent films The sig-nal modeling requires tools such as rigorous coupled wave analysis (RCWA) to predict the scatterometry signals [247] RCWA is an established calculation method for scatterom-etry of grating structures in semiconductor wafer process metrology and has been applied to interference microscopy to better understand the imaging properties of these tools [245 248 249]

Figure 52 shows a cross section of a periodic 160 nm pitch structure that was measured using interferometric scatter-ometry at a mean wavelength of 500 nm Seven parameters define the structure shape including film thickness side-wall angle (SWA in the figure) lateral width or critical dimen-sion (CD) and height Comparison with theoretical modeling using four wavelengths four angles of incidence between 30deg and 50deg and the complete 0 to 2π azimuth range available to the pupil-plane geometry provides simultaneous measure-ment of all of these sub-wavelength feature parameters in a few seconds The cross section determined experimentally by interferometric scatterometry shown to the left in figure 52 is in good agreement with the scanning-electron microscope (SEM) cross section image shown on the right The three sigma reproducibility for the seven parameters over 3 d range between 025 nm and 075 nm for dimensions and 025deg and 035deg for angles [250]

7 Summary and future work

We began this review with measurements of distance and displacement encompassing free-space stage monitoring systems methods for determining distance from fixed refer-ence points and fiber sensors for position measurement We then considered the testing of optical components for form and wavefront which extends from the earliest application of areal interference pattern evaluation for practical measure-ments the most recent developments at high lateral resolution using temporal and spatial phase shifting techniques

We next considered the evolution of interference micros-copy which today relies heavily on concepts and techniques related to optical coherence Designers of interference microscopes have learned to overcome limitations related to the sensitivity of wave optics to surface textures unresolved features and transparent film structures The theme of indus-trial or technical surface form measurements encompasses grazing incidence and infrared techniques for measuring machined ground and structured parts for applications ranging from engine component metrology to measure-ments of flatness thickness and parallelism on non-optical surfaces

The final applications area reviewed here is a significant driver for high-precision metrology semiconductor wafer processing and consumer electronics These markets have demanding requirements that challenge the full arsenal of interferometric metrology solutions from interferometric ellipsometry to scatterometry of optically-unresolved surface structures

The advertised benefits of optical interferometry such as precision speed data density and non-contact evaluation will continue to be significant factors in the development of new dimensional metrology equipment The demand for these solutions is increasing but so are the possibilities with new enabling technologies that include femtosecond lasers addi-tively-manufactured optical components and metamaterials These developments combined with the inherent advantages of wave optics will continue to generate interesting and pro-ductive developments in interferometric metrology

Figure 51 Plot of the ellipsometric parameters Delta and Psi measured as a function of wavelength and angle of incidence using the pupil-plane interferometer geometry shown in figure 50 The points are data and the lines are theoretical curves

Rep Prog Phys 82 (2019) 056101

Review

27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

27

Acknowledgments

A review paper necessarily involves the reporting of work per-formed by other researchers In addition to the many historical and open literature references I am indebted to the contrib-utions of many coworkers throughout my career in optical metrology Special thanks go to the Zygo Innovations Group and to the many skilled engineers and scientists at Zygo Cor-poration I also greatly appreciate the inspirational friends and colleagues in the larger applied optics community espe-cially those who share a passion for optics interferometry and instrument invention

ORCID iDs

Peter J de Groot httpsorcidorg0000-0002-6984-5891

References

[1] Evans C J 1989 Precision Engineering an Evolutionary View (Cranfield University Press) p 197

[2] Palmer C I 1919 Library of Practical Electricity Practical Mathematics For Home Study (New York McGraw-Hill) p 90

[3] Alder K and Evans J 2003 The Measure of All Things the Seven-Year Odyssey and Hidden Error that Transformed the World vol 71

[4] Nelson R A 1981 Foundations of the international system of units (SI) Phys Teach 19 596ndash613

[5] Hume K J 1950 Engineering Metrology (London MacDonald amp Co)

[6] Hausner M 2017 Optics Inspections and Tests a Guide for Optics Inspectors and Designers (Bellingham WA SPIE Optical Engineering Press) vol PM269 p 520

[7] de Groot P 2004 Optical metrology The Optics Encyclopedia ed T G Brown et al (New York Wiley) pp 2085ndash117

[8] Wood R W 1911 Physical Optics (New York MacMillan) pp 166ndash7

[9] Newton I 1704 Opticks or a Treatise of the Reflections Refractions Inflections and Colours of Light 4th edn ed S Smith and B Walford (Sam Smith and Benj Walford London)

[10] Fizeau H 1864 Recherches sur la dilatation et la double reacutefraction du cristal de roche eacutechauffeacute Ann Chim Phys 4 143

[11] Mantravadi M V and Malacara D 2007 Newton Fizeau and Haidinger interferometers Optical Shop Testing ed D Malacara (New York Wiley) pp 361ndash94

[12] Michelson A A and Morley E W 1887 On the relative motion of the earth and the luminiferous ether Am J Sci 34 333ndash45

[13] Michelson A A Pease F G and Pearson F 1929 Repetiion of the Michelson Morley experiment J Opt Soc Am 18 181

[14] Sigg D 2016 The advanced LIGO detectors in the era of first discoveries Proc SPIE 9960 9

[15] Harrison G R and Stroke G W 1955 Interferometric control of grating ruling with continuous carriage advance J Opt Soc Am 45 112ndash21

[16] Dukes J N and Gordon G B 1970 A two-hundred-foot yardstick with graduations every microinch Hewlett-Packard J 21 2ndash8

[17] Krohn D A MacDougall T W and Mendez A 2015 Fiber Optic Sensors Fundamentals and Applications vol PM247 4th edn (Bellingham WA SPIE Optical Engineering Press)

[18] Badami V and Abruntildea E 2018 Absolutely small sensor big performance Mikroniek 58 5ndash9

[19] Osten W 2006 Optical Inspection of Microsystems (London Taylor and Francis)

[20] Sagnac G 1911 Strioscope et striographe interfeacuterentiels Forme interfeacuterentielle de la meacutethode optique des stries Le Radium 8 241ndash53

[21] Krug W Rienitz J and Schulz G 1964 Contributions to Interference Microscopy (London Hilger and Watts)

[22] Willliam J Hacker amp Co Inc 1964 The Watson Interference Objectives Product Brochure (UK Barnet)

[23] Felkel E 2013 Scanning White-Light Interferometry Fingerprints the Polishing Process Photonics Spectra

[24] Leach R K Giusca C L and Naoi K 2009 Development and characterization of a new instrument for the traceable measurement of areal surface texture Meas Sci Technol 20 125102

[25] Hariharan P 2003 Optical Interferometry (New York Academic)

[26] Badami V and de Groot P 2013 Displacement measuring interferometry Handbook of Optical Dimensional Metrology ed K G Harding (Boca Raton FL Taylor and Francis) ch 4 pp 157ndash238

[27] Downs M J and Raine K W 1979 An unmodulated bi-directional fringe-counting interferometer system for measuring displacement Precis Eng 1 85ndash8

[28] Dorsey A Hocken R J and Horowitz M 1983 A low cost laser interferometer system for machine tool applications Precis Eng 5 29ndash31

[29] de Groot P 1997 Homodyne interferometric receiver and calibration method having improved accuracy and functionality US Patent 5663793

[30] de Lang H and Bouwhuis G 1969 Displacement measurements with a laser interferometer Philips Tech Rev 30 160ndash5

[31] Sommargren G E 1987 A new laser measurement system for precision metrology Precis Eng 9 179ndash84

[32] Demarest F C 1998 High-resolution high-speed low data age uncertainty heterodyne displacement measuring interferometer electronics Meas Sci Technol 9 1024ndash30

[33] Evans C Holmes M Demarest F Newton D and Stein A 2005 Metrology and calibration of a long travel stage CIRP Ann 54 495ndash8

[34] Holmes M L Shull W A and Barkman M L 2015 High-powered stabilized heterodyne laser source for state-of-the-art multi-axis photolithography stage control Euspenrsquos 15th Int Conf amp Exhibition Proc EUSPEN pp 135ndash6

[35] Burgwald G M and Kruger W P 1970 An instant-on laser for length measurement Hewlett-Packard J pp 14ndash6

[36] Zygo Corporation 2009 ZMI 7705 laser head Specification Sheet

[37] Sommargren G E 1987 Apparatus to transform a single frequency linearly polarized laser beam into a beam with two orthogonally polarized frequencies US Patent 4684828

[38] Hill H A 2000 Apparatus for generating linearly-orthogonally polarized light beams US Patent 6157660

[39] Hill H A and de Groot P 2002 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam US Patent 6452682

[40] Hill H A and de Groot P 2001 Apparatus to transform two nonparallel propagating optical beam components into two orthogonally polarized beam components US Patent 6236507

[41] Zanoni C 1989 Differential interferometer arrangements for distance and angle measurements principles advantages and applications VDI-Berichte 749 93ndash106

[42] Bennett S J 1972 A double-passed Michelson interferometer Opt Commun 4 428ndash30

Rep Prog Phys 82 (2019) 056101

Review

28

[43] Zygo Corporation 2009 ZMI High Stability Plane Mirror Interferometer (HSPMI)

[44] Yoda Y Hayakawa A Ishiyama S Ohmura Y Fujimoto I Hirayama T Shiba Y Masaki K and Shibazaki Y 2016 Next-generation immersion scanner optimizing on-product performance for 7 nm node Proc SPIE 9780 978012

[45] Schmidt R-H M 2012 Ultra-precision engineering in lithographic exposure equipment for the semiconductor industry Phil Trans R Soc A 370 3950ndash72

[46] Badami V G and Patterson S R 1998 A method for the measurement of nonlinearity in heterodyne interferometry American Society for Precision Engineering Annual Conf

[47] Hill H A 2000 Systems and methods for characterizing and correcting cyclic errors in distance measuring and dispersion interferometry US Patent 6137574

[48] Thijsse J Jamting Aring K Brown N and Haitjema H 2005 The detection of cyclic nonlinearities in a ZMI2000 heterodyne interferometer Proc SPIE 5879 1ndash11

[49] de Groot P J and Badami V G 2014 Revelations in the art of fringe counting the state of the art in distance measuring interferometry 7th Int Workshop on Advanced Optical Imaging and Metrology Proc FRINGE (Berlin Springer) pp 785ndash90

[50] Minoshima K Nakajima Y and Wu G 2015 Ultra-precision optical metrology using highly controlled fiber-based frequency combs Proc SPIE 9525 952502

[51] Hill H A and de Groot P 2001 Single-pass and multi-pass interferometery systems having a dynamic beam-steering assembly for measuring distance angle and dispersion US Patent 6313918

[52] Hercher M and Wijntjes G J 1991 Interferometric measurement of in-plane motion Proc SPIE 1332 602ndash12

[53] Shibazaki Y Kohno H and Hamatani M 2009 An innovative platform for high-throughput high-accuracy lithography using a single wafer stage Proc SPIE 7274 72741I

[54] Deck L de Groot P and Schroeder M 2014 Interferometric encoder systems US Patent 8670127

[55] Deck L L de Groot P J and Schroeder M 2010 Interferometric encoder systems US Patent 8300233

[56] de Groot P Badami V G and Liesener J 2016 Concepts and geometries for the next generation of precision heterodyne optical encoders Proc ASPE Annual Meeting vol 65 pp 146ndash9

[57] Wang J L Zhang M Zhu Y Ye W N Ding S Q Jia Z and Xia Y 2017 The next generation heterodyne interferometric grating encoder system for multi-dimensional displacement measurement of a wafer stage 32nd Annual Meeting of the American Society for Precision Engineering Proc ASPE Annual Meeting (ASPE)

[58] de Groot P and Liesener J 2015 Double pass interferometric encoder system US Patent 9025161

[59] de Groot P and Schroeder M 2014 Interferometric heterodyne optical encoder system US Patent 8885172

[60] Bosch T and Lescure M 1995 Selected Papers on Laser Distance Measurement (SPIE Milestone Series MS vol 115) (Bellingham WA SPIE Optical Engineering Press)

[61] de Groot P 2001 Unusual techniques for absolute distance measurement Opt Eng 40 28ndash32

[62] Benoicirct R 1898 Application des pheacutenomegravenes drsquointerfeacuterence a des deacuteterminations meacutetrologique J Phys Theacuteor Appl 7 57ndash68

[63] Poole S P and Dowell J H 1960 Application of interferometry to the routine measurement of block gauges Optics and Metrology ed P Mollet (New York Pergamon)

[64] de Groot P and Kishner S 1991 Synthetic wavelength stabilization for two-color laser-diode interferometry Appl Opt 30 4026

[65] Bechstein K-H and Fuchs W 1998 Absolute interferometric distance measurements applying a variable synthetic wavelength J Opt 29 179ndash82

[66] de Groot P J and McGarvey J A 1993 Laser gage using chirped synthetic wavelength interferometry Proc SPIE 1821 110ndash8

[67] de Groot P 1991 Interferometric laser profilometer for rough surfaces Opt Lett 16 357

[68] Kikuta H Iwata K and Nagata R 1986 Distance measurement by the wavelength shift of laser diode light Appl Opt 25 2976ndash80

[69] den Boef A J 1987 Interferometric laser rangefinder using a frequency modulated diode laser Appl Opt 26 4545ndash50

[70] Barber Z W Babbitt W R Kaylor B Reibel R R and Roos P A 2010 Accuracy of active chirp linearization for broadband frequency modulated continuous wave ladar Appl Opt 49 213ndash9

[71] Lang R and Kobayashi K 1980 External optical feedback effects on semiconductor injection laser properties IEEE J Quantum Electron 16 347ndash55

[72] Rudd M J 1968 A laser Doppler velocimeter employing the laser as a mixer-oscillator J Phys E Sci Instrum 1 723ndash6

[73] Li J Niu H and Niu Y 2017 Laser feedback interferometry and applications a review Opt Eng 56 050901

[74] de Groot P J Gallatin G M and Macomber S H 1988 Ranging and velocimetry signal generation in a backscatter-modulated laser diode Appl Opt 27 4475ndash80

[75] de Groot P J and Gallatin G M 1989 Backscatter-modulation velocimetry with an external-cavity laser diode Opt Lett 14 165ndash7

[76] Bosch T Servagent N I and Donati S 2001 Optical feedback interferometry for sensing application Opt Eng 40 20ndash7

[77] Zygo Corporation 2010 ZMItrade 77227724 Laser Operating manual

[78] Bell J A Capron B A Pond C R Breidenbach T S and Leep D A 2003 Fiber coupled interferometric displacement sensor Patent EP 0 793 079

[79] Kersey A D 1991 Interferometric optical fiber sensors for absolute measurement of displacement and strain Proc SPIE 1511 40ndash50

[80] de Groot P J Colonna de Lega X Dawson J W MacDougall T W Lu M and Troll J R 2002 Interferometer design for writing Bragg gratings in optical fibers Proc SPIE 4777 31ndash8

[81] de Groot P Deck L L and Zanoni C 2009 Interferometer system for monitoring an object US Patent 7636166

[82] Abruntildea E Badami V G Huang L and Idir M 2018 Real-time feedback for x-ray adaptive optics with an interferometric absolute distance sensor array Proc SPIE 10761 7

[83] Twyman F 1918 VI Interferometers for the experimental study of optical systems from the point of view of the wave theory Phil Mag 35 49ndash58

[84] Porter R W 1970 Mirror making for reflecting telescopes Amateur Telescope Making vol 1 ed A G Ingalls (New York Scientific American)

[85] Foucault L 1858 Description des proceacutedeacutes employeacutes pour reconnaicirctre la configuration des surfaces optiques C R Hebd Seacuteances Acad Sci 47 958ndash9

[86] Houston J B Buccini C J and OrsquoNeill P K 1967 A laser unequal path interferometer for the optical shop Appl Opt 6 1237ndash42

[87] Forman P F 1979 Interferometric examination of lenses mirrors and optical systems Proc SPIE 0163 103ndash11

[88] Wyant J C and Bennett V P 1972 Using computer generated holograms to test aspheric wavefronts Appl Opt 11 2833ndash9

[89] Schwider J and Burov R 1976 Testing of aspherics by means of rotational-symmetric synthetic holograms Opt Appl 6 83ndash8

[90] Evans C J and Kestner R N 1996 Test optics error removal Appl Opt 35 1015ndash21

[91] Zanoni C 2010 Zygo Corp measures 40 years Opt Photonics News 21 14ndash5

Rep Prog Phys 82 (2019) 056101

Review

29

[92] Bruning J H Herriott D R Gallagher J E Rosenfeld D P White A D and Brangaccio D J 1974 Digital wavefront measuring interferometer for testing optical surfaces and lenses Appl Opt 13 2693

[93] Creath K 1988 Phase-measurement interferometry techniques Progress in Optics vol 26 ed E Wolf (New York Elsevier) ch 5 pp 349ndash93

[94] Freischlad K and Koliopoulos C L 1990 Fourier description of digital phase-measuring interferometry J Opt Soc Am A 7 542ndash51

[95] de Groot P 1995 Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window Appl Opt 34 4723ndash30

[96] de Groot P J 2014 Correlated errors in phase-shifting laser Fizeau interferometry Appl Opt 53 4334ndash42

[97] Surrel Y 1996 Design of algorithms for phase measurements by the use of phase stepping Appl Opt 35 51ndash60

[98] Schreiber H and Bruning J H 2006 Phase shifting interferometry Optical Shop Testing ed D Malacara (New York Wiley) pp 547ndash666

[99] Sasaki O Okazaki H and Sakai M 1987 Sinusoidal phase modulating interferometer using the integrating-bucket method Appl Opt 26 1089ndash93

[100] Dubois A 2001 Phase-map measurements by interferometry with sinusoidal phase modulation and four integrating buckets J Opt Soc Am A 18 1972

[101] de Groot P 2009 Design of error-compensating algorithms for sinusoidal phase shifting interferometry Appl Opt 48 6788

[102] de Groot P 2011 Error compensation in phase shifting interferometry US Patent 7948637

[103] Fay M F Colonna de Lega X and de Groot P 2014 Measuring high-slope and super-smooth optics with high-dynamic-range coherence scanning interferometry Optical Fabrication and Testing (OFampT) Classical Optics 2014 OSA Technical Digest paper OW1B3

[104] Sommargren G E 1986 Interferometric wavefront measurement US Patent 4594003

[105] Deck L L and Soobitsky J A 1999 Phase-shifting via wavelength tuning in very large aperture interferometers Proc SPIE 3782 432ndash42

[106] de Groot P 2000 Measurement of transparent plates with wavelength-tuned phase-shifting interferometry Appl Opt 39 2658ndash63

[107] Deck L L 2003 Fourier-transform phase-shifting interferometry Appl Opt 42 2354ndash65

[108] Elmadih W Nefzi M and Buice E 2018 Environmental isolation Basics of Precision Engineering ed R K Leach and S T Smith (Boca Raton FL CRC Press) ch 13

[109] Schwider J Burow R Elssner K E Grzanna J Spolaczyk R and Merkel K 1983 Digital wave-front measuring interferometry some systematic error sources Appl Opt 22 3421

[110] Kinnstaetter K Lohmann A W Schwider J and Streibl N 1988 Accuracy of phase shifting interferometry Appl Opt 27 5082ndash9

[111] van Wingerden J Frankena H J and Smorenburg C 1991 Linear approximation for measurement errors in phase shifting interferometry Appl Opt 30 2718ndash29

[112] de Groot P J 1995 Vibration in phase-shifting interferometry J Opt Soc Am A 12 354ndash65

[113] de Groot P 2011 Sinusoidal phase shifting interferometry US Patent 7933025

[114] de Groot P 2006 Apparatus and method for mechanical phase shifting interferometry US Patent 7030995

[115] Deck L L 2010 Phase-shifting interferometry in the presence of vibration US Patent 7796273

[116] Deck L L 2010 Phase-shifting interferometry in the presence of vibration using phase bias US Patent 7796275

[117] Xu J Xu Q Chai L and Peng H 2008 Algorithm for multiple-beam Fizeau interferograms with arbitrary phase shifts Opt Express 16 18922ndash32

[118] Takeda M Ina H and Kobayashi S 1982 Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry J Opt Soc Am 72 156

[119] Sykora D M and Holmes M L 2011 Dynamic measurements using a Fizeau interferometer Proc SPIE 8082 80821R

[120] de Groot P 2016 Holography just a fancy word for interferometry OSA Imaging and Applied OpticsmdashDigital Holography amp 3D Imaging Proc OSA (OSA)

[121] Hariharan P 1996 Optical Holography (Cambridge Cambridge University Press)

[122] Ichioka Y and Inuiya M 1972 Direct phase detecting system Appl Opt 11 1507

[123] Sykora D M and de Groot P 2011 Instantaneous measurement Fizeau interferometer with high spatial resolution Proc SPIE 8126 812610

[124] He L Evans C J and Davies A 2013 Optical surface characterization with the area structure function CIRP Ann 62 539ndash42

[125] Malacara D 2007 Optical Shop Testing 3rd edn (New York Wiley)

[126] Sidick E 2009 Power spectral density specification and analysis of large optical surfaces Proc SPIE 7390 12

[127] Novak E Ai C and Wyant J C 1997 Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements Proc SPIE 3134 114ndash21

[128] Doerband B and Hetzler J 2005 Characterizing lateral resolution of interferometers the height transfer function (HTF) Proc SPIE 5878 587806

[129] Colonna de Lega X and de Groot P 2012 Lateral resolution and instrument transfer function as criteria for selecting surface metrology instruments Optical Fabrication and Testing OSA Proc Optical Fabrication and Testing OTu1D

[130] Takacs P Z Li M X Furenlid K and Church E L 1995 Step-height standard for surface-profiler calibration Proc SPIE 1995 235ndash44

[131] Glaschke T Deck L and de Groot P 2018 Characterizing the resolving power of laser Fizeau interferometers Proc SPIE 10829 1082905

[132] ISO 9334 2012 Optics and PhotonicsmdashOptical Transfer FunctionmdashDefinitions and Mathematical Relationships (International Organization for Standardization)

[133] Goodman J W 1996 Introduction to Fourier Optics 2nd edn (New York McGraw-Hill)

[134] Wang S 2017 Coherent phase transfer function degradation due to wave aberrations of a laser Fizeau interferometer Opt Eng 56 111711

[135] de Groot P and Colonna de Lega X 2006 Interpreting interferometric height measurements using the instrument transfer function in 5th Int Workshop on Advanced Optical Metrology Proc FRINGE (Berlin Springer) pp 30ndash7

[136] Lehmann P Xie W and Niehues J 2012 Transfer characteristics of rectangular phase gratings in interference microscopy Opt Lett 37 758ndash60

[137] Abbe E 1873 Beitraumlge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung Archiv fuumlr mikroskopische Anatomie 9 413ndash8

[138] de Groot P J 2015 Interference microscopy for surface structure analysis Handbook of Optical Metrology ed T Yoshizawa (Boca Raton FL CRC Press) ch 31 pp 791ndash828

[139] Schmit J Creath K and Wyant J C 2006 Surface profilers multiple wavelength and white light intereferometry

Rep Prog Phys 82 (2019) 056101

Review

30

Optical Shop Testing ed D Malacara (New York Wiley) pp 667ndash755

[140] de Groot P 2015 Principles of interference microscopy for the measurement of surface topography Adv Opt Photonics 7 1ndash65

[141] de Groot P J and Biegen J F 2016 Interference microscope objectives for wide-field areal surface topography measurements Opt Eng 55 074110

[142] Wyant J C Koliopoulos C L Bhushan B and Basila D 1986 Development of a three-dimensional noncontact digital optical profiler J Tribol 108 1

[143] de Groot P 2011 Phase shifting interferometry Optical Measurement of Surface Topography ed R Leach (Berlin Springer) ch 8 pp 167ndash86

[144] Dresel T Haumlusler G and Venzke H 1992 Three-dimensional sensing of rough surfaces by coherence radar Appl Opt 31 919ndash25

[145] Balasubramanian N 1982 Optical system for surface topography measurement US Patent 4340306

[146] Caber P J Martinek S J and Niemann R J 1993 New interferometric profiler for smooth and rough surfaces Proc SPIE 2088 195ndash203

[147] de Groot P Colonna de Lega X Kramer J and Turzhitsky M 2002 Determination of fringe order in white-light interference microscopy Appl Opt 41 4571ndash8

[148] Ai C and Caber P J 1995 Combination of white-light scanning and phase-shifting iterferometry for surface profile meaurements US Patent 5471303

[149] Chim S S C and Kino G S 1991 Phase measurements using the Mirau correlation microscope Appl Opt 30 2197

[150] Cohen D K Caber P J and Brophy C P 1992 Rough surface profiler and method US Patent 5133601

[151] de Groot P and Deck L 1995 Surface profiling by analysis of white-light interferograms in the spatial frequency domain J Mod Opt 42 389ndash401

[152] Evans C J and Bryan J B 1999 Structured lsquotexturedrsquo or lsquoengineeredrsquo surfaces CIRP Ann 48 541ndash56

[153] Bruzzone A A G Costa H L Lonardo P M and Lucca D A 2008 Advances in engineered surfaces for functional performance CIRP Ann 57 750ndash69

[154] Felkel E 2013 Scanning white-light interferometry fingerprints the polishing process Photonics Spectra 47 48ndash51

[155] Novak E Blewett N and Stout T 2007 Interference microscopes for tribology and corrosion quantification Proc SPIE 6616 66163B

[156] Blateyron F 2013 The areal field parameters Characterisation of Areal Surface Texture ed R Leach (Berlin Springer) ch 2 pp 15ndash43

[157] Thomas T R 2014 Roughness and function Surf Topogr Metrol Prop 2 014001

[158] Gomez C A Su R Thompson A and Leach R 2017 Optimisation of surface measurement for metal additive manufacturing using coherence scanning interferometry Opt Eng 56 111714

[159] DiSciacca J Gomez C Thompson A Lawes S D A Leach R Colonna de Lega X and de Groot P 2017 True-color 3D surface metrology for additive manufacturing using interference microscopy Proc EUSPEN Special Interest Group Meeting Additive Manufacturing

[160] Davidson M Kaufman K and Mazor I 1987 The coherence probe microscope Solid State Technol 30 57ndash9

[161] Sheppard C J R and Larkin K G 1995 Effect of numerical aperture on interference fringe spacing Appl Opt 34 4731

[162] Abdulhalim I 2001 Spectroscopic interference microscopy technique for measurement of layer parameters Meas Sci Technol 12 1996

[163] de Groot P and Colonna de Lega X 2004 Signal modeling for low-coherence height-scanning interference microscopy Appl Opt 43 4821

[164] Colonna de Lega X and de Groot P 2005 Optical topography measurement of patterned wafers Characterization and Metrology for ULSI Technology Proc ULSI Technology CP Proc ULSI vol 788 pp 432ndash6

[165] Fay M F and Dresel T 2016 Applications of model-based transparent surface films analysis using coherence scanning interferometry Proc SPIE 9960 996005

[166] Lehmann P Xie W Allendorf B and Tereschenko S 2018 Coherence scanning and phase imaging optical interference microscopy at the lateral resolution limit Opt Express 26 7376ndash89

[167] Xie W 2017 Transfer Characteristics of white light interferometers and confocal microscopes Dissertation University of Kassel

[168] Mandal R Palodhi K Coupland J Leach R and Mansfield D 2017 Application of linear systems theory to characterize coherence scanning interferometry Proc SPIE 8430 84300T

[169] Mandal R Coupland J Leach R and Mansfield D 2014 Coherence scanning interferometry measurement and correction of three-dimensional transfer and point-spread characteristics Appl Opt 53 1554ndash63

[170] Sheppard C J R Gu M Kawata Y and Kawata S 1994 Three-dimensional transfer functions for high-aperture systems J Opt Soc Am A 11 593ndash8

[171] Coupland J M and Lobera J 2008 Holography tomography and 3D microscopy as linear filtering operations Meas Sci Technol 19 074012

[172] Frieden B R 1967 Optical transfer of the three-dimensional object J Opt Soc Am 57 56ndash66

[173] Gu M 2000 Advanced Optical Imaging Theory (Springer Series in Optical Sciences vol XII) (Berlin Springer)

[174] Su R Thomas M Leach R and Coupland J 2018 Effects of defocus on the transfer function of coherence scanning interferometry Opt Lett 43 82ndash5

[175] Su R Wang Y Coupland J and Leach R 2017 On tilt and curvature dependent errors and the calibration of coherence scanning interferometry Opt Express 25 3297ndash310

[176] Leach R Giusca C Haitjema H Evans C and Jiang J 2015 Calibration and verification of areal surface texture measuring instruments CIRP Ann 64 797ndash813

[177] Haitjema H and Morel M A A 2005 Noise bias removal in profile measurements Measurement 38 21ndash9

[178] ISO 25178-604 2013 Geometrical Product Specification (GPS)mdashSurface Texture ArealmdashNominal Characteristics of Non-Contact (Coherence Scanning Interferometric Microscopy) Instruments International Standard (International Organization for Standardization)

[179] Zygo Corporation 2008 Measuring sub-Angstrom surface texture Application note AN-0002

[180] Giusca C L Leach R K Helary F Gutauskas T and Nimishakavi L 2012 Calibration of the scales of areal surface topography-measuring instruments part 1 Measurement noise and residual flatness Meas Sci Technol 23 035008

[181] Fleming A J 2013 A review of nanometer resolution position sensors Operation and performance Sensors Actuators A 190 106ndash26

[182] 2014 Understanding sensor resolution specifications and performance Lion Precision TechNote (LT05-0010)

[183] Zygo Corporation 2018 Nexview NX2 Specification sheet SS-0121

[184] Hosseini P Zhou R Kim Y-H Peres C Diaspro A Kuang C Yaqoob Z and So P T C 2016 Pushing phase and

Rep Prog Phys 82 (2019) 056101

Review

31

amplitude sensitivity limits in interferometric microscopy Opt Lett 41 1656ndash9

[185] Schmit J 2003 High-speed measurements using optical profiler Proc SPIE 5144 46ndash56

[186] de Groot P 2011 Coherence scanning interferometry Optical Measurement of Surface Topography ed R Leach (Berlin Springer) ch 9 pp 187ndash208

[187] Evans C J Troutman J Ganguly V and Schmitz T L 2013 Performance of a vibration desensitized scanning white light interferometer Int Conf on Metrology and Properties of Engineering Surfaces Paper TS1-08

[188] Badami V G Liesener J Evans C J and de Groot P 2011 Evaluation of the measurement performance of a coherence scanning microscope using roughness specimens ASPE Annual Meeting (Proc ASPE) pp 23ndash6

[189] Troutman J R Evans C J and Schmitz T L 2014 Vibration effects on an environmentally tolerant scanning white light interferometer Annual Meeting of the American Society for Precision Engineering Proc American Society for Precision Engineering (American Society for Precision Engineering) vol 59 pp 47ndash50

[190] Kujawinska M 1998 Applications of full-field optical methods in micro-mechanics and material engineering Microsyst Technol 5 81ndash9

[191] Abramson N 1969 The Interferoscope a new type of interferometer with variable fringe separation Optik 30 56ndash71

[192] Wilson I J 1983 Double-pass oblique-incidence interferometer for the inspection of nonoptical surfaces Appl Opt 22 1144ndash8

[193] Linnik V P 1941 An interferometer for the investigation of large plane surfaces Computus Rendus (Doklady) Acad Sci URSS vol 32

[194] Boland R J 2015 Grazing incidence stitching interferometry for manufacturing high precision flats Annual Meeting Proc ASPE vol 62 pp 383ndash8 (ASPE)

[195] Birch K G 1973 Oblique incidence interferometry applied to non-optical surfaces J Phys E Sci Instrum 6 1045

[196] Birch K G 1973 Grazing incidence interferometry applied to non-optical surfaces NPL Report MOM 4 1ndash42

[197] Hariharan P 1975 Improved oblique-incidence interferometer Proc SPIE 14 2

[198] Dresel T Schwider J Wehrhahn A and Babin S V 1995 Grazing incidence interferometry applied to the measurement of cylindrical surfaces Opt Eng 34 3531ndash5

[199] de Groot P 2000 Diffractive grazing-incidence interferometer Appl Opt 39 1527ndash30

[200] de Groot P Colonna de Lega X and Stephenson D 2000 Geometrically desensitized interferometry for shape measurement of flat surfaces and 3D structures Opt Eng 39 86

[201] Barus M 1910 The interference of the reflected diffracted and the diffracted reflected rays of a plane transparent grating and on an interferometer Science 31 394ndash5

[202] Jaerisch W and Makosch G 1973 Optical contour mapping of surfaces Appl Opt 12 1552ndash7

[203] Jacquot P M and Boone P M 1990 Two holographic methods for flatness testing with subwavelength or multiple-wavelength sensitivities Proc SPIE 1212 207ndash19

[204] Yeskov D N Koreshev S N and Seregin A G 1994 Holographic reference glass for surface planeness testing Proc SPIE 2340 10

[205] Chang B J Alferness R and Leith E N 1975 Space-invariant achromatic grating interferometers theory Appl Opt 14 1592ndash600

[206] Munnerlyn C R and Latta M 1968 Rough surface interferometry using a CO2 laser source Appl Opt 7 1858ndash9

[207] Kwon O Wyant J C and Hayslett C R 1980 Rough surface interferometry at 106 microm Appl Opt 19 1862ndash9

[208] Colonna de Lega X de Groot P and Grigg D 2002 Instrument de mesure de planeacuteiteacute drsquoeacutepaisseur et de paralleacutelisme sur piegraveces industrielles usineacutees 3rd Colloquium for Controcircles et Mesures Optiques pour lrsquoIndustrie Proceedings of the Socieacuteteacute Franccedilaise drsquoOptique (Socieacuteteacute Franccedilaise drsquoOptique) pp 1ndash6

[209] Olchewsky F 2017 Caracteacuterisation des eacutecoulements instationnaires 3D par tomographie holographique numeacuterique multidirectionnelle PhD Dissertation Universiteacute Lille

[210] Vest C M 1979 Holographic Interferometry (Wiley Series in Pure and Applied Optics) (New York Wiley)

[211] Ostrovsky Y I Butusov M M and Ostrovskaya G V 1980 Interferometry by Holography (Springer Series in Optical Sciences) (Berlin Springer)

[212] Osten W Faridian A Gao P Koumlrner K Naik D Pedrini G Singh A K Takeda M and Wilke M 2014 Recent advances in digital holography Appl Opt 53 G44ndash63

[213] Osten W and Ferraro P 2006 Digital holography and its application in MEMSMOEMS inspection Optical Inspection of Microsystems (Boca Raton FL CRC Press) pp 351ndash425

[214] Stetson K A and Brohinsky W R 1985 Electrooptic holography and its application to hologram interferometry Appl Opt 24 3631ndash7

[215] Leith E N 2007 Holography The Optics Encyclopedia (New York Wiley) pp 773ndash800

[216] Stetson K A 2015 50 years of holographic interferometry Proc SPIE 9442 11

[217] Oshida Y Iwata K Nagata R and Ueda M 1980 Visualization of ultrasonic wave fronts using holographic interferometry Appl Opt 19 222ndash7

[218] de Groot P J Dolan P J Jr and Smith C W 1985 Holographic imaging of ultrasonic standing waves in methanol liquid nitrogen and liquid helium-II Phys Lett A 112 445ndash7

[219] Boettcher U Li H de Callafon R A and Talke F E 2011 Dynamic flying height adjustment in hard disk drives through feedforward control IEEE Trans Magn 47 1823ndash9

[220] Harker J M Brede D W Pattison R E Santana G R and Taft L G 1981 A quarter century of disk file innovation IBM J Res Dev 25 677ndash90

[221] Stone W 1921 A proposed method for solving some problems in lubrication Commonw Eng 1921 115ndash22

[222] Fleischer J M and Lin C 1974 Infrared laser interferometer for measuring air-bearing separation IBM J Res Dev 18 529ndash33

[223] Best G L Horne D E Chiou A and Sussner H 1986 Precise optical measurement of slider dynamics IEEE Trans Magn 22 1017ndash9

[224] Ohkubo T and Kishigami J 1988 Accurate measurement of gas-lubricated slider bearing separation using visible laser interferometry J Tribol 110 148ndash55

[225] Lacey C Adams J A Ross E W and Cormier A J 1992 A new method for measuring flying height dynamically Proc DiskCon 1992 pp 27ndash42 (IDEMA)

[226] de Groot P Dergevorkian A Erickson T and Pavlat R 1998 Determining the optical constants of read-write sliders during flying-height testing Appl Opt 37 5116

[227] de Groot P 1998 Optical properties of alumina titanium carbide sliders used in rigid disk drives Appl Opt 37 6654

[228] de Groot P 1998 Birefringence in rapidly rotating glass disks J Opt Soc Am A 15 1202ndash11

[229] de Groot P J 2016 Challenges and solutions in the optical measurement of aspheres Ultra Precision Manufacturing of Aspheres and Freeforms OptoNet Workshop

Rep Prog Phys 82 (2019) 056101

Review

32

[230] Scott P 2002 Recent developments in the measurement of aspheric surfaces by contact stylus instrumentation Proc SPIE 4927 199ndash207

[231] Fan Y J 1998 Stitching interferometer for curved surfaces PhD Dissertation Technische Universiteit Eindhoven

[232] Thunen J G and Kwon O Y 1983 Full aperture testing with subaperture test optics Proc SPIE 0351 19ndash27

[233] Tricard M Forbes G and Murphy P 2005 Subaperture metrology technologies extend capabilities in optics manufacturing Proc SPIE 5965 59650B

[234] Tang S 1998 Stitching high-spatial-resolution microsurface measurements over large areas Proc SPIE 3479 43ndash9

[235] Shimizu K Yamada H and Ueyanagi K 1999 Method of and apparatus for measuring shape US Patent 5960379

[236] Colonna de Lega X Dresel T Liesener J Fay M Gilfoy N Delldonna K and de Groot P 2017 Optical form and relational metrology of aspheric micro optics Proc ASPE vol 67 pp 20ndash3

[237] Dresel T Liesener J and de Groot P J 2017 Measuring topograpy of aspheric and other non-flat surfaces US Patent 9798130

[238] Colonna de Lega X Dresel T Liesener J Fay M Gilfoy N Delldonna K and de Groot P 2017 Lessons learned from the optical metrology of molded aspheres for cell phone cameras 8th High Level Expert MeetingmdashAsphere Metrology (Physikalisch-Technische Bundesanstalt)

[239] Raymond C 2005 Overview of scatterometry applications in high volume silicon manufacturing AIP Conf Proc 788 394ndash402

[240] Pluta M 1993 Measuring techniques Advanced Light Microscopy vol 3 (New York Elsevier)

[241] Shatalin S V Juškaitis R Tan J B and Wilson T 1995 Reflection conoscopy and micro-ellipsometry of isotropic thin film structures J Microsc 179 241ndash52

[242] Feke G D Snow D P Grober R D de Groot P J and Deck L 1998 Interferometric back focal plane microellipsometry Appl Opt 37 1796ndash802

[243] Colonna de Lega X M and de Groot P 2009 Interferometer with multiple modes of operation for determining characteristics of an object surface US Patent 7616323

[244] Davidson M P 2006 Interferometric back focal plane scatterometry with Koehler illumination US Patent 7061623

[245] Ferreras Paz V Peterhaumlnsel S Frenner K and Osten W 2012 Solving the inverse grating problem by white light interference Fourier scatterometry Light 1 e36

[246] Colonna de Lega X 2012 Model-based optical metrology Optical Imaging and Metrology ed W Osten and N Reingand (New York Wiley) ch 13 pp 283ndash304

[247] Moharam M G and Gaylord T K 1982 Diffraction analysis of dielectric surface-relief gratings J Opt Soc Am 72 1385ndash92

[248] Totzeck M 2001 Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields Optik 112 399ndash406

[249] de Groot P Stoner R and Colonna de Lega X 2006 Profiling complex surface structures using scanning interferometry US Patent 7106454

[250] de Groot P Colonna de Lega X and Liesener J 2009 Model-based white light interference microscopy for metrology of transparent film stacks and optically-unresolved structures Proc FRINGE 6th International Workshop on Advanced Optical Metrology ed W Osten and M Kujawinska (Berlin Springer) pp 236ndash43

Rep Prog Phys 82 (2019) 056101