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156 J. Opt. Soc. Am./Vol. 72, No. 1/January 1982

Fourier-transform method of fringe-pattern analysis forcomputer-basedtopography and interferometryMitsuo Takeda, Hideki Ina* and Seiji Kobayashi

University of Electrocommunications, 1-5-1, Chofugaoka, Chofu, Tokyo, 182, JapanReceived May 7, 1981; revised manuscript received August 27, 1981

A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a non-contour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the objector wave-front form, which has not been possible by the fringe-contour-generation techniques. The method hasadvantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivi-ty. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

INTRODUCTIONIn various optical measurements, we find a fringe pattern ofthe formg(x,y) = a (x,y) + b(x,y) cos[27rfox 0(x,y)], (1)

where the phase 0(x, y) contains the desired information anda (x, y) and b x, y) represent unwanted irradiance variationsarising from the nonuniform light reflection or transmissionby a test object; in most cases a(x, y), b(x, y) and 0(x, y) varyslowly compared with the variation introduced by the spa-tial-carrier frequency fo.The conventional technique has been to extract the phaseinformation by generating a fringe-contour map of the phasedistribution. In interferometry, for which Eq. (1) representsthe interference fringes of tilted wave fronts, the tilt is set tozero to obtain a fringe pattern of the formgo(x, y) = a(x, y) + b(x, y) cos[o(x,y)], (2)

which gives a contour map of O(x, y) with a contour interval27r. In the case of moir6 topography,l for which Eq. (1) rep-resents a deformed grating image formed on an object surface,another grating of the same spatial frequency is superposedto generate a moire pattern that has almost the same form asEq. (2) except that it involves other high-frequency erms thatare averaged out in observation. Although these techniquesprovide us with a direct means to display a contour map of thedistribution of the quantity to be measured, they have fol-lowing drawbacks: (1) The sign of the phase cannot be de-termined, so that one cannot distinguish between depressionand elevation from a given contour map. (2) The sensitivityis fixed at 2w7recause phase variations of less than 2 7rcreateno contour fringes. (3) Accuracy is limited by the unwantedvariations a (x, y) and b(x, y), particularly in the case ofbroad-contour fringes. Fringe-scanning techniques2 havebeen proposed to solve these problems, but they requiremoving components, such as a moving mirror mounted on atranslator, which must be driven with great precision andstability.We propose a new technique that can solve all these prob-lems by a simple Fourier-spectrum analysis of a noncontourtype of fringe pattern, as given in Eq. (1).

PRINCIPLE AND OPERATIONFirst, a noncontour type of fringe pattern of the form givenin Eq. (1) is put into a computer by an image-sensing devicethat has enough resolution to satisfy the sampling-theoryrequirement, particularly in the x direction. The input fringepattern is rewritten in the following form for convenience ofexplanation:g(x, y) = a(x, y) + c(x, y) exp(27rifox)

+ c*(x,y) exp(-27rifox), (3)with

c(x, y) = ('/2 )b(x, y) exp[i 0(x, y)], (4)where * denotes a complex conjugate.Next, Eq. (3) is Fourier transformed with respect to x bythe use of a fast-Fourier-transform (FFT) algorithm, whichgives

G(f, y) = A(f, y) + C(f-fo, y) + C*(f + fo, y), (5)where the capital letters denote the Fourier spectra and f isthe spatial frequency in the x direction. Since the spatialvariations of a(x, y), b(x, y), and 1(x, y) are slow comparedwith the spatial frequency fo, the Fourier spectra in Eq. (5) areseparated by the carrier frequency fo, as is shown schemati-cally in Fig. 1(A). 3 We make use of either of the two spectraon the carrier, say C(f - fo, y), and translate it by fo on thefrequency axis toward the origin to obtain C(f, y), as is shownin Fig. 1(B). Note that the unwanted background variationa(x, y) has been filtered out in this stage. Again using theFFT algorithm, we compute the inverse Fourier transform ofC(f, y) with respect to f and obtain c (x, y), defined by Eq. (4).Then we calculate a complex logarithm of Eq. (4):

log[c(x, y)] = log[('/ 2)b(x, y)] + ip(x, y). (6)Now we have the phase q(x, y) in the imaginary part com-pletely separated from the unwanted amplitude variation b(x,y) in the real part. The phase so obtained is indeterminateto a factor of 2w. In most cases, a computer-generated func-tion subroutine gives a principal value ranging from -7r to 7r,as, for example, is shown in Fig. 2(A). These discontinuitiescan be corrected by the following algorithm. We determine

0030-3941/82/010156-05$01.00 1981 Optical Society of America

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(A)

C(f.Y)y

(B)Fig. 1. (A) Separated Fourier spectra of a noncontour type of fringepattern; (B) single spectrum selected and translated to the origin.The y axis is normal to the figure.

W(xy)

-Tr_ _'_ _ X (A)

4Tr - - -

Y4W - _ __ _

O7n XjX ____y

Fig. 2. (A) Example of a phase distribution having discontinuitiesthat are due to the principal-value calculation; (B) offset phase dis-tribution for correcting the discontinuities in (A); (C) continuedprofile of the phase distribution. The y axis is normal to thefigure.an offset phase distribution 0O x, y) that should be added tothe discontinuous phase distribution OdX,A) to convert it toa continuous distribution Xc x, y):

Xc (X, Y) = 10d X, Y) + to (X, Y). (7)The first step in makingthis determination is to compute thephase difference

5 d (xi, A) = Od(Xi, - -d (Xi--1, Abetween the ith sample phasditributin hreceding it, withthe suffix running from 1 to N to coverall the samplepoints.Since the variation of the phase is slow compared with thesamplinginterval, the absolute value of the phase differenceI ofd Xi, Y) ds much less than 2 ,at points whero addphasedistribution is continuous. But it becomes almost 27r at

points where the 27rphase jump occurs. Hence, by setting anappropriate criterion for the absolute phase difference, say0.9 X 27r, we can specify all the points at which the 2w7rhasejump takes place and also the direction of each phase jump,positive or negative, which is defined as corresponding to thesign of And (Xi, y). The second step is to determine the offsetphase at each sample point sequentially, starting from thepoint x0 = 0. Since only a relative phase distribution needsto be determined, we initially set kox(xo, y) = 0.4 Then weset f0 x(xi, y) = 0 x (x 0, y) for i = 1, 2, 3_ . , k-1 until thefirst phase jump is detected at the kth sample point. If thedirection of the phase jump is positive (as marked by t in thefigure), we set Oox(Xk, y) = kox(Xk-1, y)- 2r, and if it isnegative (as marked by D),e set 0ox(Xk, y) = O(Xk l, y) +2r. Again,westarttosetb/ox(xi,y) = XOx(xk,y) ori = k +1,i =k + 2_ , i = m - 1, until the next phase jump occursat the mth sample point, where we perform the same 27rad-dition or subtraction as at the kth sample point, with k nowbeing replaced with m. Repeating this procedure of 27rphaseaddition or subtraction at the points of phase jump, we candetermine the offset phase distribution, as shown in Fig. 2(B),the addition of which to kd (X, y) gives a continuous phasedistribution k,(x, y), as is shown in Fig. 2(C). In the case ofmeasurement over a full two-dimensional plane, a furtherphase-continuation operation in the y direction is necessarybecause we initially set qOx(x0 ,y) =0 for all y without respectto the phase distribution in the y direction. It is sufficientto determine an additional offset phase distribution in the ydirection, kOY(x,y), on only one line along the y axis, say, onthe linethrough the point x = XL, L beingarbitrary. This canbe done by the same procedure as was described for the x di-rection, with the initial value now being set at XoY(XL, yo) =0. The two-dimensional offset phase distribution is thengiven by

'o (x,y) = 0o x(X,y) - qOoX(xL,y)+ 4o (xL,y). (8)In Eq. (8),(.X(x, y) -XOX(XL, y) represents the differenceofthe offset phase between the points (x, y) and (XL, y), andboY(XL, y) that between points (XL, y) and (XL, yO), so thatfO x, y) givesa relative offset phase distribution defined asthe difference from the initial value at (XL, Yo)-

EXPERIMENTSThe validity of the proposed method was examined in ex-periments using a Michelson interferometer with a He-Nelaser source. Figure 3 shows5 an example of the interferencefringes of tilted wave fronts put into a microcomputer (DigitalLSI-11/2) by a solid-state line-image sensor (ReticonRL1024H) with 1024 photodiode elements separated from oneanother by 15 Arm;a sensor of this type is particularly suitedfor our purpose because it has high resolution and accuracyin the position of elements and a number of elements that isconvenient for an FFT algorithm. Note the presence in Fig.3 of unwanted irradiance variations corresponding to a (x, y)and b(x, y) in Eq. (1).

Before computingby the FFT algorithm, we weighted thedata by the multiplication of a hanning window, as is shownin Fig. 4, to eliminate the influence of the discontinuity of thedata at the both ends. Figure 5 shows the computed spa-tial-frequency spectra corresponding to Eq. (5); the spectraare separated by the carrier frequency fo = 3 lines/mm. Only

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158 J. Opt. Soc. Am./Vol. 72, No. 1/January 1982imaginary part is shown in Fig. 6(b). Figure 6(a) shows con-ventional contour fringes generated by inserting a heat sourcein one of the arms of the tilt-free interferometer. The mea-surement shown in Fig. 6(b) was made over a range of the linesegment marked by H-iin the picture. Note that in this ex-

Fig. 3. Intensity distribution of the interference fringes of tilted wavefronts taken into the microcomputer.

(a)Od(xY)

.0no~j

s

a'fSI

Fig. 4. Distribution of the interference fringes weighted by ahanning window w(x) = 1 - cos(2irx/D), where D is the range of themeasurement (15.3 mm). ..6 6.6i 9.6 '6.62 533.6 599.34 666.60 733... 703.32 46.58 4 .2 I .

(b)fV0 3 lines/mm

0.00 66.61 133.32 449.66066.64 333.30 364.66 466.62 533.26 564.44 666.10 733.26 746.62 466.56 33.04 4.64

Fig. 5. Computed spatial-frequency spectra. The zero-frequencyspectrum is clipped to enhance the detail of the spectra on the car-riers.

one spectral sideband was selected and shifted by f0 towardthe origin, and its inverse Fourier transform was computedagain by the FFT algorithm to obtain c(x, y) in Eq. (4). Thecomplex logarithm of c(x, y) was then computed; the dis-continuous phase distribution Od(X, y) obtained from the

(c)Fig. 6. (a) Conventional contour fringes showing the index distri-bution around the heat source; (b) discontinuous phase distributionobtained by measurement over a range of the line segment markedbyH in (a); (c) corrected continuous-phase distributions representinga complete profile of the phase value over the range marked by l-4 in(a).

, i ' :r "",.' .S..1 1. # '. 8 sin.'1E E......11 , l| r! | *

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Vol. 72, No. 1/January 1982/J. Opt. Soc. Am. 159

(a)09(x,y)

IIoVA4

3 96960 .32 l93.98 2365433 3.30399.96465.6253i3.26 s99.3 669.63733.29 739.92 866.59 933.24 939.90

(V

x

tb)Fig. 7. (a) Contour fringes showing the residual aberration of theinterferometer; (b) phase distribution (less than 27r) measured overa range of the line segment marked by [ in (a).

o .00 66.96 3i3.3219.99 269.64 333.30 399.i6 996.62333.23599.99 666.60 733.26 799.92 996.56 933.24 993.60y~ .; ... .. ." ....... ... ..

which gives a complete profile of the phase measured over therange of the linesegmentmarkedby Hin Fig. 6(a). Note thatthe profile can be determined uniquely without ambiguityabout the distinction between elevation and depressionof thewave front form.To demonstrate the sensitivity of the measurement, onemore example is given in Fig. 7. Figure 7(a) shows contour-type fringes obtained when the heat source was removed; itrepresents a residual aberration of the interferometer. Hereagain, the measurement was made over a range of the linesegment marked by H in the picture, and the result is shownin Fig. 7(b). Note that a phase distribution of less than 27r isclearly detected by the proposed method, whereas it cannotbe observed in the picture of the contour-type fringes. Theresidual aberration of the interferometer is stored in thememory of the computer and is subtracted from the mea-surement data to correct he error it has caused. This permitsthe use of optical elements of rather low quality even for themeasurement of phase distributions less than 2wx.To see the effect of noise, the same object was measuredtwice with a 1-min interval between measurements; the dif-ference in the results was computed and is shown in Fig. 8. Ascan be seen in the figure, the fluctuation that is due to noiseis less than X/30, except at the ends of the measurement range.The increase of errors at the ends is due to the use of thehanning window. Because the fringe amplitude becomes zeroin these regions, as is seen in Fig. 4, the determination of phasebecomesquite sensitive to noise.CONCLUSIONA Fourier-transform method of computer-based fringe-pat-tern analysishas been proposed and verified by experiments.The method has the sensitivity for detectingphase variationsof less than 27rand can be applied to subwavelength inter-ferometry without a fringe-scanning process. The accuracyhas been improved by the complete separation of the phasefrom other unwanted irradiance variations. By use of thismethod, clear distinction can be made between elevation anddepression, even from a single fringe pattern. When themethod is applied to topography, it permits fully automaticmeasurement without man-machine communication, as isneeded for the fringe-order assignment in computer-aidedmoir6 topography; further, because only one selected spectrumis used, the method has the additional merit that the mea-surement is not disturbed by higher-order harmonics, whichare included in a nonsinusoidal grating pattern and give riseto spurious moire fringes in the case of moire topography.

The authors thank T. Yatagai of the Institute of Physicaland Chemical Research for helpful discussions and H. Satoand T. Mimaki of the University of Electrocommunicationsfor their interest and encouragement.* Present address, Optical Products Development Division,Canon Inc., 53, Imaikamicho, Nakahara, Kawasaki, Kana-

gawa, 211, Japan.REFERENCES1. See, for example, D. M. Meadows, W. 0. Johnson, and J. B. Allen,"Generation of surface contours by moire patterns," Appl. Opt.9, 942-947 (1970); H. Takasaki, "Moir6 topography," Appl. Opt.

XFig. 8. Effect of noise on the fluctuation of the measured phasedistribution.

ample the position of 2wphase jumps coincides with that ofdark fringes. The discontinuous distribution in Fig. 6(b) wascorrected by adding an offset phase, as described in the pre-ceding section. Figure 6(c) shows the continuous distribution,

TY/15 (X/30)

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Hi- . . . .. ' . . . . . . . . . . . . . !

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so

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160 J. Opt. Soc. Am./Vol. 72, No. 1/January 19829,1467-1472 (1970). Although these papers describe methods thatuse a shadow of a grating, our proposed method applies to a pro-jected grating image as in projection-type moir6 topography; fora discussion of projection-type moir6 topography, see, for example,M. Idesawa, T. Yatagai, and T. Soma, "Scanning moire methodand automatic measurement of 3-D shape," Appl. Opt. 16,2152-2162 (1977).2. See, for example, J. H. Bruning, "Fringe scanning interferometers,"in Optical Shop Testing, D. Malacara, ed. (Wiley, New York,1978), pp. 409-437.

3. In all figures, the variation in the y direction is not illustrated be-cause it is not necessary for the explanation of the principle.4. The superscript x denotes that f0 x(x, y) is an offset phase forcorrecting discontinuities in the x direction along a line for whichy is fixed; the same is denoted by the superscript y for the y di-rection. The offset phase without a superscript means thatdiscontinuities can be corrected by it in both x and y directions.5. In all figures, note only the scales indicated by large letters andignore the fine letters and scales on the axes drawn by the graphicplotter subroutine.

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