solution to the shearing problem

ooea

Solution to the shearing problem

Clemens Elster and Ingolf Weingartner

Lateral shearing interferometry is a promising reference-free measurement technique for optical wave-front reconstruction. The wave front under study is coherently superposed by a laterally sheared copyof itself, and from the interferogram difference measurements of the wave front are obtained. Fromthese difference measurements the wave front is then reconstructed. Recently, several new and efficientalgorithms for evaluating lateral shearing interferograms have been suggested. So far, however, allevaluation methods are somewhat restricted, e.g., assume a priori knowledge of the wave front understudy, or assume small shears, and so on. Here a new, to our knowledge, approach for the evaluationof lateral shearing interferograms is presented, which is based on an extension of the difference mea-surements. This so-called natural extension allows for reconstruction of that part of the underlyingwave front whose information is contained in the given difference measurements. The method is notrestricted to small shears and allows for high lateral resolution to be achieved. Since the method usesdiscrete Fourier analysis, the reconstructions can be efficiently calculated. Furthermore, it is shownthat, by application of the method to the analysis of two shearing interferograms with suitably chosenshears, exact reconstruction of the underlying wave front at all evaluation points is obtained up to anarbitrary constant. The influence of noise on the results obtained by this reconstruction procedure isinvestigated in detail, and its stability is shown. Finally, applications to simulated measurements arepresented. The results demonstrate high-quality reconstructions for single shearing interferograms andexact reconstructions for two shearing interferograms. © 1999 Optical Society of America

OCIS codes: 120.3180, 120.5050, 120.4630.

1,2,5
1. Introduction
Interferometry is the established state-of-the-artmeasurement technique for investigating the topog-raphy of high-quality surfaces and for characterizingimaging errors ~wave-front aberration! of lenses andbjectives. Generally, a light wave influenced by thebject under test is coherently superposed by a refer-nce wave. From the interference pattern gener-ted the wave aberration1,2 or the topography of a

surface3 can be determined. A disadvantage of thismeasurement principle is that a reference wave hasto be provided. This reference wave itself introducesadditional aberrations.4

Lateral shearing interferometry offers the oppor-tunity to reconstruct a wave front without use of a

C. Elster [email protected]! is with the Physikalisch-Technische Bundesanstalt Berlin, Abbestrasse 2-12, D-10587 Ber-lin, Germany. I. Weingartner [email protected]! iswith the Physikalisch-Technische Bundesanstalt Braunschweig,Bundesallee 100, D-38116 Braunschweig, Germany.

Received 3 February 1999; revised manuscript received 11 May1999.

0003-6935y99y235024-08$15.00y0© 1999 Optical Society of America

5024 APPLIED OPTICS y Vol. 38, No. 23 y 10 August 1999

reference wave. However, the reconstructionproves to be a highly sophisticated data analysisproblem. The wave front under study is coherentlysuperposed by a laterally sheared copy of itself and,from the interferogram, difference measurements ofthe wave front are obtained. From these differencemeasurements the wave front itself is reconstructed.To obtain difference measurements with high signal-to-noise ratio, large shear values are desirable. Thedrawback of this promising measurement techniqueis the mathematical and computational complexity ofthe corresponding analysis. Moreover, there is someinherent loss of information, since only differencemeasurements are known. To be competitive withinterferometric methods based on the use of a refer-ence wave, high-precision reconstructions from lat-eral shearing interferograms are demanded.

Owing to the development of new and efficient al-gorithms ~cf. Refs. 6–30!, lateral shearing interferom-etry has recently met with growing interest.However, all these evaluation methods are somewhatlimited; i.e., they either assume a priori knowledge ofthe wave front under study, are restricted to smallshears, assume that the shear equals the spacing ofthe measurement points, or yield approximate solu-tions only. As a consequence, so far, shearing inter-

oFrtof

pf

tpwtpcgt

nts

ferometry generally cannot compete with theestablished interferometric methods based on the useof a reference wave.

One of the evaluation methods frequently appliedis the polynomial method.6,13,16,29 It models the un-derlying wave front by a polynomial whose unknowncoefficients are determined by least squares. Thedegree of the polynomial, however, has to be chosenin advance. If the degree selected is too small, thepolynomial model can prove to be poor. However, anexcessively large degree can lead to difficulties thatare due to ill conditioning or overfitting. An approx-imation based on splines, which should be preferredfor their numerical stability and better convergenceproperties, has not been reported so far to our knowl-edge.

The polynomial method usually makes use ofZernike polynomials. By use of Fourier polynomialsas suggested in Ref. 8, however, exact reconstructionsare obtained for the special case in which the shearequals the spacing of the measurement points.

The comparison method takes each point of theinterferogram into account and leads to a set of cor-responding equations.10,15,19,24 For the evaluation ofone shearing interferogram this method would implythat an underdetermined system of equations has tobe solved and thus is usually applied for the analysisof several shearing interferograms. In this case asystem of overdetermined equations is to be solved orapproximated in the case of noisy data. However, toapply this method, the lateral shear must be equal tothe spacing of the interference pattern measurementpoints considered. This implies either small lateralresolution or a small lateral shear and, thus, a smallmeasurement signal ~deflection of the interferencefringes or modulation of the interference pattern, re-spectively!.27

Another method uses Tikhonov regularization23 tobtain a reconstructed wave front that is smooth.or this method an appropriate regularization pa-ameter has to be chosen for good results to be ob-ained. Since this method leads to low-pass filteringf the frequency response, the whole Fourier trans-orm of the wave front is somewhat biased.

Recently a method, believed to be new, has beenroposed that uses the so-called shearing transferunction in a two-step process.17,28 In the first step,

the application of the shearing transfer function tothe sheared data leads to a reconstruction withstrong shear-periodic ~s-periodic! disturbances. Inhe second step the shearing transfer function is ap-lied to the sheared data multiplied by a smoothindow function, which results in good approxima-

ions of the underlying wave front for a small centralart of the reconstruction. Combination of the re-onstructions obtained in the two steps then yieldsood overall reconstructions with high lateral resolu-ions for larger shears as well.

In this paper we present what we believe to be aew and completely different approach. Our idea iso reduce the general reconstruction problem to thepecial case of a periodic problem that is easily solved.

This reduction is achieved by suitable—hereaftercalled natural—extension of the differences.

In Section 2 the shearing problem is discussed withrespect to the properties of the underlying wave frontand the difference measurements. In Subsection2.A the case of an infinite extension of both underly-ing wave-front and difference measurements is dealtwith, and in Subsection 2.B the special case of peri-odic problems is discussed and explicitly solved byharmonic analysis. The general problem is posed inSubsection 2.C. In Subsections 3.A and 3.B naturalextension of the differences is introduced, and it isshown how this extension reduces a general problemto a periodic one. Reconstructions based on this nat-ural extension are shown to be stable with respect tonoise in Subsection 3.C.

The application of natural extension to the analysisof two shearing interferograms with suitably chosenshearing parameters is studied in Section 4. In Sub-section 4.A it is shown that exact reconstruction ofthe underlying wave front—up to an arbitraryconstant—at all evaluation points is achieved for ex-act data. The influence of noise on the results ob-tained by this reconstruction procedure is studied indetail in Subsection 4.B, where good stability resultsare derived.

In Section 5 we demonstrate the performance of thenew evaluation method by applying it to simulateddifference data. Both noiseless and noisy data aresimulated and analyzed. The results show high-quality reconstructions for one shearing and excel-lent results for two shearing interferograms. Theapplication of the new reconstruction method is dem-onstrated for shearing interferometry only. Themethod is, however, well suited to the analysis ofdifference measurements in other fields of metrologyas well, for example, in dimensional metrology.

2. Shearing Problem

The shearing problem is, generally speaking, the taskof reconstructing a wave front f ~x! from differencemeasurements,

Df ~x! 5 f ~x 1 sy2! 2 f ~x 2 sy2!,

as obtained by lateral shearing interferograms.Here we describe some aspects of this problem. Inparticular, the properties of f ~x! and Df ~x! with re-spect to their lateral extension and their periodicityare discussed. The purpose is to explain the originof the difficulties of the reconstruction of a wave frontof only finite extension and no inherent periodicity,which is the usual case. Periodic problems are dis-cussed in more detail, and an explicit solution is de-rived. In Section 3 we propose a new, to ourknowledge, approach to overcoming the difficulties ofthe shearing problem and offer a solution by so-callednatural extension of the differences. This naturalextension effectively reduces the general problem to aperiodic one.

10 August 1999 y Vol. 38, No. 23 y APPLIED OPTICS 5025

ft

5k

ra

tr

5

A. Measurements with Infinite Extension

Assume the ~hypothetical! case in which the waveront and the wave-front difference have infinite ex-ension. By performing Fourier transforms we get

F@ f ~x 1 sy2!#~n! 5 *2`

`

f ~x 1 sy2!exp~22pinx!dx

5 *2`

`

f ~x!exp~22pinx!exp~ipns!dx,

F@ f ~x 2 sy2!#~n! 5 *2`

`

f ~x 2 sy2!exp~22pinx!dx

5 *2`

`

f ~x!exp~22pinx!exp~2ipns!dx,

where s denotes the shear, F is the Fourier transform,x is a length coordinate, and n is the spatial fre-quency. It directly follows that

F@ f ~x!#~n! 5 T~n!F@Df ~x!#~n!,

T~n! 51

exp~ipns! 2 exp~2ipns!

51

2i sin~pns!. (1)

T~n! represents the well-known so-called shearingtransfer function,8,17 which is undefined for sin~pns!

0. As a consequence, F@ f ~x!#~ks21! remains un-nown for k [ Z with Z 5 $0, 61, 62, . . .%, and no

information about these s-periodic parts can be ob-tained from the difference function.

If it can, however, be assumed that F@ f ~x!#~ks21!emains finite for all k [ Z, f ~x! can be recovered inccordance with

f ~x! 5 *2`

`

exp~2pinx!T~n!F@Df ~x!#~n!dn,

where

T~n! 5 HT~n!, n Þ ks21, k [ Z0, otherwise .

B. Periodic Measurements

Some problems are inherently periodic. Examplesinclude properties of divided circles,31 roundness ofcylinders and spheres, i.e., generally speaking, mea-surements that are linked with the full circle 2p 5360°. In these cases the wave front has a periodicitywith the full circle of 2p. Because of this periodicity,the wave-front differences also have a periodicity of2p and, even more importantly, the difference mea-surements can also be determined for the full circle.

For the analysis of general p-periodic problems;e.g., p 5 2p, f ~x! is represented in Fourier termsaccording to

f ~x! 5 (k52`

`

fkck~x!, (2)


where

ck~x! 5exp~2pikxyp!

Îp,

fk 5 ~ck, f !,with

~u, v! :5 *0

p

u*~x!v~x!dx,

and p denotes complex conjugation.From Eq. ~2! it directly follows that Df ~x! can be

written according to

Df ~x! 5 (k52`

` FexpSipksp D 2 expS2

ipksp DGfkck~x!

5 (k52`

`

2i sin~pksyp! fkck~x!. (3)

Note that Df ~x! can contain nonvanishing Fouriercomponents only for those k with sin~pksyp! Þ 0.

The crucial point now is that Df ~x! can be obtained,i.e., measured, over the whole interval @0, p#, owing tothe inherent periodicity of the problem. Hence theFourier coefficients 2i sin~pksyp! fk in Eq. ~3! can bedetermined from the difference function according to

2i sin~pksyp! fk 5 ~ck, Df !,

and all Fourier coefficients fk with sin~pksyp! Þ 0 ofhe wave front can be reconstructed. However, theemaining Fourier coefficients fk with sin~pksyp! 5 0

cannot be reconstructed, since they correspond tos-periodic parts of the wave front, which are totallylost, owing to the shearing operation; i.e., Dck~x! 5ck~x 1 sy2! 2 ck~x 2 sy2! [ 0 holds for all k withsin~pksyp! 5 0. By setting these Fourier coefficientsto zero, we obtain a solution that contains no ~arbi-trarily chosen! s-periodic part.

For p-periodic problems the reconstruction of thatpart of the wave front whose information is availablecan hence be carried out easily by the harmonic anal-ysis proposed.

C. General Problem

The evaluation of lateral shearing interferogramsgenerally leads to the problem of reconstructing awave front f ~x! over the interval @0, p# based on mea-surements of the wave-front differences,

Df ~x! 5 f ~x 1 sy2! 2 f ~x 2 sy2!

over the interval @sy2, p 2 sy2#,

where s denotes the shear chosen. @0, p# is the ap-erture limiting the wave front f ~x! to be recon-structed, @sy2, p 2 sy2# is the interval over which thewave-front difference Df ~x! can be measured and thatis smaller than the domain of the wave front. This isthe central problem, since the shearing transfer func-tion cannot be applied here. If, e.g., Df ~x! is simplyextended by zero within the intervals @0, sy2#, @p 2sy2, p# and the harmonic analysis proposed in Sub-

l

f

f

f

c

f

A

i

sc

section 2.B is applied, wrong results with strongs-periodic disturbances in the reconstructed wavefront f ~x! are obtained.17

3. Solution by Natural Extension

Natural extension of the difference function is intro-duced.32 This natural extension allows for the gen-eral shearing problem of Subsection 2.C to be reducedto a p-periodic problem that can easily be treated asdescribed in Subsection 2.B. This means that thenatural extension to be introduced below allows fordifferences of the wave front under study to be cal-culated within the intervals @0, sy2# and @p 2 sy2, p#from the difference measurements given within theinterval @sy2, p 2 sy2#, and these calculated differ-ences equal differences that would occur if the under-lying wave front were first p-periodically extendedand then sheared. In other words, natural exten-sion furnishes information about Df ~x! over the wholeinterval @0, p#, which is already contained in the dif-ferences given within the interval @sy2, p 2 sy2#.

In Subsection 3.A the special case in which p is amultiple of s and the underlying wave front is p-peri-odic, i.e., satisfies f ~p! 5 f ~0!, is treated. Note thatthis is only part of the assumption of p-periodic prob-lems, since the difference measurements in this caseare given only over the limited interval @sy2, p 2 sy2#.The natural extension then reduces this problem to ap-periodic problem that is easily solved in accordancewith Subsection 2.B. In Subsection 3.B it is shownhow we can reduce the general case to the special caseof Subsection 3.A by suitably extending and changingthe difference measurements themselves to satisfy thespecific requirements of Subsection 3.A. The stabilitywith respect to noise in the difference measurements isdiscussed in Subsection 3.C.

A. Natural Extension when p Is a Multiple of s

The idea of the natural extension of Df ~x! is as fol-lows: If the underlying wave front f ~x! is extendedp-periodically to fp~x!, say, then the difference func-tion Df ~x! can be extended to ~the p-periodic function!Dfp~x! according to

Dfp~x! 5 fp~x 1 sy2! 2 fp~x 2 sy2!, (4)

which leads, in particular, to a natural extensionDfp~x! of Df ~x! over the whole interval @0, p#. In thespecial case in which p is a multiple of s this naturalextension can be explicitly determined from the dif-ference function Df ~x! ~given over the interval @sy2,p 2 sy2# only! and thus reduces the general problemto a p-periodic problem.

The natural extension Dfp~x! is explicitly calcu-ated according to

Dfp~x! 5 5Df ~x!, sy2 # x # p 2 sy2

2 (l51

pys21

Df ~x 1 ls!, 0 # x , sy2

2 (l51

pys21

Df ~x 2 ls!, p 2 sy2 , x # p

(5)

rom the given difference function Df ~x! for x [ @sy2,p 2 sy2#. To show this, note first that Dfp~x! is welldefined for x [ @0, p# when Df ~x! is given for x [ @sy2,p 2 sy2# only; i.e., Dfp~x! can be uniquely calculatedor x [ @0, p# when Df ~x! is given for x [ @sy2, p 2

sy2#. Furthermore, the following relation holds:

ck~x! 1 (l51

pys21

ck~x 6 ls! 5 ck~x!1 2 exp~62pik!

1 2 expS62piks

p D 5 0

for all k with sin~pksyp! Þ 0,

rom which

k~x! 5 2 (l51

pys21

ck~x 6 ls!

for all k with sin~pksyp! Þ 0 (6)

ollows. Since the natural extension of Dfp~x! ac-cording to Eq. ~4! contains only Fourier coefficientsfor k with sin~pksyp! Þ 0 ~cf. Subsection 2.B!, it fol-lows immediately from Eq. ~6! that Eq. ~5! coincideswith Eq. ~4! for x [ @0, p# and that Eq. ~5! thus equalsthe natural extension.

As in the case of p-periodic problems the Fouriercoefficients fk for all k with sin~pksyp! Þ 0 can thensimply be determined according to

fk 5 ~ck, Dfp!y@2i sin~pksyp!#, (7)

with Dfp~x! calculated from Df ~x! according to Eq. ~5!.s already explained in Subsection 2.B, for all k with

sin~pksyp! 5 0 no information about the correspond-ng Fourier coefficients fk can be obtained through

Df ~x!.

B. General Case when p Is Not a Multiple of s

The general case in which p is not a multiple of s andthe underlying wave front f ~x! does not satisfy f ~p! 5f ~0! can be reduced to that treated above. In a firsttep the original problem is extended to that of re-onstructing a wave front over a larger interval @0, p9#

with a p9 . p chosen as a multiple of s. This is doneby extension of the given difference measurementssmoothly and arbitrarily from the interval @sy2, p 2sy2# to the interval @sy2, p9 2 sy2#. Then, by sub-traction of a suitably determined constant from the~smoothly extended! difference measurements, theunderlying ~and also smoothly extended! wave frontis changed to a p9-periodic one. The problem ob-tained in this way is then solved over @0, p9#, and thesubtraction of the constant from the difference mea-surements is accounted for. Finally, only the part ofthe solution with x [ @0, p# is retained.

In detail, let p9 . p denote the smallest real num-ber that satisfies p9 5 rs where r is a suitable naturalnumber. Then define the smooth extensionDfsmooth~x! of Df ~x! according to


s

ats

li

d

w

ffipo

ct

5

Note that the solution of the related problem—reconstruction of fsmooth~x! from Dfsmooth~x!—alsoolves the original problem for x [ @0, p# in the sense

that

fsmooth~x 1 sy2! 2 fsmooth~x 2 sy2! 5 Df ~x!

holds for x [ @sy2, p 2 sy2#. Since Df ~x! is typicallyextended over a short part only, i.e., p9 2 p ,, p, andsince this extension is performed smoothly, the re-construction fsmooth~x! obtained is expected to givesimilarly good results as when p is a multiple of s ~cf.lso the simulation results in Section 5!. Note fur-her that this smooth extension changes the corre-ponding Fourier transforms only slightly.Next, since typically fsmooth~p9! Þ fsmooth~0!, a re-

ated problem is constructed and solved first. Thedea is to look for a solution f2~x! to a related problem

and that differs from fsmooth~x! by a linear function cxwhere the constant c is chosen such that f2~x! satis-fies f2~p9! 5 f2~0!. For this purpose define

Df2~x! 5 Dfsmooth~x! 2 cs

for a suitable constant c, which is chosen as describedbelow. Clearly, if f2~x! is a solution to this relatedproblem, then

fsmooth~x! 5 f2~x! 1 cx (9)

is a solution to the ~smoothly extended! original prob-lem. Note that by addition of a linear function cx toa function the corresponding difference is changed bythe constant cs.

To determine a suitable constant c, one may ap-proximate fsmooth~x! by a linear function in the least-squares sense, i.e., by defining c as the mean value ofDfsmooth~x! divided by the shear s. Another way is to

efine c according to

c 5 @Dfsmooth~sy2! 1 Dfsmooth~sy2 1 s!

1 · · · 1 Dfsmooth~sy2 1 ~p9ys 2 1!s!#yp9. (10)

As can easily be seen, the above expression equals@ fsmooth~p9! 2 fsmooth~0!#yp9, and hence the problemcan be exactly changed into a p9-periodic one, at leastfor noiseless data. Note, however, that this estimateis less robust with respect to noise than the one pro-posed above.

Note finally that when the original problem ischanged in the way described above the final solutionwill generally contain some s-periodic part, since infact f2~x! contains no s-periodic part but the linearfunction cx does. This procedure relies on the factthat the Fourier expansion of f2~x! generally con-verges much more rapidly than that of fsmooth~x!

ithin ~0, p9!, which in turn leads to better recon-struction results.


C. Influence of Random Errors on the Solution

To show that the reconstruction algorithm based onnatural extension according to Subsection 3.A is sta-ble with respect to random errors, it is sufficient toshow that the obtainable Fourier coefficients fk asdefined by Eq. ~7! are stable. To show this, note firstthat the absolute of sin~pksyp! is bounded from belowor all k with sin~pksyp! Þ 0 ~indeed, there is only anite number of different values of sin~pksyp!, sinceis a multiple of s!. Furthermore, the inner product

f Dfp~x! and ck~x! can be written according to

~ck, Dfp! 5 *0

p

Dfp~x!c*k~x!dx

5 2*0

sy2

(l51

pys21

Df ~x 1 ls!c*k~x!dx

1 *sy2

p2sy2

Df ~x!c*k~x!dx

2 *p2sy2

p

(l51

pys21

Df ~x 2 ls!c*k~x!dx

5 *sy2

p2sy2

Df ~x!c*k~x!$1 2 exp@if~x!#%dx,

where f~x! is a suitable phase. Since the absolute of$1 2 exp@if~x!#% is bounded by 2, it directly followsthat the inner products ~ck, Dfp! are stable with re-spect to noise, from which it follows that the Fouriercoefficients fk obtained by natural extension are sta-ble.

The above argument strictly holds only if the spe-cific requirements of Subsection 3.A are satisfied. Inthe general case the uncertainty of the estimatedlinear trend has to be taken into account. In thecase of small shear values, i.e., s ,, p9, the least-squares estimate of this trend seems preferable tothat proposed in Eq. ~10!.

4. Exact Reconstruction from Two ShearingInterferograms

The natural extension Dfp~x! allows for those Fouriercoefficients fk of the underlying wave front f ~x! to bealculated, about which information is availablehrough Df ~x!. The remaining Fourier coefficients

cannot be determined. By combining the analysis ofseveral shearing interferograms for different shears,we can largely reduce the number of unknown fk.Moreover, to reconstruct any wave front f ~x! up to anarbitrary constant at N discrete evaluation points,the results from only two interferograms are needed.This can be done by appropriate choice of the shear-ing parameters s1, s2; the number of measurement

Dfsmooth~x! 5 HDf ~x!, sy2 # x # p 2 sy22Df ~p 2 sy2! 2 Df{p 2 sy2 2 @x 2 ~p 2 sy2!#%, p 2 sy2 , x # p9 2 sy2 . (8)

Fx

h

t

w

wtrv

wtsTqn

points; and the spacing of the measurements D.Note that this reconstruction algorithm does notmake assumptions, as regards the underlying wavefront f ~x!, such as smoothness or the demand thatf ~p! should equal f ~0!. In the following this exactreconstruction procedure is described briefly ~cf. Ref.33 for a detailed description!.

A. Analysis of Two Shearing Interferograms

Denote by r, n two natural numbers with no commondivisor, and define N 5 rn, D 5 pyN, s1 5 nD, s2 5 rD,ns1

5 ~r 2 1!n, ns25 ~n 2 1!r. N denotes the number

of evaluation points of the wave front; D is the spacingof measurement points; ns1

, ns2are the number of

measured sheared values for the two shearing inter-ferograms; and s1, s2 are the two shearing parame-ters. For ease of notation D 5 1 is assumed; that is,p 5 N, s1 5 n, and s2 5 r.

For the given differences of the wave front

yaj :5 Df ~za

j! 5 f ~zaj 1 sjy2! 2 f ~za

j 2 sjy2!,

a 5 0, . . . , nsj2 1, j 5 1, 2

at the measurement points

zaj 5 sjy2 1 a, a 5 0, . . . , nsj

2 1, j 5 1, 2,

calculate ~natural extensions!

vaj :5 H ya

j, a 5 0, . . . , nsj2 1

2 (l51

Nysj21

ya2lsj

j, a 5 nsj, . . . , N 2 1

,

j 5 1, 2,

as well as the corresponding Fourier coefficients

vkj :5

1N (

l50

N21

vlj expS2

2piklN D ,

k 5 0, . . . , N 2 1, j 5 1, 2. (11)

or both interferograms the same evaluation pointsl 5 l, l 5 0, . . . , N 2 1 of the wave front to be

reconstructed are used with yaj 5 fa1sj

2 fa; fl 5f ~xl!, l 5 0, . . . , N 2 1 denote the values of the wavefront f ~x! at the evaluation points xl. As shown inRef. 33 the relations

vkj 5 fkFexpS2piksj

N D 2 1G ,

k 5 0, . . . , N 2 1, j 5 1, 2, (12)

old where fk, k 5 0, . . . , N 2 1 denote the Fouriercoefficients of fl, l 5 0, . . . , N 2 1. Since s1 and s2have no common divisor, for all k 5 1, . . . , N 2 1 atleast one of the two numbers @exp~2piks1yN! 2 1#,@exp~2piks2yN! 2 1# differs from zero; i.e., fk can bedetermined for all k 5 1, . . . , N 2 1, which implieshat fl, l 5 0, . . . , N 2 1 can—apart from an arbitrary

constant—be exactly reconstructed.

B. Analysis of Reconstruction Method Stability

To analyze the stability of the reconstruction method,it is assumed that the given data ya

j,s are perturbedby noise according to

yaj,s 5 ya

j 1 haj, a 5 0, . . . , nsj

2 1, j 5 1, 2,

where the random variables haj are independent and

satisfy

E~haj! 5 0, E~ha

j!2 5 s2.

As shown in Ref. 33, one then obtains

(k51

N21

E~ufk 2 fku2! 5 d2s2,

where the results of the two shearing experimentsare combined according to

fk 5ws1

~k! fk1,s 1 ws2

~k! fk2,s

ws1~k! 1 ws2

~k!, k 5 1, . . . , N 2 1,

(13)

ith

fkj,s 5 50, if sin@p~ksjyN!# 5 0

vkj,s

exp~2piksjyN! 2 1, otherwise

,

k 5 1, . . . , N 2 1, j 5 1, 2,

with vkj,s as the Fourier coefficients obtained from

the ~noisy! data according to Eq. ~11! and appropriateeights as given in Ref. 33. It is shown in Ref. 33

hat the noise amplification d remains small; i.e., theeconstruction procedure is stable. In Table 1 somealues for d are given.

5. Simulation Results

The performance of the reconstruction on the basis ofnatural extension is demonstrated by evaluation ofsimulated difference measurements. The test func-tion ~Fig. 1! mainly shows the behavior of typical

ave-front aberrations ~defocusing, spherical aberra-ion, coma! and additional strong disturbances byeveral harmonic components and two sharp peaks.his test function therefore contains high spatial fre-uencies. Note that the results stated below foroisy data are typical results.

Table 1. Noise Amplification Values d for Some Pairs of SelectedShears s1, s2

s1 s2 N d

23 29 667 1.072 331 662 5.27

29 71 2059 1.1917 19 323 1.013 41 123 1.60

15 16 240 0.98


1Ndft

se~wctdp8

a

a

5

A. Simulation Results for One Shear

First, the reconstruction of the wave front is studiedwhen only a single interferogram is evaluated. Fig-ure 1 shows the test function used for the reconstruc-tion problem and the data function with a shear s 5

y~17, 5!, where p 5 1 is chosen in arbitrary units.ormal random numbers with zero mean and stan-ard deviation 1 3 1023 were added to the differenceunction. To demonstrate the practical aspects men-ioned above, the shear s 5 1y~17, 5! was used, i.e., p

is not a multiple of s, and, moreover, the test functionis not p periodic. ns 5 991 differences of the testwave front were taken equally spaced with D 5 @1 21y~17, 5!#y990. For the reconstruction, first, the


imulated difference measurements were smoothlyxtended and detrended according Eqs. to ~8! and10!. Then the solution based on natural extensionas calculated, with all Fourier coefficients that

ould not be calculated set to zero. Figure 2 showshe corresponding reconstruction. The maximumifference between the test wave front and the ap-roximated reconstruction obtained is approximately3 1023, and the mean reconstruction error is ap-

proximately 4 3 1023.

B. Simulation Results for Two Shears

Second, the reconstruction by evaluation of two suit-able shearing interferograms was studied where,again, p 5 1 is chosen in arbitrary units. The two

Fig. 1. Test function ~1! used for the reconstruction problem anddata function ~2! with a shear s 5 1y~17, 5!. Normal randomnumbers with zero mean and standard deviation 1 3 1023 weredded to the data function.

Fig. 2. Test function ~dotted curve! and reconstruction from data function ~Fig. 1!. The maximum difference between the test wave frontnd the approximated reconstruction obtained is approximately 8 3 1023, and the mean reconstruction error is approximately 4 3 1023.

Fig. 3. Test function ~1! used for the reconstruction problem anddata functions ~2 and 3! with shears s 5 1y15 and s 5 1y16,respectively. Normal random numbers with zero mean and stan-dard deviation 1 3 1023 were added to the data functions.

10. R. L. Frost, C. K. Rushforth, and B. S. Baxter, “Fast FFT-based

3a

shears s1 5 1y15 and s2 5 1y16 with ns15 224 and

ns25 225 values of the difference of the test wave

front were in both cases taken equally spaced withD 5 1y~15 3 16!. For exact difference measure-ments up to machine precision the reconstructedwave front does not differ—apart from an arbitraryfunction—by more than 3 3 10215 from the test wavefront at all evaluation points.

To study the influence of noise, normal randomnumbers with zero mean and standard deviation 1 31023 were then added to the data functions. Thetwo data functions obtained and the original testfunction are shown in Fig. 3. The correspondingmaximum reconstruction error obtained from thesenoisy data is approximately 2 3 1023, and the meanerror is approximately 1 3 1023. The correspondingreconstruction together with the original test func-tion are shown in Fig. 4.

References1. V. Ronchi, “Le frangie di combinazione nello studio delle su-

perficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9–35~1923!.

2. W. J. Bates, “A wavefront shearing interferometer,” Proc.Phys. Soc. London 59, 940–952 ~1947!.

3. G. Schulz, “Ein Interferenzverfahren zur absoluten Eben-heitsprufung langs beliebiger Zentralschnitte,” Opt. Acta 14,375–388 ~1967!.

4. M. Born and E. Wolf, Principles of Optics ~Pergamon, Oxford,UK, 1989!.

5. V. Ronchi, “Forty years of history of a grating interferometer,”Appl. Opt. 3, 437–450 ~1964!.

6. S. Baumer, “Quantitative Mikro-Messtechnik mit einem Lat-eral Shearing Interferometer,” Ph.D. dissertation ~Optics In-stitute of Berlin Technical University, Berlin, 1995!.

7. H. von Brug, “Zernike polynomials as a basis for wave-frontfitting in lateral shearing interferometry,” Appl. Opt. 36,2788–2790 ~1997!.

8. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of awave front from difference measurements using the discreteFourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 ~1986!.

9. D. L. Fried, “Least-squares fitting a wave-front distortion es-timate to an array of phase-difference measurements,” J. Opt.Soc. Am. 67, 370–375 ~1977!.

Fig. 4. Test function and reconstruction from data functions ~Fig.!. The maximum difference between the test wave front and thepproximated reconstruction obtained is approximately 2 3 1023,

and the mean reconstruction error is approximately 1 3 1023.

algorithm for phase estimation in speckle imaging,” Appl. Opt.18, 2056–2061 ~1979!.

11. D. C. Ghiglia and L. A. Romero, “Direct phase estimation fromphase differences using elliptic partial differential equationsolvers,” Opt. Lett. 14, 1107–1109 ~1989!.

12. F. Guse, “Auswertung von Messungen mit Optimierungs-verfahren—demonstriert an Interferometrie und Ellipsomet-rie,” Ph.D. dissertation ~Optics Institute of Berlin TechnicalUniversity, Berlin, 1996!.

13. G. Harbers, P. J. Kunst, G. W. R. Leibbrandt, “Analysis oflateral shearing interferograms by use of Zernike polynomi-als,” Appl. Opt. 35, 6162–6172 ~1996!.

14. R. H. Hudgin, “Wavefront reconstruction for compensated im-aging,” J. Opt. Soc. Am. 67, 375–378 ~1977!.

15. B. R. Hunt, “Matrix formulation of the reconstruction of phasevalues from phase differences,” J. Opt. Soc. Am. 69, 393–399~1979!.

16. G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-frontanalysis with high accuracy by use of a double-grating lateralshearing interferometer,” Appl. Opt. 35, 6151–6161 ~1996!.

17. S. Loheide and I. Weingartner, “New procedure for wavefrontreconstruction,” Optik 108, 53–62 ~1998!.

18. R. J. Noll, “Phase estimates from slope-type wavefront sen-sors,” J. Opt. Soc. Am. 68, 139–140 ~1978!.

19. M. P. Rimmer, “Method for evaluating lateral shearing inter-ferometer,” Appl. Opt. 13, 623–629 ~1974!.

20. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberra-tions using a lateral-shear interferometer having variableshear,” Appl. Opt. 14, 142–150 ~1975!.

21. F. Roddier and C. Roddier, “Wavefront reconstruction usingiterative Fourier transforms,” Appl. Opt. 30, 1325–1327 ~1991!.

22. H. Schreiber and J. Schwider, “Lateral shearing interferome-ter based on two Ronchi gratings in series,” Appl. Opt. 36,5321–5324 ~1997!.

23. M. Servin, D. Malacara, and J. L. Marroquin, “Wave-frontrecovery from two orthogonal sheared interferograms,” Appl.Opt. 35, 4343–4348 ~1996!.

24. W. H. Southwell, “Wavefront estimation from wavefront slopemeasurements,” J. Opt. Soc. Am. 70, 998–1006 ~1980!.

25. H. Takajo and T. Takahashi, “Least-squares phase estimationfrom the phase difference,” J. Opt. Soc. Am. A 5, 416–425~1988!.

26. X. Tian, M. Itoh, and T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry,”Appl. Opt. 34, 7213–7220 ~1995!.

27. K. Freischlad, “Sensitivity of heterodyne shearing interferom-eters,” Appl. Opt. 26, 4053–4054 ~1987!.

28. S. Loheide and I. Weingartner, “Verfahren zur Ermittlungeiner optischen, mechanischen, elektrischen oder anderenMessgrosse,” German patent 197 05 609.1 ~15 May 1997!.

29. I. Weingartner and H. Stenger, “A simple shear-tilt inter-ferometer for the measurement of wavefront aberration,”Optik 70, 124–126 ~1985!.

30. J. B. Saunders, “Measurement of wavefronts without a refer-ence standard. Part 1. The wavefront shearing interferom-eter,” J. Res. Natl. Bur. Stand. Sect. B 65B, 239–244 ~1961!.

31. A. Fricke and I. Weingartner, “Verfahren zur Ermittlung einerWinkelteilung aus der Winkelteilungsdifferenz, bestimmt ausder Differenz zweier Winkelpositionen mit jeweils konstantemWinkelabstand,” German patent 197 20 968.8 ~patent pending!.

32. C. Elster, S. Loheide, and I. Weingartner, “Verfahren zurErmittlung einer optischen, mechanischen, elektrischen oderanderen Messgrosse,” German patent 198 33 269.6 ~patentpending!.

33. C. Elster and I. Weingartner, “Method for exact wave-frontreconstruction from two lateral shearing interferograms,” J.Opt. Soc. Am. A ~in press!.


solution to the shearing problem

Documents