Fourier transform method for measurement ofthin film thickness by speckle interferometry

Canan Karaaliog˜ luYani SkarlatosBogazici UniversityFizik BolumuBebek, Istanbul 80815Turkey

Abstract. The surface profile of an Al thin film and its thickness areobserved by electronic speckle pattern interferometry (ESPI). The Mich-elson interferometer is used as our basic interferometric system to obtaininterference fringes on a CCD camera. These interference fringes de-pend on the path differences due to the surface contours of the thin film.The interference fringes are analyzed with the fast Fourier transformmethod and a wrapped phase is obtained. An unwrapping procedure isused to obtain a continuous phase. Results on thickness measurementare presented. © 2003 Society of Photo-Optical Instrumentation Engineers.[DOI: 10.1117/1.1572498]

Subject terms: Fourier transforms; thin films; speckle interferometry.

Paper 020238 received Jun. 11, 2002; revised manuscript received Oct. 25,2002; accepted for publication Oct. 28, 2002.

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1 Introduction

Optical techniques are powerful for deformation analysContour measurement by interferometry is used widelydetermine the shape of a surface. Information aboutsurface of a static object can be obtained from interferefringes whose contour lines or fringes characterize theface on which they are formed. There are two basic meods for measurement of surface shape or height profileoptical contouring: phase shifting and the Fourier traform.

The fast Fourier transform~FFT! method has been useby many researchers, such as de Angelis et al. to meathe focal lengths of lenses1 and the index of refraction otransparent liquid materials,2 Spagnolo et al. to obtain surface topography of ancient stone artworks3,4 or to analyzemicroclimate variation on artworks,5 and Wu et al. to quan-tify immunoreactive terminal area apposed to nerve cel6

Thin film thickness is an important parameter in maapplications. In this paper, we obtain a surface topografor thin films by using the FFT method. This methodused for the first time in the literature to measure the thiness of a thin film. The fringe patterns were captured bCCD camera. The deformed surface information is encointo these fringes. Contour fringes were displayed onmonitor and then analyzed with the FFT constructimethod.

This speckle technique does not require the projectiona grating and often no preparation is required to be ablget interference patterns on the specimen surface. Thobviously a big advantage in studying thin film specimewhich are small, delicate, and difficult to handle. In caswhere no substrate surface is exposed, a portion of themust be removed by etching or some other means to cra step at the edge of the film. Furthermore, a reflectcoating must be applied in cases where the film and/orsubstrate are not sufficiently reflective. The speckle inferometry method used here has the advantage thatequipment required consists of only a laser, some sim

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optical components, a digital camera, and a computer.patterning or marking of the specimen is necessary.

2 Principle of the Method „Based on Ref. 7 …

If structured light consisting of parallel stripes, or fringeis projected onto a surface, then the surface acts as a pmodulator, with the amount of modulation at any point dpending on the height of the surface at that point. In variooptical measurements, we find a fringe pattern of the fo

f ~x,y!5a~x,y!1b~x,y!cos@2pu0x1f~x,y!#, ~1!

where the phasef(x,y) contains the desired informatioand a(x,y) and b(x,y) represent unwanted irradiancvariations arising from the nonuniform light reflection otransmission by a test object.8

The Fourier transform method for fringe pattern analyrequires an added high-spatial-carrier frequency. In tcase, the distinct and equally spaced sinusoidal frinserve as the carriers. If the spatial carrier frequency is prerly selected, the contour can be reproduced using theverse Fourier transform and phase unwrapping techniqThe method has the advantage of using only one interence pattern during processing. The resulting fringesanalyzed with automatic phase unwrapping, and the baground intensity variations and speckle noise canreduced.7

For carrier fringes generated parallel to they axis, thecarrier signal being 1-D and involving only thex compo-nent, can be described by

f ~x!5a~x!1b~x!cos@2pu0x1f~x!#. ~2!

For the purpose of Fourier fringe analysis, the inpfringe pattern can be conveniently written in the followinform:

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f ~x!5a~x!1c~x!exp~ i2pu0x!1c* ~x!exp~2 i2pu0x!,~3!

wherec(x)5(1/2)b(x)exp@if(x)# and the term containingthe asterisk denotes its complex conjugate. In this caseFourier transform off (x) would be

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exp~2 i2pux! f ~x!dx, ~4!

F~u!5A~u!1C~u2u0!1C* ~u1u0!. ~5!

Since the spatial variations ofa(x), b(x), andf(x) areslow compared with the spatial frequencyu0 , the Fourierspectra in Eq.~5! are separated by the carrier frequencyu0 .

The functions in Eq.~5! denote Fourier transforms of thcorresponding quantities in the spatial domain andu0 is thespatial frequency in thex direction. The functionA(u) rep-resents the contribution to the spectra from low-frequebackground illumination and it is centered around the zeorder termu50, C(u2u0) and C* (u1u0) are centeredaroundu5u0 and u52u0 , where u0 is the spatial fre-quency of the interferometric fringe pattern under evaltion. SpectraA(u) andC* (u1u0) are eliminated by bandpass filtering, thus isolating the functionC(u2u0), whichis subsequently translated byu0 toward the origin to giveC(u). The inverse Fourier transform ofC(u) with respectto u yields the equationc(x)5(1/2)b(x)exp@if(x)#. Fromthis equation, the phase can be calculated pointwise byverting the relation

tanf~x!5Im@c~x!#

Re@c~x!#, ~6!

where Im and Re denote the imaginary and real partsc(x), respectively.

The phase calculated by Equation~6! is wrapped be-tween2p and1p. Then, a final unwrapping procedurenecessary to reconstruct the continuous phase funcf(x). Subsequently, the height of the object point is etracted from the unwrapped phase data as

z~x!5l

4pf~x! ~7!

wherel is the wavelength of the light source.8–12

Phase unwrapping is carried out to convert the disctinuous phase to a continuous one by adding or subtrac2p to the wrapped phase. This removes the discontinuito yield a continuous phase map related to the measuremparameter.7 Phase values can be measured with a depth~z!resolution greater thanl/200 with the FFT method,3 whichcan accomplish fully automatic distinction between a dpression and an elevation of the object shape, requirefringe order assignments or fringe center determinatiand no interpolation between fringes as it gives height dtribution at all the picture elements over the object imag9

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3 Description of the Experimental Setup andResults Obtained

We evaporated Al as a thin film onto a glass substraThen, a second Al evaporation was performed througstainless steel mask approximately 0.2 mm from the sstrate to obtain a thin strip on the first Al layer. A schemadiagram of the Al thin strip is shown in Fig. 1. We peformed the experiment to measure the second strip’s thness. When we measured its thickness by a commercavailable interferometric thickness monitor~VarianAscope* Interferometer model no. 980-4006!, we foundthat it changed approximately from 284.7 to 261.2 nm, dpending on the exact location. Naturally this method wnot noncontact as the Fizeau plate of the thickness monpresses against the sample during measurement.

We used a He-Ne laser with a wavelengthl5632.8 nm as our light source. The distances betweenobject beamsplitter and the mirror beamsplitter are notportant in our experiment since the main point is justobtain the interference pattern caused by the Al strip. Tpictures were taken of the film surface of size 1.230.945 cm. We obtained reference and deformed statesliding the glass substrate, which was mounted to a micrlock. When we illuminated an area that included the partAl strip’s part, the resultant interference pattern was o‘‘deformed state.’’ When the area that did not include thestrip was illuminated, the resultant interference pattern wour ‘‘reference state.’’

We used a commercial digital still camera~Sony, CVX-V18NSP! to directly capture digital images of the speckfields. A schematic diagram of the experiment is shownFig. 2. Light from the laser is expanded by the beampander~BE! and is incident on the beamsplitter~BS!. Thetwo beams from the BS then fall on the object and tmirror. The speckle fields created by the object andmirror travel in opposite directions and interfere with eaother in the image plane of the digital still camera afpassing through the camera lens. The digital still camhas a 5763768-pixel charge-coupled device~CCD! arrayas its recording element. A neutral density filter~THOR-LABS Mounted Metallic ND filter, D50.4, 0.5! wasplaced in front of the camera to reduce the intensity ofincoming light, and thus enhance the images. The aperand shutter speed of the camera were selected approprito have the maximum light intensity without saturating tcamera. The exposure time of the camera was set at 1s. The image acquisition card converts the images dire

Fig. 1 Schematic of the Al thin film.

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Karaalioglu and Skarlatos: Fourier transform method . . .

into portable network graphics~PNG! image files, whichare in turn converted into JPEG images since the larequire much less memory space in the hard disk ofcomputer. Subsequently, a special software was useconvert the images from RGB color mode to gray-scmode. We thus have 5763768-pixel images, where eacpixel is represented by 1 byte of data~256 gray levels!. Onesuch image is shown in Fig. 3.

A median filter with a 333-pixel matrix was used toreduce electronic noise in the captured image, i.e., this fiwas applied to the interference patterns. This filter sorts9 pixels in each 333 neighboring region and replaces eacenter pixel from the source image by the median valueits eight neighbors. The effect is to reduce all pixels thatdarker or brighter than their neighbors, and thus remnoise. This is a linear filter operation in which no informtion is lost from the original image.

The optical system provides information regardichanges in the optical path length of the interferomesystem. These path length changes are related to defotions of the test object as fringes indicating contoursconstant displacement. The contour spacing is given by

Fig. 2 Schematic diagram of the experimental setup for FFTmethod: BS, beamsplitter; BE, beam expander; ND filter, neutraldensity filter; PC, personal computer.

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teger multiples of the laser wavelength used in the interometer.

The fringe patterns were digitized with an 8-bit grascale, corresponding to 256 levels. A median filter was uto reduce the noise in the pattern; then a 1-D FFT algoritwas applied to each row of 512 pixels to obtain a powspectrum of spatial frequencies. The data points were csen 512 pixels for one row instead of 768 pixels, whichthe capability of the camera, since the FFT requires theof one row to be a power of 2. Different spatial signafrom the original image were represented as different fquencies at various distances from the center of the pospectrum, with the concentration of the lowest frequenccloser to the center and the higher frequencies farther afrom it. When an inverse FFT with a filter procedure wemployed, the information in the frequency domain wtransformed back to the spatial domain. This operationstored the original image with high frequencies suppressmaking the image of the terminal area more promineFinally, we obtained four plots; wrapped and unwrappphase plots for reference and deformed states. While Fi

Fig. 3 Interference pattern obtained from the thin film used in ourexperiment.

Fig. 4 Wrapped and unwrapped phase plots for deformed surface (for the ‘‘deformed state’’).

Karaalioglu and Skarlatos: Fourier transform method . . .

Fig. 5 Wrapped and unwrapped phase plots for flat surface (for the ‘‘reference state’’).

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shows the plots from the deformed surface, Fig. 5 shothe plots from the flat surface.

The number of sawteeth in the wrapped phase plotequal to the number of fringes~i.e., the spatial frequency othe pattern! in our interference patterns.

Using Eq. ~7!, we found the thickness to be 26062 nm between the pixel values from 280 to 420. Usithe mentioned Varian thickness monitor on the same lotion, we found this value to be 261.263 nm.

4 Discussion and Conclusion

Our work represents a new application of a well-knowtechnique. We have measured the thickness of an Alfilm with a 2-nm accuracy using the FFT method. Tmethod, explained in Sec. 2, generally assumes a lincarrier phase function,8 which introduces errors since thcarrier fringes are in general not equally spaced andeven straight, due to unavoidable environmental distortias well as aberrations and imperfections of optielements.13,14 In practice, unwanted environmental distubances are expected to introduce deviations from lineain the linear carrier phase that one would ideally achie

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This nonlinearity causes a noise, especially in the begning and end parts of our difference phase plots. Therefwe can accept the points from pixel number 20 to pixnumber 500 as the thin film’s surface and ignore the edgThe noise also results in a loss of calibration, which caua shift in the elevation of the flat portion of the samplearound 4.5~;200! nm. Thus, thickness measurements mbe taken with reference to the flat surface.

Obtaining equally spaced fringes is an important main the case of FFT analysis. If they are not nearly equal,FFT approach does not work well. The fringe pattern donot need to be stabilized, i.e., the small unwanted vibratido not affect the result as far as only one captured frinpattern is enough for calculations.

Thex axis for the FFT is along a row of the digital arraThe angle between fringe lines and thex axis is irrelevantfor the calculations. Although our fringes are not exacparallel ~Fig. 3!, the surface topography can still be otained from them. Nonparallelism results from the undformed surface not being optically flat due to imperfectioon the substrate and the evaporated reflective coatingfact, the variation of the film contour can easily be seen

Fig. 6 Difference in the unwrapped phase plots, i.e., the surface profile for one row of the interferencepattern.

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Fig. 3. The intensity variations in any one row are suciently sinusoidal to calculate the FFT for that row.

In the FFT method, we could obtain the exact surfatopography of a thin film with 1-D calculations. Even if thfringes appear equispaced and parallel, the low distortiothem contains information about the object. The FFTproach works well if the noise and the useful signal aseparable in the interference spectrum, so that with filteroperation the noise can be eliminated. The differencetween phase values of interference patterns from theformed and flat sections of a sample give us the absoamount of deformation. The fringe pattern does not havebe stabilized unlike in the phase shifting method, sincepicture is enough to analyze the surface.

The site of the deformation is clearly visible in the righhand side plots of Figs. 4 and 5. The right-hand side pof these figures are the unwrapped phase plots, and tplots show the surface shapes. The shapes are plotted oincline since they are found by adding each adjacent phvalue to each other as a requirement of unwrapping produre ~see Fig. 6!. The reference flatness is not the mocrucial part of this method as long as we are lookingjust the difference between flat and deformed surfacefind the deformation amount. The flatness is importantobtain nearly equispaced interference fringes. However,critical point is the filtering process. If we had a calibratsample, we could correct the method for yielding morecurate results. In this case, we could obtain the thicknvalues with high accuracy. Moreover, the speckle methhas the important advantage of being noncontact; thusformation of the films during thickness measurementavoided.

When we used the interferometer to measure thefilm’s thickness, we found that it varied from 284.7261.2 nm. We could not measure thickness at a particpoint exactly by using this apparatus because of itslateral resolution. However, the FFT method enables msurement of thickness at any specified point on the fisince 1 pixel51.26 cm/768. Moreover, when we use ththickness monitor, every measurement deforms thefilm’s surface slightly, whereas no deformation is causedthe FFT method.

This new application is important in the area of thin filthickness measurements for the following reasons: Ieasy to perform and understand; all of the apparatus usethe experiment is standard and relatively inexpensive;nally, this measurement does not cause any extra defotion on a thin film’s surface since this method is absolutnoncontact. In the future, better results can be obtaiwith improved CCD cameras and new software. Largpowers of 2 can be included in the calculations, and larsurface topographies can be obtained with higher restion.

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AcknowledgmentsWe would like to thank Pietro Ferraro for his helpful sugestions.

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Canan Karaaliog˜ lu received her BS degree in physics and educa-tion from Marmara University, Istanbul, Turkey, in 1997, and her MSdegree in physics from Bogazici University, Istanbul, Turkey, in2001. Her studies focused on measurement of thin film thicknessusing different optical methods during study for her MS degree. In2001 she began her PhD program at Stevens Institute of Technol-ogy, Hoboken, New Jersey where she is studying Ti:sapphire lasersand tetrahertz spectroscopy. She has been a research assistantwith the Ultrafast Laser Spectroscopy and High-Speed Communica-tion Laboratory at Stevens Institute of Technology since June 2002.

Yani Skarlatos received his BS degree in electrical engineeringfrom Robert College in Istanbul, Turkey, and his PhD degree in en-gineering and applied science from Yale University in 1970 and1974, respectively. His professional career since then has alternatedbetween academia and industry. He is currently a professor of phys-ics with Bogazici University, Istanbul. His research interests includeoptics, the optoelectronic properties of amorphous materials, andchaotic behavior in polymers.