Numerical correction of referencephases in phase-shifting interferometry byiterative least-squares fitting

Geon-Soo Han and Seung-Woo Kim

We present a new computational algorithm of phase-shifting interferometry that can effectively eliminatethe uncertainty errors of reference phases encountered when we obtain multiple interferograms. Thealgorithm treats the reference phases as additional unknowns and we determine their exact values byanalyzing interferograms using the numerical least-squares technique. A series of simulations provethat this algorithm can improve measuring accuracy because it is unaffected by the nonlinear and randomerrors of phase shifters.

Key words: Phase-shifting interferometry, optical metrology, correction of reference phases, surfaceprofile measurements.

Introduction

Phase-shifting interferometry has been widely usedfor the ultraprecision measurement of surface pro-files.1-3 Its metrological principle is to measure thewave front reflected off the test surface relative to thewave front reflected off a reference surface by analyz-ing multiple interferograms produced by the twowave fronts with different reference phases. Onecan introduce the reference phases by using a piezo-electric-type phase shifter that moves the test orreference surface to change the relative optical pathdifference. In practice, because of deterministic andrandom errors that are inherent in the piezoelectricphase shifter, the actual reference phases deviatefrom the intentional values so that a suitable compen-sation scheme is needed if measuring accuracy is to beenhanced.

Previous research4 -9 for this purpose has beenconcentrated mainly on deterministic linear compen-sations of the phase shifter prior to or after interfero-grams are obtained. In practical measurement envi-

The authors are with the Department of Precision Engineeringand Mechatronics, Korea Advanced Institute of Science and Tech-nology, 373-1, Kusung-dong, Yusung-gu, Taejon 305-701, SouthKorea.

Received 27 September 1993; revised manuscript received 10January 1994.

0003-6935/94/317321-05$06.00/0.© 1994 Optical Society of America.

ronments, however, the phase shifter exhibits aconsiderable amount of nonlinear behavior and alsothe random errors caused by thermal and vibrationaldisturbances often appear significant. In this studya new computational algorithm of phase-shiftinginterferometry is suggested in which the referencephases are taken as additional unknowns and theirexact values are determined from interferograms soas to minimize deterministic and random measure-ment errors simultaneously.

Phase-Shifting InterferometryFigure 1 illustrates the basic optical configuration forphase-shifting interferometry. The intensity of adetected interference pattern at a detector positionedat point x, y can be expressed as

I(x, y) = A 2(x, y) + B2(x, y) + 2A(x, y)B(x, y)cos ¢(x, y),

(1)

where A(x, y) and B(x, y) are the amplitudes of thewave fronts from the test beam and the referencebeam, respectively, and +(x, y) is the phase to bemeasured. The phase P(x, y) cannot be determinedfrom a single intensity since there are two additionalunknowns, A(x, y) and B(x, y). Phase-stepping andintegral-bucket techniques take multiple interfero-grams with different reference phases, so that theintensity of thejth interferogram appears in the form

1 November 1994 / Vol. 33, No. 31 / APPLIED OPTICS 7321

interference | ( 1i|pattern L ? 1 hi

B.s.

lightsource

I I

range. These modified algorithms however turn outto be effective only for the linear errors of thereference phases but, as in the 5-bucket(7r/2) algo-rithm, the nonlinear errors cannot be compensated.10

di, 62 di

I I I I iI I IIiI I. . . 11I II l III I [I L

referencemirror

h(x,y)

test surface

Fig. 1. Optical configuration for phase-shifting interferometry.B.S., beam splitter.

of

I (x, y) = D(x, y){l + y(x, y)sinc(A/2)cos[)(x, y) + 8y]}

= D(x, y){1 + V(x, y)cos[C(x, y) + b]}. (2)

In Eq. (2), 8j (forj = 1, 2, .. ., m and 81 = 0) is definedas the reference phase that represents the amount ofphase shift in each measurement. The A equals thephase-shift difference from one measurement to thenext for the integral-bucket technique, i.e., A = aj -bj-i, and becomes 0 for the phase-stepping technique.By applying trigonometric rules and omitting (x, y)for convenience, we can rewrite Eq. (2) as

Ij = D + DV cos 4 cos Aj - DV sin sin 8j. (3)

If the reference phases are equispaced in the intervalbetween 0 and 2iT, i.e., oj = 27r(j - 1)/m forj = 1, . . .,m, then phase (4 can be readily decided by'

-= -tan'[( Ij sin i)( Ij cos . (4)

This m-bucket algorithm assumes that the actualvalues of A& are correctly induced as intended; anyerrors in bj are reflected in the computed 4). Thelinear component of the errors can be compensated byadopting the modified algorithm6 in which one candetermine phase 4) by using two sets of m - 1 bucketswith a Tr/2 offset. If m = 5, this algorithm may bereferred to as the 5-bucket (Tr/2) algorithm that canbe expressed in the simple form of

Determination of Actual Reference Phases

To improve the measuring accuracy of phase-shiftinginterferometry, it becomes clear that the actual val-ues of the reference phases should be precisely deter-mined from interferograms, i.e., the 8j's of Eq. (2)should be regarded as additional unknown variablestogether with D, V, and 4). In this case, however, thesolutions for the unknowns cannot be uniquely deter-mined if the interferometric intensities of only asingle detector are to be considered. Regardless ofthe number of buckets taken, as seen in Eq. (3), thenumber of unknowns turns out to be m + 2, i.e., D, V,(4) 82, 83 . . , Xm which is always larger by two thanthat of the given m equations.

To determine the 8j's, the assumption should bemade that they do not vary from one detector toanother. Assuming that the 8j's are equally inducedto all the detectors, Eq. (3) can be extended as

Iij = Di + DiVi cos(i + aj), (6)

where the subscript i is adopted to identify thedetectors. Now the unknown variables are rear-ranged as follows:

Iij = Di + DiVi cos()i + aj)

= Di + DiVi cos 4)i cos - DiVi sin X)i sin 8j

= Di + Ci cos A - Si sin A. (7)

If n is the total number of detectors, then the totalnumber of individual equations involved in Eq. (7)turns out to be n x m with 3n + m - 1 unknowns.Therefore, in order to have unique solutions for theunknowns, the number of equations should be largerthan that of the unknowns, i.e.,

2nmŽ3n+m-1 or nŽ l+ 3 (8)

Once the above condition is met, the unknowns canbe determined numerically by using the least-squarestechnique. Now the error function E can be definedas follows:

ltan 2(I4 - I2)-= tan 11 - 2I3 + 15

Besides Eq. (5) we found two other algorithms thatwe intend to use to reduce the uncertainty of thereference phases; one7 determines phase 4) by averag-ing two different results obtained by using the initialm - 1 buckets and the last m - 1 buckets among totalm buckets, whereas the other8 assumes that thereference phases are equispaced over an arbitrary

where Iij represents the actual intensity measured atthe ith detector in thejth interferogram. By substi-tuting Eq. (7) into Eq. (9), the e becomes

n m

e = 1 I (Di + Ci cos by - Si sin by - Jj)2. (10)

7322 APPLIED OPTICS / Vol. 33, No. 31 / 1 November 1994

(5)n m

= 1 E (j - ij)2,i=1 j=1

(9)

| x l -

I a,� _

Or it can be expressed as

n m

E = IEi= Ej,i=1 j=1

(11)

where Ei and ej denote the subsums, respectively,represented by

m

Ei = E (Di + C cos 8 - Si sin y - Iij)2, (12)j=1

m

E = A (Di + Ci cos Sj -Si sin 8y lj)2. (13)

Then the unknowns should be decided so as tominimize error e, i.e.,

aE OE OE

aac i aSi

Table 1. Simulation Parameters

Number of detectorsNumber of buckets

n = 200m = 5

Actual reference phases A = (j - 1) (j = 1, 2,... , 5)

in which the coefficients Al, A2, A3, and A4 are definedas

Al = z (ij -Di)Si,i=

n

A2 = E (Iij - Di)Ci,i=1

aEo aEi E=or - - -= 0, (14)aD, aC Si

_E = a = 0.a 8 a j

(15)

Substituting Eqs. (12) and (13) into Eq. (14) gives thesolutions of Di, Ci, and Si in the form of a 3 x 3 matrixequation such as

Di 51 cos j

Qi = 8 hi CS 8j j(Cos 8j) 2

-Si L sin Aj I cos Aj sin 8j- ii

LVij sin .

in which the notation E represents

m

I -j=l

X sin Aj

I cos Aj sin 8

1(sin 8j) 2i

A3 = 1: -cisi,i=n

n

A4 = - 1: (S,2 - CQ2).2i=1

(18)

(19)

(20)

(21)

Numerical Algorithm

Now the 8j's and the other unknowns may be deter-mined by numerical computation in an iterativemanner. Let the superscript k be the iteration num-ber, i.e., k = 0, 1, . . . , then the necessary numericalprocedures may be summarized as follows:

J Step 1: Assume initial guess for IStep 2: Compute Dik, Cih, and Si by substituting

A> into Eq. (16) so as to minimize the error ei.(16) Step 3: Using the Dik, Ck, and Sik obtained in the

previous step, decide 8jk+1 so as to satisfy Eq. (17) thatminimizes error ej.

Step 4: Check the convergence of A1 k+i with thecondition jjk+i - jk < K, where K is a predeter-mined small constant. If the convergence is notsatisfied, then increment iteration number k and goto Step 2.

Similarly, the solution for 8j can also be determinedfrom Eq. (15) in the form of

Al cos 8j + A2 sin Aj + A3 cos 28j + A4 sin 28j = 0

10

0Is

Io

0

.0

411jCl -5

-10

0 40 80 120 160 200

measurement length(x 1Om)

Fig. 2. Ideal two-dimensional surface profile.

(17)

- 5-

') 3-V1

- 2-0c

8) 1-I

-2-*0x)C -3-

o -4-

E -5-

0 30 60 90 120 1'

-6,

iteration number

Initial reference phase: () 6 = j-1)

I -(Il)6 = (j1r-l)

Actual reference phase : = 3-(j-I) (j=1,2.5)

Fig.3. Convergence paths of reference phases (a- -_-8j).

1 November 1994 / Vol. 33, No. 31 / APPLIED OPTICS 7323

. _ ~~~~~~~~~~~~~~~~~~~~~~~~.

_ _I

.26. .

.04 _

A, .I,

.5

-6, .

-6, .2

50150

<o

v)0

aC)

C

C)

C)

-oC)0

E

0

-3.

-4.

_5

0 30 60 90

iteration number

6,

6.

6,5

6,,

-62

-6,

-6.

-6,,

120 1'

Initial reference phase: (I) bJ = - (j- )

(Il), =- lo -l)

Actual reference phase : 6 = j3-(j-l) (j=1,2,...,5)

Fig. 4. Convergence paths of reference phases (, A - 8j).

Step 5: Finally, compute Dik+l, Cik+l, and Sik+lwith ajk+l by Eq. (16) and then determine phase + byusing the relation of 4) = tan-(Sik+1/Cjk+1).

It is possible that the values of 8j obtained by theabove numerical process may not be the real solutions.In fact, two sets of solutions for Eq. (10) exist thatminimize error e. Let Di, Ci, Si, 8ji be the set of realsolutions, then its conjugate set {Di, Ci, -Si, - 8} alsosatisfies Eq. (10) since Si sin Bj = -Si sin(-8j). Ittherefore becomes necessary to constrain the rangesof Sj to be all positive or all negative in actual phaseshifting so that the numerical solutions may besimply verified by checking their signs. In addition,the initial guesses for 8jo in Step 1 should be as close totheir actual values as possible to avoid the conver-gence to conjugate solutions, and this can be achievedin practice by taking the intended reference phases asthe initial guesses.

Discussion

To evaluate the performance of the suggested algo-rithm, a computer simulation was performed with anideal two-dimensional test surface profile as shown inFig. 2. With the parameters summarized in Table 1,five interferograms of 200 detectors were generated

0.6

0.5

0.4

0.3

.8 0.2

0.l.

0 20 40 60 80 100

input command (Vr5r)

Fig. 5. Exemplary hysteresis curve of the piezoelectric actuator.

0.6

C

C.)

Zo o5 0

I C

0 c)UU:2 X

O5

0.5

0.4

0.3

0.2

0.1

0.0 vI

1 2 3 4 5 6

shift number

Fig. 6. Error of the reference phase that is due to piezoelectricnonlinearity.

with the actual reference phases of 8j = 3X(j - 1)/40,i.e., 8l = 0, 82 = 3X/40, 83 = 6A/40,... 85 = 12X/40.Then, assuming that the S.'s are unknown, thecomputational algorithm was applied to the interfero-grams with different sets of initial guesses for j.Figure 3 shows how the computed reference phasesconverge to their actual values as iteration proceeds.Two different sets of initial guesses were tested:Ajo = X(j - 1)/10 for curves I and 8j' = X(j - 1)/20for curves II. In either case, good convergence wasobtained. On the other hand, as shown in Fig. 4, ifthe initial guesses are taken as negative values asopposed to the actual phases, the computed referencephases converge to their conjugate solutions. It maybe concluded from these simulations that good conver-gence to exact solutions can be achieved only whenthe V's are taken with the same signs of the actualreference phases.

Another simulation was performed to verify howthe suggested algorithm can practically eliminatehysteresis errors of a piezoelectric phase shifter.Figure 5 shows an exemplary hysteresis curve of apiezoelectric actuator with 3% nonlinearity. If theactuator is assumed to be linear and is used for phaseshifting, the actual reference phases deviate from theintentional values as illustrated in Fig. 6. If thesame test surface of Fig. 2 is tested with the conven-

01

0..

0

U)

V)

0

-1

0 40 80 120 160 200

measurement ength(x 10ptm)

Fig. 7. Surfaceerrors.

profile errors that are duo to reference phaso

7324 APPLIED OPTICS / Vol. 33, No. 31 / 1 November 1994

if~~~~~~~~~~~~~~~~~v -~~~~~

D~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. -,

. r H~~~~~~~~~~I

:O

tional phase-shifting algorithms of Eqs. (4) and (5),measurement errors reach a maximum of approxi-mately /100 and /200, respectively, as demon-strated in Fig. 7. This new algorithm reduces theerror to a negligible level for which the limit is in factdetermined by computational truncation errors.

ConclusionA new computational algorithm of phase-shiftinginterferometry has been suggested to eliminate effec-tively the uncertainty errors of the reference phasesencountered in obtaining multiple interferograms.A series of simulations proved that the algorithm hasgood numerical convergence and can improve mea-surement accuracy because it is unaffected by thenonlinear and random errors of phase shifters.

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A. D. White, and D. J. Brangaccio, "Digital wavefront measur-ing interferometer for testing optical surfaces and lenses,"Appl. Opt. 13, 2693-2703 (1974).

2. J. C. Wyant, "Interferometric optical metrology: basic prin-ciples and new systems," Laser Focus 18, 65-71 (1982).

3. J. C. Wyant and K. Creath, "Recent advances in interferomet-ric optical testing," Laser Focus/Elect. Opt. 21, 118-132(1985).

4. C. L. Koliopoulos, "Interferometric optical phase measure-ment techniques," Ph.D. dissertation (Optical Science Center,University of Arizona, Tucson, Ariz., 1981).

5. Y. Y. Cheng and J. C. Wyant, "Phase shifter calibration inphase-shifting interferometry," Appl. Opt. 24, 3049-3052(1985).

6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spola-czyk, and K. Merkel, "Digital wave-front measuring interfer-ometry: some systematic error sources," Appl. Opt. 22,3421-3432 (1983).

7. C. Ai and J. C. Wyant, "Effect of piezoelectric transducernonlinearity on phase shift interferometry," Appl. Opt. 26,1112-1116(1987).

8. P. Carr6, "Installation et utilisation du comparateur photo-6letrique et interferentiel du Bureau International des Poids etMesures, " Metrologia 2, 13-23 (1966).

9. K. Kinnstaetter, A. W. Lohmann, J. Schwider, and N. Streibl,"Accuracy of phase shifting interferometry," Appl. Opt. 27,5082-5089 (1988).

10. J. van Wingerden, H. J. Frankena, and C. Smorenburg,"Linear approximation for measurement errors in phaseshifting interferometry," Appl. Opt. 30, 2718-2729 (1991).

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