step height measurement using two-wavelength phase ...kcreath/...v26p2810_step_height.pdf · step...

Step height measurement using two-wavelengthphase-shifting interferometry

Katherine Creath

Two-wavelength phase-shifting interferometry is applied to an interference phase-measuring microscopeenabling the measurement of step features. The surface is effectively tested at a synthesized equivalentwavelength Xeq = ;\ab/I Xa - XbI by subtracting phase measurements made at visible wavelengths Xa and Xb.

The rms repeatability of the technique is X/1000 at the equivalent wavelength. To improve the precision ofthe data, the phase ambiguities in the single-wavelength data are removed using the equivalent wavelengthresults to determine fringe orders. When this correction is made, a measurement dynamic range (featureheight/rms repeatability) of 1O4 is obtainable. Results using this technique are shown for the measurement ofan optical waveguide and a deeply modulated grating.

1. Introduction

Phase-measurement techniques at a single wave-length have the built-in assumption that the wavefrontincident on the detector array does not change by morethan one-half of the measurement wavelength betweenadjacent detector elements. This is equivalent to re-quiring less than one-half of a fringe per detector spac-ing or one-quarter of the measurement wavelength insurface height change between adjacent pixels whentesting a surface in reflection. The problem in mea-suring step heights using an optical profiler is deter-mining fringe-order numbers when removing phaseambiguities because the phase cannot be sampled suf-ficiently across the discontinuity. The left side of Fig.1 shows a step whose height is greater than a quarter ofa wavelength. On the right is a possible fringe patternfor this step. Given this information, we do not knowhow to match the fringe-order numbers on the twosides of the discontinuity. Figure 2(a) shows a blackand white photograph of white light fringes for a stepwhere it is obvious that this step is more than a coupleof fringes high. As soon as a narrow bandpass filter isintroduced to define the measurement wavelength asshown in Fig. 2(b), the fringe orders become ambigu-ous, and the height of the step cannot be determined atthis wavelength.

The author is with Wyko Corporation, 1955 East Sixth Street,Tucson, Arizona 85719.

Received 6 March 1987.0003-6935/87/142810-07$02.00/0.© 1987 Optical Society of America.

To determine the step height in the example of Fig.1, the optical profiler measurement can be desensitizedby creating a beat wavelength from two measurementsat shorter wavelengths. This solution is shown in Fig.3. Measurements are made at Xa and Xb yielding fringeorders m and n. When these measurements are com-bined, the result is as if the surface had been measuredat a longer Xeq giving fringe order numbers p. Thistechnique is known as two-wavelength phase-shiftinginterferometry.1-5 Using this technique along withcorrecting ambiguities in the single-wavelength data,the dynamic range of an optical profiler can be extend-ed without sacrificing its high measurement precision.

The upper limit to the height ranges which can bemeasured is dictated mostly by the depth of focus ofthe objective being used. The ultimate height limitdepends on the temporal and spatial coherence of thesource. With the 1oX microscope objective used inthese examples, the measurement range is limited to+5 gm. Many kinds of sample easily fall within thisrange which are not measurable using a single wave-length. Applications of this technique include themeasurement of steep slopes, step heights, and roughsurfaces. This paper explains the two-wavelengthtechnique for measuring step heights and shows theresults of testing step features in an optical waveguideand a grating.

11. Theory

Phase-shifting interferometry is a well-known tech-nique which has been used with great success in opticalprofilers.6 ,7 For the calculations in this paper, an algo-rithm developed by Carr6 is used.8 This algorithmcalculates the phase independent of the amount ofphase shift so that the profiler does not need recalibra-

2810 APPLIED OPTICS / Vol. 26, No. 14 / 15 July 1987

(A)

T> Y14 f

1 X V

(B)

Fig. 1. (A) Step height too large to measure with a single wave-length X. (B) Fringes in region of step show possible ambiguity in

fringe order.

mm-l-lm m+1

Xb

eq l

P

m+5

n+4

Ip+1

Fig. 3. Synthesis of beat wavelength Aeq using shorter wavelengthsXa and Xb.

tion each time the wavelength is changed. The phaseinformation is obtained by shifting the phase of onebeam in the interferometer by a known amount andmeasuring the intensity in the interferogram for manydifferent phase shifts. The intensity for each dataframe is integrated over the time it takes to move areference mirror linearly through a 2a change in phase(usually near 90°) with a piezoelectric transducer(PZT). Four frames of intensity data are recorded inthis manner:

(a)

A(x,y) = I11 + y cos[o(x,y) - 3arII;

B(x,y) = It1 + y cos[O(x,y) - a;

C(x,y) = I1l + y cos[/(xy) + ]);

D(x,y) = I,11 + y cos[o(xy) + 3a]};

(1)

where Io is the average intensity, and y is the modula-tion of the interference term. The phase 0 is calculat-ed using

(b)

(c) (d)Fig. 2. Black and white photographs of (A) white light fringes of an optical waveguide, (B) same fringes with a narrowband filter in place, (C)

white light fringes of a grating, and (D) fringes of grating with narrowband filter in place.

15 July 1987 / Vol. 26, No. 14 / APPLIED OPTICS 2811

Otn, [(A-D) +(B -QC][3(B -C)- (A- Df)]l (2)= tank/1 (B + C) - (A + D) J

at each detected point in the interferogram. The cal-culation of the phase is independent of the actualamount the phase is shifted as long as a is linear andconstant.

Extension of the measurement range of phase-shift-ing interferometry over a larger dynamic range is ob-tained by subtracting measurements taken at two dif-ferent wavelengths3 :

¢,(XY) = 27rOPD(xy) = a(XY) - /b(XY), (3)\eq

where q5a and 4,b are the phases measured for Xa and Xb.The difference in the phases measured at the twowavelengths yields the phase associated with an equiv-alent wavelength Xeq. Results from a test using Eq. (3)are the same as if the surface were measured using anequivalent wavelength:

Aeq , 1\ (4)

OPD in Eq. (3) refers to the optical path differencebetween the wavefront reflected off the test surfaceand reference surface. In the case of the optical profil-er, the OPD is twice the difference between the testsurface and reference surface. Thus the surfaceheight is given by

1 F0eq(XY)XeqlH(xy) = - 9 * (5)

To remove 27r ambiguities from the equivalent wave-length data, the phase difference between two adjacentpixels of the equivalent phase must be less than 7r (eq/2 in OPD, Xeq/4 on the test surface). 27r phase ambigu-ities are smoothed after subtracting the two measuredphases by comparing adjacent pixels and adding orsubtracting multiples of 2 until the difference be-tween adjacent pixels is <7r. The wavefront's phasebetween adjacent pixels must not change by more than7r (one-half wave of OPD) at the equivalent wavelengthfor the 27r ambiguities to be handled correctly. Thesensitivity of the test can be varied by changing the twoilluminating wavelengths.

To measure the phase at an effective wavelength equsing two shorter wavelengths under computer con-trol, the following algorithm is used. First, the com-puter takes four frames of data at XBa while shifting thephase in the interferometer. Next, the phase shifter isreturned to its original position, the illumination isswitched from a to Xb, and four more data frames aretaken while shifting the phase the same as for Xa. Thephases qHa and b are calculated modulo 27r using Eq.(2) and then subtracted to yield 4,eq modulo 27r. Phaseambiguities are then removed using an integrationroutine, and surface height data are calculated usingEq. (5). This approach is very simple and easily im-plemented using existing phase-measurement opticalprofilers.

There is a reduction in signal-to-noise ratio (SNR)at the equivalent wavelength relative to the smaller

•7 AX7 ^4 AS

Fig. 4. Wavelength bandpass used for calculation in Eq. (6).

measurement wavelengths because the SNR is wave-length dependent.1 5 This scaling is referred to as anerror magnification and is proportional to the ratio ofthe two wavelengths. Thus a SNR of 1000 at 0.5 ,mbecomes a SNR of 50 at 10 ,um.

The equivalent wavelength data can be used to re-move ambiguities in one of the single-wavelengthphase measurements by determining the correct fringeorders.' As long as the noise in the equivalent wave-length phase is less than one-half of the visible mea-surement wavelength in OPD, the number of 27r termsto add to the single-wavelength measurement can bedetermined by direct point-by-point comparison withthe integrated equivalent-wavelength phase. A cor-rected phase is simply determined by comparing (Xeq/Xvis)4,eq to vis and finding an integral number of 27rterms to add to 4, vis, This correction extends the dy-namic range of the shorter visible wavelength and in-creases the SNR of the measurement by the ratio Xeq/Xvis over the equivalent wavelength measurement.

Since the optical profiler used in this work usesnarrowband illumination, the effects of the source'swavelength bandwidth need to be determined. As-suming a wavelength spread as shown in Fig. 4, theintensity in the interferogram can be written as

(6)I = Jv2 IO + cos 12A ) dX'

where AX is the wavelength bandwidth and X is theaverage wavelength. When AX2 << X2, evaluation ofthis integral yields

i rOPD ) c 27r PD I=Io 1+ysinc ,; )cost X, ) , (7)

where sinc(x) = sin(x)/x, and the coherence length b isdefined as 2/AX. The only effect of integrating overthe wavelength bandwidth is to reduce the modulationof the interference fringes by the sinc factor. Other-wise, the interference fringes are defined at the averagewavelength. If we assume a wavelength X of 600 nmand a wavelength bandwidth AX of 10 nm, the coher-ence length is 36 m, the modulation is reduced to 0.99for an OPD of 1 gm. For a bandwidth of 100 nm, themodulation is reduced to 0.88. A plot of fringe modu-lation (contrast) vs wavelength bandwidth for differ-ent OPD values is shown in Fig. 5.

Even though the wavelength bandwidth does notaffect the measurement, the average or effective wave-length of the source may cause errors because theequivalent wavelength is very dependent on the differ-


- - VW, Al

10.80.6 .- OD1~

Fringe 0.4 _0- OPD=1 gmContrast 0.2 .- OPD=5gm

O * se **iq1 U- OPD=10g±m..0.20 ~ f..iflf0 25 50 75 100Wavelength Bandwidth

(nm)

Fig. 5. Fringe contrast vs wavelength bandwidth for differentOPDs using sinc(7rOPD/6,) terms of Eq. (7) at 600 nm.

Output to computer

Image sensor

Light Field Aperture yepieceAsource stop

Spectralfilter

I PZT transducerMirau microscopVe . (microprocessor controlled)

obetie,~ Reference surface

Mirau interferometerL. . _Beamsplitter plate

Test surface

Fig. 6. Schematic of optical profiler using a Mirau interferometer.

ence between the two measurement wavelengths. Anerror of 1 nm in wavelength for sources at 600 and 630nm yields an equivalent wavelength of 12.6,gm with anerror of +0.42 ,tm. If the phase ambiguities in thesingle-wavelength data are corrected using the equiva-lent wavelength data as described above, errors due tothe difference in the wavelengths do not matter. Thusit is best to use the equivalent wavelength data only asa way to reconstruct the wavefront made at a singlewavelength.

Finally, to measure step heights accurately, the sam-ple must have the same material on both sides of thestep providing the same phase change on reflection. Ifdifferent materials are on opposite sides of the step,the height will not be correct unless the phase changeon reflection for both materials is taken into account.

III. Experiment

Figure 6 shows the optical layout for the opticalprofiler used in these experiments. It consists of aMirau interferometer attached to an optical micro-scope. A PZT is used to shift the reference surfacerelative to the test surface producing an intensity mod-ulation which is measured using a 1024-element Fair-child linear detector array. The measurement wave-length is determined by a narrowband interferencefilter placed in front of the white light source. Thefilters used in this paper had 10-nm bandwidths with

Table I. Equivalent Wavelengths Using 10 nm Bandpass InterferenceFilters

Xeq (tm) 0.6509 0.6316 0.6117 0.5774

0.6509 - 21.30 10.16 5.110.6316 21.30 - 19.41 6.730.6117 10.16 19.41 - 10.300.5774 5.11 6.73 10.30 -

(a)

(b)

Fig. 7. Scanning electron micrographs of 1-pim high step in opticalwaveguide.

center wavelengths at 650.9, 631.6, 611.7, and 577.4nm. The equivalent wavelengths possible using thisfilter set are shown in Table I. The chromatic aberra-tion of this system is found to be within the noise of themeasurement by scaling and subtracting surfaceheight measurements taken at different wavelengths.A 1OX objective is used for all measurements.

The first object tested is an optical waveguide con-sisting of a l-,gm high nickel stripe on a silicon sub-strate overcoated with a thin layer of gold. Scanningelectron micrographs of this feature are shown in Fig.7, and photographs of the fringes are shown in Figs.2(a) and (b). Figures 8(A) and (B) show measure-


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Fig. 8. Profile of optical waveguide at (A) 650.9 nm and (B) 577.4 nm where step is incorrect. (C) Profile combining measurements of (A) and(B) to obtain correct step (eq = 5.11 jm). (D) Repeatability of 5.11-jim measurement.

ments made at 650.9 and 577.4 nm. These profilesshow that the steps in the grating could not be deter-mined because they are >X/4. When the modulo-27rphase at 577.4 nm is subtracted from the modulo-27rphase at 650.9 nm, the result is the profile in Fig. 8(C),which is effectively measured at 5.11 gm. In this casethe step heights can be determined unambiguously.When the test is repeated on the same location of thegrating, the difference between two sets of data at 5.11Am shows an rms of 6 nm, which is X/800 [shown in Fig.8(D)].

Next a gold-overcoated grating with a modulationdepth of 1.3 gm is measured. Photographs of interfer-ence fringes taken through the microscope are shownin Figs. 2(c) and (d). Figure 9(A) shows the measure-ment made at 650.9 gim, which does not resemble asquare grating. When this measurement is combinedwith one made at 611.7 nm, the grating is correctlyrepresented at 10.16 m in Fig. 9(B). This measure-ment has an rms repeatability of X/1400 [see Fig. 9(C)].A measurement at an equivalent wavelength of 19.41/um (combining 631.6 and 611.7 nm) shown in Fig. 9(D)has the same step height as measured at 10.16 m but isnoisier because of the SNR scaling.

To improve the noise in the measurement, the pro-file at 10.16gAm is used to remove the ambiguities in the650.9-nm profile of Fig. 9(A). The corrected profile is

shown in Fig. 10(A). It is noticeably less noisy thenFig. 9(B) and shows the same step height. The differ-ence between two corrected measurements taken at650.9 nm shows an rms measurement repeatability of0.35 nm (or /230) shown in Fig. 10(B). This meansthe step height of 1.3gAm is measurable to a precision of0.35 nm yielding a dynamic range of 3700. Using theequivalent wavelength data to determine fringe orderincreases the measurement precision by the ratio of thewavelengths.

IV. Conclusions

Two-wavelength techniques are very valuable forthe testing of steep surfaces such as steps because ofthe variable measurement sensitivity attainable bychanging wavelengths. The technique'presented inthis paper is very straightforward and easy to imple-ment. The wavelength bandwidth of a source used inphase-measurement interferometry only affects thefringe modulation, whereas the difference between thetwo visible wavelengths may cause errors in determi-nation of the step height because the equivalent wave-length is very sensitive to this difference. To increasethe precision of the two-wavelength measurement andreduce errors due to uncertainties in the difference ofthe measurement wavelengths, the single-wavelengthphase data are corrected for 27r ambiguities using the


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Repeatability of 10.16-jim measurement. (D) Profile of grating at 19.41 jm.

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integrated two-wavelength phases to determine fringe Referencesorders. The measurement precision is X/1000 at the 1. Y.-Y. Cheng, and J. C. Wyant, "Two-Wavelength Phase Shifting

equivalent wavelength. Visible wavelength precision Interferometry," Appl. Opt. 23, 4539 (1984).

is attainable by using the equivalent wavelength mea- 2. Y.-Y. Cheng, and J. C. Wyant, "Multiple-Wavelength Phase-

surement to determine the fringe orders for the single- Shifting Interferometry," Appl. Opt. 24, 804 (1985).

wavelength data. With this technique, a dynamic 3. K. Creath, Y.-Y. Cheng, and J. C. Wyant, "Contouring Asphericrange of 104 is possible for the measurement of step Surfaces Using Two-Wavelength Phase-Shifting Interferome-

heights. try," Opt. Acta 32, 1455 (1985).


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4. K. Creath and J. C. Wyant, "Testing Aspheric Surfaces Using try," Appl. Opt. 24, 1489 (1985).Two-Wavelength Phase-Shifting Interferometry," J. Opt. Soc. 7. J. C. Wyant, and K. N. Prettyjohns, "Three-dimensional SurfaceAm. A 2(13), P58 (1985). Metrology Using a Computer-Controlled Noncontact Instru-

5. K. Creath, and J. C. Wyant, Direct Phase Measurement of ment," Proc. Soc. Photo-Opt. Instrum. Eng. 661, 292 (1986).Aspheric Surface Contours," Proc. Soc. Photo-Opt. Instrum.Eng. 645, 101 (1986). 8. P. Carr6, "Installation et utilisation du comparateur photoelec-

6. B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, "Measurement trique et interferential du Bureau International des Poids etof Surface Topography of Magnetic Tapes by Mirau Interferome- Mesures," Metrologia 2, 13 (1966).

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