Spatial phase-shift interferometry—a wavefront analysis technique for three-dimensional topometry

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2498 J. Opt. Soc. Am. A/Vol. 22, No. 11 /November 2005 Wolfling et al.

Spatial phase-shift interferometry—a wavefrontanalysis technique for three-dimensional

topometry

Shay Wolfling

Nano-Or Technologies, Yodfat 1, Lod 71291, Israel, and Department of Applied Physics,The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Emmanuel Lanzmann

Nano-Or Technologies, Yodfat 1, Lod 71291, Israel

Moshe Israeli

Nano-Or Technologies, Yodfat 1, Lod 71291, Israel, and Department of Computer Science,Technion Israel Institute of Technology, Haifa 32000, Israel

Nissim Ben-Yosef

School of Computer Science and Engineering, Department of Applied Physics,The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Yoel Arieli

Nano-Or Technologies, Yodfat 1, Lod 71291, Israel, and Department of Physics,Jerusalem College of Technology, Jerusalem 91160, Israel

Received January 3, 2005; accepted March 14, 2005

We describe a new wavefront analysis method, in which certain wavefront manipulations are applied to a spa-tially defined area in a certain plane along the optical axis. These manipulations replace the reference-beamphase shifting of existing methods, making this method a spatial phase-shift interferometry method. We dem-onstrate the system’s dependence on a defined spatial Airy number, which is the ratio of the characteristicdimension of the manipulated area and the Airy disk diameter of the optical system. We analytically obtain theresulting intensity data of the optical setup and develop various methods to accurately reconstruct the in-spected wavefront out of the data. These reconstructions largely involve global techniques, in which the entirewavefront’s pattern affects the reconstruction of the wavefront in any given position. The method’s noise sen-sitivity is analyzed, and actual reconstruction results are presented. © 2005 Optical Society of America

OCIS codes: 120.5050, 070.6110, 180.6900.

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. INTRODUCTIONhe importance of high-accuracy wavefront analysis forurface measurements’ applications in microelectronicsnd optical components, as well as for applications in la-er communication and target designation, stimulates ex-ensive research in new quantitative methods for wave-ront phase measurements.1–3 We introduce and develop aew, to our knowledge, interferometry technique for sur-ace measurements and wavefront analysis, which is aeneralization and combination of two different types ofechniques—the phase-shift interferometry and wave-ront manipulation techniques, such as the Zernike phaseontrast. We name this method the spatial phase shiftSPS).

One of the most widespread technologies for topometryeasurements is the phase-shift interferometry tech-

1084-7529/05/112498-12/$15.00 © 2

ique. In this technique an interference is generated be-ween a beam reflected from an inspected object and a ref-rence beam reflected from a reference object. A relativeovement induced between the two beams causes a

hase shift.4 The resulting intensities are changing inter-erence patterns, which are measured at different refer-nce translations. These interference patterns depend onhe object’s surface as well as on the surface and positionf the reference mirror. From several interference pat-erns, taken at different positions of the reference mirror,he object’s surface is obtained with high accuracy. Therexist various algorithms to calculate the object’s surfacerom several measurements of interference patterns.5,6

he implementations of these interferometric methodsave known limitations, including vibration sensitivityue to the need to use a reference beam, operation com-

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Wolfling et al. Vol. 22, No. 11 /November 2005 /J. Opt. Soc. Am. A 2499

lexity, the need for dedicated imaging objectives incorpo-ating the reference plane, dependence on the referenceuality, and large footprints.Other approaches for topometry measurements are

ased on Zernike’s phase contrast method to detect phaseariations of a wavefront, by transforming them tohanges in intensity that can be viewed. In this tech-ique, invented by Zernike,7 a phase-changing plate is in-erted at the focal plane of an imaging lens, applying ahase delay to the central spatial region of the Fourierlane. With an imaging lens, an enhanced contrasted im-ge is obtained in the image plane, where the phaseariations of the object are transformed into intensityariations. The original phase-contrast method developedy Zernike has several major limitations. The first limita-ion is a requirement for a uniform amplitude in the en-ire imaging channel—a uniform intensity distribution ofhe illumination and complete transparency of the in-pected object. A second limitation is that the phase of thenspected object can be calculated only when it is verymall (where the viewed changes in intensity are linear tohis phase). A third assumption of the phase-contrastethod is that the phase delay is applied in an infinitely

mall region of the Fourier plane. These limitations makehe phase contrast a nonquantitative method.

A variety of reviews of the phase-contrast methodxist,8,9 and several directions exist to broaden it by ex-anding the phase delay suggested by Zernike to severalonsecutive constant phase delays.10–13 These methodsvercome the uniform amplitude and small-phase limita-ions described above and use the phase-shift interferom-try formalism to obtain quantitative wavefront measure-ents from the intensities’ output of the phase-contrast

onfiguration. However, all these methods still make theernike approximation that the phase delay is applied inn infinitesimally small region of the Fourier plane lo-ated precisely at the dc of the wave reflected by the ob-ect. This assumption permits the adoption of a phase-hifting formalism that is based on a constant referenceeam but limits the accuracy of the wavefront measure-ent. It should be noted that modulating the phase in the

ptical path in order to reconstruct the reflected wave-ront is not limited to the Zernike contrast method and ispplied in other methods as well. One such method is thescillating pyramid sensing,14 in which a four-facetedlass pyramid is placed in the focal plane, angularly sepa-ating the field into four parts. Another method is thehase-diversity method,15 in which a focal-plane imageegraded by unknown aberrations is collected togetherith additional images that include known aberrations.In this article we describe a new wavefront reconstruc-

ion method, the SPS interferometry, based on a uniqueptical setup and reconstruction algorithm. This methodelaxes the Zernike approximation of existing methodsnd permits wavefront reconstruction with minimal ap-roximations. The implementation of this method openshe way to a highly accurate surface metrology capability.fter describing the optical scheme in Section 2, we willnalytically characterize in Section 3 the intensitiesormed on the image sensor. Section 4 presents a normal-zation of the system’s important parameters to a dimen-ionless number. In Section 5 we detail the various stages

n the reconstruction of the wavefront from the resultingntensities. Section 6 addresses the reconstruction’s sen-itivity to noise and overall accuracy, and Section 7 con-ludes and summarizes the article.

. SPATIAL PHASE-SHIFT OPTICAL SETUPn a previous article we introduced a general formalism of

newly developed SPS interferometry technique.16 Theptical setup implementing this method17 is described inig. 1.A spatially filtered and collimated monochromatic light

ource of wavelength � is illuminating an inspected reflec-ive object via a beam splitter and generates a normallyncident plane wave. The wavefront reflected from the ob-ect is denoted f�x ,y�=A�x ,y�exp�i��x ,y��. In first approxi-

ation, the complex amplitude of this wavefront in alane immediately adjacent to the object plane encodeshe height topography (height map) of the object. Indeed,he height map h�x ,y� is related to the phase map � of theeflected wavefront by the equation

��x,y� = − 4�h�x,y�

�mod 2�, �1�

here mod 2� is the phase modulo 2�. The height mapan be reconstructed unambiguously from the phase map,rovided that the pixel-to-pixel height differences are lesshan � /4. To determine the field of view, a field stop D isnserted in the object plane. It ensures that A�x ,y� (andherefore also f) vanishes outside the domain D, i.e.,�x ,y�=0 if �x ,y��D. No particular assumption is made onhe geometrical shape of the field stop. The area of theeld stop is denoted by A�D�.The object is placed in the front focal plane of the first

maging lens, having a focal length L, such that the wave-ront reflected from the object is Fourier transformed byhe lens. At the imaging lens’s back focal plane, an opticalanipulator (OM), such as a liquid-crystal-based device,

perates phase and amplitude manipulations in a certainart of the plane. The OM is composed of two parts: anctive area � and a complementary nonactive area �c.he area of the active part � is denoted by A��. No par-icular assumption is made on the geometrical shape ofhe active area �. It can be circular, rectangular, or anyther shape. The phase delay actively introduced between

ig. 1. General characteristic optical setup of the SPS interfer-metry technique.

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he two parts � and �c is denoted by �, and the amplitudeodification of the active area (or apodization factor),hich is the amplitude transmittance ratio of the two ar-as, is denoted by k. The apodization factor is indepen-ently controlled from the phase delay �. Once manipu-ated by the OM, an additional lens performs the inverseourier transform of the transformed wavefront and com-letes the imaging. Various different manipulations �� ,k�re applied by the OM, interfering the spatial parts of thenspected wavefront one with the other and generatingifferent interference patterns. The resulting intensitiesre collected by a CCD camera, which measures the vari-us intensities at every pixel.

Thus, although the system is based on the physicalrinciple of interference, no reference beam and no mov-ng parts are used. This suggests that the SPS methodas some key advantages, such as insensitivity to vibra-ions and air turbulence, as well as the ability to obtainhase-shift interferometry accuracy with minimal ap-roximations and fast reconstructions. Furthermore, suchn optical setup permits general wavefront measurement,ncluding imaged wavefronts formed by any aberration-orrected optical system, introduced as the imaging lensf the setup.

From now on, throughout the article, we narrow theiscussion and present the analysis only for the caseshere the active area � is central and the nonactive areac is surrounding it. In Sections 3 and 4 we introduce the

ormalism used to model the optical setup above.

. IMAGE FORMATION IN SPATIALHASE-SHIFT INTERFEROMETRY

n the optical setup described in Section 2, the field im-ediately in front of the focal plane containing the OM is

roportional to F�u ,v�, where F designates the Fourierransform of the original wavefront f�x ,y� and �u ,v� rep-esent the spatial frequencies. In the Fourier plane, thehysical dimensions �x� ,y�� are given by x�=u�L, y�v�L. At the Fourier plane, the wavefront is multipliedy H�x� ,y��, the transmission function of the OM,

H�x�,y�� = k exp�i��G��x�,y�� + �1 − G��x�,y��

= G��x�,y��k exp�i�� − 1� + 1, �2�

here G� is the characteristic function of the active areathat equals 1 in � and vanishes outside �. Assuming for

implicity that the two imaging lenses have the same fo-al length, the field at the image plane is proportional to

�k exp�i�� − 1�S�x,y� + f�x,y�. �3�

e have defined

S�x,y� � f�x,y� � g�x,y� �4�

s the two-dimensional convolution of the complex ampli-ude of the original wavefront f�x ,y� with the rescaled in-erse Fourier transform g�x ,y� of the characteristic func-ion G� of the active area �. Explicitly, the function g�x ,y�s given by the equation

g�x,y� = F−1�G��u�L,v�L��, �5�

here F−1 stands for the inverse Fourier transform. Weote two interesting particular cases. In the first case,here � is a disk with diameter �, g�x ,y� is given by

g�x,y� =1

2� �

�L�2

�

J1��

�L�x2 + y2�

��

�L�x2 + y2

, �6�

here J1 is a Bessel function of the first kind.In the second particular case, where � is a square of

ide length �, g�x ,y� is given by

g�x,y� = � �

�L�2

sinc� �

�Lx�sinc� �

�Ly� . �7�

nder the restriction of a central active area � consideredn this article, the function S can be interpreted as theow pass of the original wavefront f�x ,y�, containing theower frequencies of the function f�x ,y� that pass throughhe active area � of the OM. Formula (3) shows that theeld at the image plane is the interference of two waves:he wave diffracted by the object and the wave of its low-ass function S multiplied by the constant �k exp�i��−1�.n the SPS scheme, the S function can be considered theeference beam, and it plays a role similar to that of thexternal reference beam in standard phase-shift interfer-metry (PSI) theory.

Nevertheless, there are several crucial differences be-ween the PSI and the SPS theories. First, as discussed inection 2, the SPS is a common-path method, where theeference beam is part of the main beam. Therefore, aspposed to PSI, the SPS is intrinsically insensitive to vi-rations in the Z direction (perpendicular to the wave-ront). Second, the SPS method can be applied to anyavefront. A third difference is that the reference beam in

he SPS is object dependent and depends in a global man-er on the object. As will be seen in Section 5, this globalharacter of the SPS theory has deep implications. One ofhem is that the SPS is less sensitive to noise. Anothermplication is that reconstruction algorithms are heavierhan those for the PSI. In the limit case of the Zernike ap-roximation, where the phase-shifting element in the fo-al plane is assumed to be a disk of infinitely small di-ension �, formula (6) shows that g is a constant function

roportional to �� /�L�2, and so the low-pass function S isconstant proportional to �A�D�A�� /�2L2f, where f is

he average value of the original wavefront f over the ap-rture stop. It should be noted that, even in this limitase, the constant �S� has to be reconstructed, a step notresented by the existing methods.Returning to the general case, it follows from expres-

ion (3) that the image intensity grabbed by the CCD isiven by

I��x,y� = ��k exp�i�� − 1�S�x,y� + f�x,y��2. �8�

e present the analysis for a fixed apodization factor knd varying phase delays � and thus denote the differentntensities by I �x ,y�. From Eq. (8) it can be seen that,
�

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hen k=1 and �=0, there is no indication of the phaseata in the resulting intensity grabbed by the CCD. Thentensity then reduces to the amplitude reflectance of thebject, as in standard imaging:

I0�x,y� = �f�x,y��2 = A2�x,y�. �9�

e recall that no light is reflected from the object outsidehe field stop: f�x ,y�=0 outside the domain D. However, inhe image plane, beyond the image of the domain D, thentensity is given by

I��x,y� = �k exp�i�� − 1�2�S�x,y��2. �10�

e refer to this intensity, produced by the low-pass func-ion S, as the halo. The halo intensity is a function of thehase shift � and of the apodization coefficient k of theM.In an alternative expression to Eq. (8), the phase-anipulated intensities grabbed by the CCD can be ex-

ressed using the complimentary function (or high-passunction) R= f−S of the original wavefront:

I��x,y� = �k exp�i��S�x,y� + R�x,y��2. �11�

hroughout this article, we illustrate the SPS theory us-ng computer simulations. The simulative target objecte shall repeatedly consider consists of four pads of dif-

erent heights and reflectances, as shown in Fig. 2. Theptical parameters used for the simulation are �740 nm, focal length L=220,000 nm, and the field stop

ig. 2. Four-pad simulative object of different heights and in-ensity reflectances: �reflectance, height�= �1,−40 nm� , �0.2,

100 nm� , �0.28,80 nm� , �0.76,0 nm�. (a) Three-dimensional viewf the object. (b) Reflectance map.

ig. 3. SPS simulative intensities. (a) Standard imaging (k=1,=0). (b) Phase-manipulated intensities (k=0.45, �=�). Observehe halo around the field stop in (b).

s a disk of diameter d=1.31 mm. The central area of theM is assumed to be a small disk of diameter �160 �m. The attenuation factor is k=1/�0.2 for all thehase delays (the intensity of the light passing throughhe nonactive area of the OM is attenuated by a factor of/5). The resulting SPS intensities (in the image plane)

or this target object are shown in Fig. 3. Note the haloround the image of the field stop in Fig. 3(b).In the next sections we show that the wavefront f�x ,y�

an be reconstructed with a very high accuracy from ateast three different intensity maps I��x ,y� obtained fromifferent phase shifts �.

. INTRODUCING THE SPATIALIRY NUMBER

n this section we show that several critical parameters ofhe setup can be renormalized to a unique dimensionlessumber. To do so, we will rescale both the field stop in thebject plane and the active area of the OM in the Fourierlane to unit areas. For the object plane normalization,etting the field stop to a unit area is equivalent to res-aling the object plane (in each dimension) by a factor ofA�D�. We denote the dimensionless coordinates of theescaled system by �x , y� so that �x ,y�= ��A�D�x ,�A�D�y�.n the dimensionless coordinate system, the function�x ,y� becomes f�x ,y�= f��A�D�x ,�A�D�y�= f�x , y�. Simi-arly, S�x , y�=S�x ,y�. According to these definitions, in theimensionless coordinate system, Eq. (4) becomes S�x , y�f�x , y�� g�x , y�, where

g�x, y� = A�D�g��A�D�x,�A�D�y� = F−1�G��u�L,v�L��

= F−1�G�� L

�A�D�u,

�L

�A�D�v� . �12�

e turn now to the normalization of the Fourier plane.or this normalization, setting the active area � of theM to a unit area � is equivalent to rescaling the Fourierlane (in each dimension) by a factor of �A��. Using thisormalization, we can express G� as

�� L

�A�D�u,

�L

�A�D�v�

= G�� L

�A�D�A��u,

�L

�A�D�A��v� . �13�

e define a unitless number as the characteristic dimen-ion of the manipulated area �A��, normalized by theiry disk size of the system �L /�A�D�. Since this number

s the spatial area being manipulated, expressed in unitsf the Airy disk diameter, we name it the spatial Airyumber (SAN). Mathematically, the SAN is expressed as

SAN =�A��A�D�

�L. �14�

t follows from Eqs. (13) and (14) that expression (12) cane rewritten as

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g�x, y� = F−1�G�� u

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he definition of the normalized low-pass function S�x , y�nd Eq. (15) show that the normalized low-pass functionepends on the object, on the SAN, and on the geometricalhapes of � and D but does not depend directly on theirarticular sizes. It follows that the SPS intensities [Eq.8)] depend (only) on these factors and naturally on thehase shift � and the apodization coefficient k. Equation15) shows that, when SAN→0, g degenerates to a nearlyonstant function in the field stop g�x , y��SAN2. There-ore, the same conclusion holds for the low-pass S�x , y�

SAN2�Df. This is the classical Zernike approximation.hroughout the remainder of the article, all the functions

nvolved are assumed to be rescaled. In particular, theathematical domain D is now assumed to be of unit

rea. However, the ˜ conventation adopted in this sectiono specifically designate rescaled functions is abandonedor the sake of clarity.

. RECONSTRUCTING THE WAVEFRONTn this section we detail specific methods to reconstructhe original wavefront reflected from the object within aery good accuracy. In a previous publication,16 research-rs suggested a direct estimation of the reflected phase,ased on an iterative method, generalizing Fienup’s itera-ive Fourier transform algorithm.18 However, this sug-ested method did not converge for all instances. In thisrticle we present new reconstruction algorithms that aressentially not iterative and converge for all instances.enoting the phase of the complex function S�x ,y� by�x ,y� and the amplitude by C�x ,y� such that S�x ,y�C�x ,y�exp�i��x ,y��, Eq. (8) can be rewritten as

I��x,y� = ��k exp�i�� − 1�C�x,y�exp�− i�x,y�� + A�x,y��2,

�16�

here �x ,y� is phase difference �x ,y��x ,y�−��x ,y�.s can be seen in Eq. (16), the intensity does not directlyepend on the phase of the object ��x ,y� but on the func-ion �x ,y� only. This implies that ��x ,y� cannot be recon-tructed directly and locally from the intensities, andome global mathematical techniques have to be used.heoretically, only the two independent unknown func-ions A�x ,y� and ��x ,y� have to be determined, and sonly two intensity images are needed [naturally, A�x ,y�an also be obtained directly using any imaging setup].owever, the amplitude and the phase C�x ,y� ,��x ,y� de-end in a global and complex way on f�x ,y�. To simplifyhe reconstruction process (and to increase the accuracy),�x ,y�= �S�x ,y�� is treated and reconstructed essentiallys an independent unknown at the cost of using three in-ensity images instead of two.

We divide the reconstruction of the inspected wavefrontnto four major steps, reviewed in the corresponding sub-ections of this section. In Subsection 5.A we computerom at least three intensities three local functions de-ending solely on C�x ,y�, A�x ,y�, and �x ,y�. In Subsec-ion 5.B we detail the reconstruction of C�x ,y�. In Subsec-

ion 5.C the reconstructions of A�x ,y� and �x ,y� areetailed. Finally, in Subsection 5.D, the function ��x ,y� iseconstructed from the three functions C�x ,y�, A�x ,y�, and�x ,y�. Since ��x ,y�=�x ,y�+��x ,y�, this concludes the re-onstruction.

. Basic Local Functions of the Reconstructionhe first step of the reconstruction is the computation of

hree local functions that extract the relevant informationrom the different intensity images.

. Main Expressiono simplify the intensities’ dependence on the controllednd known phase delays � and separate the coordinatesx ,y� dependence, we express Eq. (16) as

I��x,y� = 0�x,y� + c�x,y�cos � + s�x,y�sin �, �17�

here

0�x,y� = A2�x,y� + �1 + k2�C2�x,y�

− 2A�x,y�C�x,y�cos �x,y�,

c�x,y� = 2kA�x,y�C�x,y�cos �x,y� − 2kC2�x,y�,

s�x,y� = 2kA�x,y�C�x,y�sin �x,y�. �18�

sing standard linear least-squares techniques, one caneconstruct the functions 0 ,c ,s from at least three dif-erent intensity maps I�1

, . . . ,I�j, j�3. The local 3�3 ma-

rix that has to be inverted for the linear least-squaresrocedure depends only on the phases �1 , . . . ,�j and not onhe coordinates �x ,y�. Hence the computations involvedre quick and purely local.

. Alternative Expression and Comparison with Phase-hift Interferometryt should be noted that the functions 0 ,c ,s can be ex-ressed in an alternative way, which is more similar tohe PSI form. Recall that R�x ,y� designates the high passf the wave f�x ,y�, and set R�x ,y�=B�x ,y�exp�i �x ,y��. Us-ng Eq. (11), one has

I��x,y� = �k2C2�x,y� + B2�x,y��

+ 2kC�x,y�B�x,y�cos�� + �x,y� − ��x,y��.

�19�

herefore, comparing Eqs. (17) and (19), we obtain

0�x,y� = k2C2�x,y� + B2�x,y�,

c�x,y� = 2kC�x,y�B�x,y�cos� �x,y� − ��x,y��,

s�x,y� = 2kC�x,y�B�x,y�sin� �x,y� − ��x,y��. �20�

vidently, expression (19) is similar to that of the PSIheory. This means that, using a standard PSI reconstruc-ion method (such as the four-bucket method5), one caneconstruct locally and very easily �x ,y�−��x ,y�, the dif-erence between the phase of the high pass and the phasef the low pass. However, the phase ��x ,y� of the originalavefront cannot be deduced directly, locally, and accu-

ately from the difference �x ,y�−��x ,y�, even if ��x ,y� is

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ssumed to be a constant as it is in the Zernike approxi-ation. Indeed, since f=S+R, we obtain

�x,y� = ��x,y� − ��x,y�

= arg�1 +B�x,y�

C�x,y�exp i� �x,y� − ��x,y�� . �21�

ence, the ratio �B�x ,y�� / �C�x ,y�� has to be determined inrder to first compute �x ,y� and then ��x ,y�.

. Reconstructing C„x ,y…fter obtaining the functions above, we describe in thisubsection a few alternative methods for the reconstruc-ion of C�x ,y�. As will be demonstrated, this cannot beone by purely local computations, and a global arguments required.

. Analytical Argumentn an ideal system, the function S can be obtained ana-ytically: Since S is the low pass of the wavefront function, it has to be a smooth function and therefore can behown to be analytical. Since C2�x ,y� is therefore alsonalytical and can be directly obtained outside the fieldtop by

C2�x,y� =1

1 + k20�x,y� = −1

2kc�x,y�, �22�

t would follow that this should be enough to obtain2�x ,y� over the field stop. However, such an approach isoise sensitive, making it impractical.

ig. 4. Various profiles of the functions C±, C= �S�, and B /k�R� /k. (a) Reconstructed functions C− (dashed curve) and C+

dotted curve) for the basic example. (b) Low-pass C (solid curve)nd attenuated high-pass B /k (dashed–dotted curve) for the ba-ic example. Note that C (solid curve) can be reconstructed out ofhe two local functions C± in (a) using a smoothness argument.c) Same as (a) but CCD noise was added to the simulation. Ob-erve that a simple smoothness argument cannot be used any-ore to reconstruct C. Note also that the noise is amplified in ar-

as where C± are close to each other. (d) Low-pass C (solid curve)nd attenuated high-pass B /k (dashed–dotted curve) of a per-ectly flat mirror. Note that, within the field stop, both C and Bre smooth.

. Smooth Continuation Algorithmse thus turn now to developing an alternative way to

ompute the function C2�x ,y� within the field stop. Fromormulas (18) it follows that X= �kC�x ,y��2 satisfies theecond-degree equation

X2 − 0X +c

2 + s2

4= 0. �23�

s a result, at any position �x ,y� ,C�x ,y� equals one of thewo solutions:

C±�x,y� =1

k�20�x,y� ± �0

2�x,y� − c2�x,y� − s

2�x,y��1/21/2.

�24�

n general, the actual function C coincides globally nei-her with the function C− nor with C+ but is a patchworkf the two solutions C−,C+ at different locations x ,y [seeig. 4(a)]. A global smoothness argument is necessary tolue adequately the local solutions C−,C+ [see Fig. 4(b)] tobtain the actual low-pass function C. It turns out that,ithin the field stop, this smoothness argument is notnough to build unambiguously C out of C−,C+ [see Fig.(d) for the case in which both the low pass and the highass are smooth functions]. Therefore, one has to add theomplementary condition (22) outside the field stop (seeppendix A for details). Namely, C is a smooth functionver the whole plane satisfying Eq. (22) outside the fieldtop, and that can be constructed within the field stop byppropriately gluing C±. On the basis of these properties,ifferent algorithms can be applied to reconstruct C. Fornstance, one can develop a region-growing-typelgorithm that starts with a region � where theunctions �C±�� are distant enough and iteratively con-truct the smoothest possible function by gluing C±.

The smooth continuation algorithms are also noiseensitive—the idea that around any point �x ,y� at leastne of the two functions C± is locally perfectly smootholds only in an ideal noise-free world. In practice, nonef the functions C± is really locally smooth [see Fig. 4(c)].nother drawback of these methods is that in transitionegions, where the solutions C± nearly meet, errors aremplified, and generally none of the local solutions C± re-lly coincides with the genuine function C [Fig. 4(c)]. Ob-iously, the gluing idea is problematic in these regions.he sensitivity to noise of the local functions C± is dis-ussed more in detail in Section 6. These arguments indi-ate that a robust noise-insensitive algorithm has to be aslobal as possible.

. Global Polynomial Approximatione present in this subsection another approach to recon-

truct C= �S�, which is global in nature and is thereforeess noise sensitive than the previous methods. This glo-al method is based on the fact that if the SAN is smallnough, then the low-pass S can be very accurately ap-roximated in the field-stop domain by a polynomial of aow degree. Indeed, it follows from Eq. (15) and the facthat the active area � is bounded that the function g cane approximated by a polynomial of low degree if the SANs small enough. Moreover, the convolution of an arbitrary

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unction by a polynomial is a polynomial of the same de-ree. Naturally, if S is a polynomial, then C2= �S�2 is also aolynomial. This polynomial has to satisfy the second-egree equation in C2 [Eq. (23)]. The main idea of this sec-ion is that, instead of determining C2 locally, the (global)olynomial coefficients that best fit Eq. (23) are obtained.his is a nonlinear optimization problem, and severaltandard optimization algorithms exist and can be ap-lied to solve this problem.19

Naturally, polynomial approximation techniques areumerically and computationally meaningful only for lowegree. We have found out that, for reasonably smallANs, approximating S by a degree-four or even a degree-wo polynomial is a very good approximation that ensures/400 (nanometer level in topography) accuracy in theverall reconstruction. This approximation is also muchetter than the Zernike approximation, equivalent in thisontext to approximating S by a constant function. Theernike approximation is effective only for a very smallAN and can thus induce error of tens of nanometers. Us-

ng this algorithm with a degree-six (degree-four) approxi-ation, one can reconstruct the function C for the simu-

ative target of Fig. 2 with an accuracy better than 0.5%5%), leading to an overall reconstruction accuracy betterhan 1 nm �2 nm�. The same object reconstructed usinghe standard Zernike approximation results in a 47% ac-uracy for the function C and more than 100 nm errors inhe overall reconstruction.

The algorithm described in this subsection, althougheing noise insensitive, is computationally heavier thanhe ones discussed in Subsection 5.B.2, and its complexityncreases rapidly with the SAN. The smooth continuationlgorithms, on the other hand, accommodate well with auch higher SAN but are more sensitive to noise. The de-

ision of which method to use depends on the specific sys-em parameters.

. Obtaining the Phase Difference �nce the amplitude of the low-pass C= �S� is obtained, theext step is the reconstruction of the amplitude of the re-ected wave A�x ,y� and the reconstruction of the differ-nce �x ,y� between the phase of the reflected wave andhe phase of the function S. This requires only local com-utations. From Eqs. (18), it can be seen that the ampli-ude A�x ,y� is given by

A�x,y� = �0�x,y� +1

kc�x,y� + �1 − k2�C2�x,y� 1/2

.

�25�

t should be noted that in all cases, except when k=1,here Eq. (25) reduces to

A�x,y� = �0�x,y� + c�x,y��1/2, �26�

he function C is needed to compute the amplitude A. Itollows from Eq. (18) that the phase �x ,y� can be com-uted from C and 0 ,c ,s using the equation

�x,y� = arg�c�x,y� + 2kC2�x,y� + is�x,y��, �27�

here the argument of a complex number denotes itshase.

. Phase of the Low Passnother major part of the reconstruction algorithm (and

ts final step) is the reconstruction of the phase of the low-ass ��x ,y�. Once ��x ,y� is computed, ��x ,y� is obtainedsing �=+�. Equation (16) shows that the intensityrabbed by the CCD is not directly affected by �. Thus,nce again, global arguments are necessary to obtain thisunction. Figure 5 shows a profile of the phase ��x ,y� forhe simulative target described in Fig. 2. The goal of thisubsection is to describe how can this phase be recon-tructed.

The starting point for the reconstruction of ��x ,y� is theonvolution formula (4) defining the S function. We definehe following function:

��x,y� �A�x,y�exp�i�x,y��

C�x,y�. �28�

ince at this step of the reconstruction we have alreadybtained C, A, and , the function � is known. Using thisefinition, we can rewrite Eq. (4) as

S = ��S� � g. �29�

his is a linear integral equation in S. Once solved, thehase � of the S function is obtained. In the remainder ofhis subsection we will present various alternatives forhe reconstruction of ��x ,y�, based on several approachesor solving Eq. (29) above.

. Direct Discretization Approachhe first alternative for the reconstruction of � is basedn discretizing equation (29). Let �xi ,yj�, 1� i, j�m, be aiscrete regular grid of a rectangular domain containinghe field stop. Since Eq. (29) is a linear integral equation

ig. 5. Two profiles: (a) phase of the low-pass � for the basic ex-mple [the phase is expressed in nanometers using formula (1)],b) residual error after the global polynomial approximation al-orithm is applied.

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n S, it can be written in a form of operators. A discreteersion of this form of Eq. (29) on this domain is thus

s = Ls, �30�

here s is the m2 vector si,j=S�xi ,yj� and L is the m2

m2 matrix Li,jk,l=��xk ,yl�g�xi−xk ,xj−xl�. The vector s is

n eigenvector of eigenvalue 1 of the matrix L. There areany well-known methods to compute eigenvectors19

any of these methods can be applied to reconstruct s).he direct discretization approach advantage is that itsomplexity does not depend very much on the SAN. Onhe other hand, its complexity increases dramaticallyith m, and some sparse-matrix arguments are quicklyecessary. Nevertheless, this approach is powerful forpecific cases.

. Global Polynomial Approximationnother alternative for the reconstruction of � utilizes

he polynomial approximation of the S function. Wetated in Subsection 5.B.3 that the low-pass function S is

polynomial of low degree, providing that the SAN ismall enough. Once more, for a relatively small SAN, theolynomial approximation idea is the key to an effectiveoise-insensitive global solution of Eq. (29). Briefly, if theAN is small enough, then the function g�x ,y� is a poly-omial of low degree. We substitute g�x ,y� by its polyno-ial approximation gn�x ,y� of degree n in the integral

quation, and we look for the polynomial S=�i+j�nsi,jxiyj

hat satisfies

S = ��S� � gn. �31�

hus, the complexity of the problem has been consider-bly reduced. Instead of trying to compute the function Sn each location as in the direct discretization method, oneooks for only a few coefficients. Applying this algorithmdegree eight) to the simulative target of the phase, we re-onstructed the phase � with a peak-to-peak error ofbout 1 nm [see Fig. 5(b)].

. More General Approximations we discussed in Subsection 5.B.3, polynomial approxi-ations are stable techniques only for low-degree polyno-ial approximations. This forces one to keep the SAN

mall enough. Actually, the global polynomial approxima-ion method described above can be improved in such aay that similar accuracy can be obtained while oneorks with polynomials of much lower degree (half in

ome cases!), and therefore working with higher SANs isossible. Roughly speaking, the idea is to approximate theonvolutant function g in the convolution formula S= f�gy a wider class of functions than the class of polynomialshat still ensures that the function S is a polynomial. Aetter polynomial approximation of S can be obtained byetter approximating the convolutant g (see details in Ap-endix B).

. NOISE SENSITIVITY ANDECONSTRUCTION ACCURACYhen the method developed in Section 5 is implemented

nto a physical system, there are many potential sources

f error. These errors include the quality of the phase-hifting element (and in particular the nonuniformity ofhe phase shift in the active area of the OM), the uncer-ainty on the effective value of the phase shifts �s, theuality of the optical elements, the noise and quantizationf the CCD camera, the coherent noise induced by the la-er (parasite diffraction patterns, fringes, speckles), etc.hese errors, depending on their nature, affect the recon-truction process in different ways. For instance, the CCDoise or the laser coherent noise just add high-frequencyoise to the system, whereas errors in the phase-shiftalue � or low-quality optical elements induce model de-iation that can be difficult to detect and compensate.ne of the challenges of the SPS method is the construc-

ion of a reliable confidence map for the overall recon-truction process: a map that can indicate erroneous pix-ls or measurements. As opposed to PSI, whereomputations are local, the global character of the SPSethod implies that a group of erroneous pixels can theo-

etically affect the measurement over the whole field stop.o constructing a reliable confidence map is an extremelyelicate task in the SPS theory. In this section, we presentbrief analysis of the reconstruction sensitivity to noise,

ropose a few possible techniques to determine measure-ent quality, and, finally, discuss the overall reconstruc-

ion accuracy that can be achieved. Many error detectionrocedures can be inserted at different steps of the recon-truction. Each of these measurement quality indicatorsr confidence maps has its own interpretation and sensi-ivity to specific noises.

. Contrast Modulation Maphe SPS intensity grabbed by the CCD camera variesith the phase delay �. From Eq. (19), we define (by anal-gy with the phase-shift theory), the contrast modulationap as

M�x,y� =�Max

�

I��x,y� − Min�

I��x,y��Max

�

I��x,y� + Min�

I��x,y�=

2kC�x,y�B�x,y�

k2C2�x,y� + B2�x,y�.

�32�

t follows immediately from Eqs. (18) that the modulationap can also be expressed using the functions 0 ,c ,s by

he equation

M�x,y� = �� c�x,y�

0�x,y� 2

+ � s�x,y�

0�x,y� 2�1/2

. �33�

he contrast map is a very important indicator of mea-urement quality. Measurements with low-contrastodulation result in a low signal-to-noise ratio. Very low

ontrast can also introduce other effects such as surfaceisplanarity and algorithms’ instability. Surprisingly, ase shall see in Subsection 6.C, very high contrast intro-uces other problems.The contrast modulation is a direct function of the SAN

umber introduced in Section 4. For a small SAN, thearger the SAN, the higher is the contrast modulation ofhe SPS intensity images, but the heavier are the recon-truction computations. The contrast modulation also de-reases for a very high SAN. Figure 6 shows the depen-

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ence of the contrast modulation map on the SAN for theimulative target described in Fig. 2. This dependencean be understood by recalling that the contrast dependsn the ratio of the two interfering beams, one passinghrough the active area of the OM and the other passinghrough the nonactive area. It should be noted thathanges in the apodization factor k (in reconstructing dif-erent objects) are intended to optimize the contrast, byeducing the intensity passing through the nonactive areao match it to the intensity passing through the activerea. Since the quality of reconstruction is sensitive tohanges in contrast, it follows that it is also sensitive tohanges in k.

. Reconstruction’s Noise Sensitivity of the Localunctions �0 ,�c ,�sn this subsection we analyze the noise sensitivity of theinear least-squares procedure involved in the computa-ion of the functions 0 ,c ,s, described in Subsection.A. We also show that a confidence map can be naturallyssociated with this process. We denote, respectively, by�x ,y� and I�x ,y� the 3D and N-dimensional vectors

0�x ,y� ,c�x ,y� ,s�x ,y��T and �I�1�x ,y� , . . . ,I�N

�x ,y��T,here T denotes a transposed vector. We introduce also

he matrix

P � �1 cos �1 sin �1

] ] ]

1 cos �N sin �N� . �34�

y definition of the least-squares procedure, at any loca-ion �x ,y� ,��x ,y� is the vector V that minimizes the Eu-lidean norm �I�x ,y�−PV�2:

�I�x,y� − P��x,y��2 = MinV

�I�x,y� − PV�2. �35�

he sensitivity to noise of the linear least-squares proce-ure depends on the so-called condition number20 ��U� ofhe matrix U=PTP. By definition, the condition number�U� of the symmetric matrix U is simply the ratio of theighest eigenvalue by the smallest eigenvalue of U. Actu-

ig. 6. Average within the field stop of the contrast modulationap M�x ,y� of Eq. (32) for the simulative target described in Fig.

.

lly, the condition number ��U� does not only control theensitivity to noise but also the sensitivity to errors in �i.he smaller the condition number, the more robust is the

east-squares procedure. It can be shown (see Appendix C)hat, in general, ��U��2 and that the minimum value�U�=2 is reached if (but not only if) �1 , . . . ,�N are chosenqually distributed:

�p = �0 +2��p − 1�

N, 1 � p � N. �36�

rom the minimization property [Eq. (35)], it is clear thathe term �I�x ,y�−P��x ,y��2 for every location �x ,y� is aatural confidence map for the least-squares process. Ofourse, to really play the role of a confidence map, thisap has to be suitably normalized.

. Noise Sensitivity of the Reconstruction of C„x ,y…e discuss in this subsection the sensitivity to noise ofq. (23) and, in particular, of the two local solutions C±

2

efined by Eq. (24). As we have discussed in Subsection.B.2, in an ideal error-free situation, C2 is a smooth func-ion that can be reconstructed by adequately gluing thewo solutions C±

2. Noise and errors significantly compli-ate this idyllic situation. For instance, high-frequencyoise such as CCD camera noise, coherent noise, or lightcattering of certain diffusive materials introduce high-requency noise to both C±. In such cases, none of theunctions C± are locally smooth, even through C still al-ost coincides (up to high-frequency noise) with one of

he two functions C± [see Fig. 4(c)]. The noise in transi-ion regions where the functions C± are close to each others amplified [see Fig. 4(c)]. Indeed, the solutions of aecond-degree equation are much more sensitive to noisehen the discriminant d of the equation is small. If � is anrror term added to the discriminant d of a second-degreequation, then the resulting error term in the solutions ofhe equation is proportional to � if ��d and �� if ��d.

We define d�x ,y��02�x ,y�−c

2�x ,y�−s2�x ,y� as the dis-

riminant of the specific Eq. (23). It is evident that theiscriminant d�x ,y� and the contrast modulation map�x ,y� are connected by the relation:

1 − M2�x,y�

02�x,y�

= d�x,y�. �37�

his implies in particular that d�x ,y��0 if M�x ,y��1:he discriminant of Eq. (23) is small where the contrastodulation map is high. In other words, when the con-

ig. 7. (a) Residual height map error (scale in nanometers) (b)rofile of the error map.

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rast is very high, the noise is amplified, and the smoothontinuation algorithm is difficult to apply.

. Simulative Target: Accuracy of the Reconstructionecall the simulative target and the optical parameterssed for the simulation described in Section 3. The SAN

or this example equals 1.3. With algorithms discussed inhe previous sections, the height map of the basic examples reconstructed with accuracy better than 1.5 nm peak toeak, and 1.5 Å in average, over the whole field-stop do-ain. The residual error map is shown in Fig. 7.

. SUMMARYe have described in detail a wavefront reconstructionethod, based on a unique optical setup, utilizing an op-

ical manipulator in the optical axis instead of the con-entional interferometry reference beam. The resultingntensities of the optical setup were analytically obtainednd analyzed for their dependence on various system pa-ameters. A dimensionless number, named the spatialiry number, was introduced and shown to substantiallyovern these intensities. Specific reconstruction algo-ithms were developed to accurately obtain the phase andmplitude of a wavefront out of the optical setup’s result-ng intensities. Various alternative methods, for differentystem parameters, were developed. The reconstruction’s

oise sensitivity and accuracy were also analyzed, andeveral techniques to determine measurement qualityere introduced. It should be noted that, although no the-retical proof of uniqueness of the reconstruction solutions provided, it was applied to many tens of objects, and inll cases, without any exceptions, the correct solution wasbtained.

The described wavefront measurement method, includ-ng the optical scheme and the reconstruction algorithms,as implemented into an industrial measurement tool, inhich a 3D measurement sensor is integrated with ahite-light microscope. This 3D metrology tool is cur-

ently being used to measure various objects of differenteights and shapes. Figure 8 depicts such a measurementf a 88 nm certified step height, by VLSI standards. Threeifferent intensities [Figs. 8(a)–8(c)] are the output of theptical scheme as collected by the CCD camera. These in-ensities are analyzed to obtain the wavefront reflectedrom the measured object. The phase of this wavefront issed to characterize the height topography of the objectFig. 8(d)]. Actual measurements of various objects anddditional certified step-height standards show a higheasurement accuracy (better than 1 nm) and repeatabil-

ty (better than 1 Å 1�), as could be expected by the the-retical considerations. The implementation of thisethod and the details of the metrology tool, as well as

ig. 8. Chrome-plated standard of 88 nm step height, as measured by the SPS method. (a)–(c) Three intensities are recorded by theCD camera, where the X and Y axes are in CCD pixels and the bar is in gray levels. (d) The resulting height map of the inspected object,s reconstructed by the system, is presented as a 3D view, where the X and Y axes are in micrometers and the bar is the height inanometers. (e) A profile across the sample indicates its step height.

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he analysis of its performance, will be discussed in futureublications.

PPENDIX A: IN THE FIELD STOP,MOOTHNESS ARGUMENT IS NOTNOUGHithin the field stop, the smoothness property of C is not

nough to build unambiguously C out of the two localunctions C−, C+. In order to understand this phenom-non, we note that in the SPS theory, the low-pass S andhe high-pass R= f−S of the wavefront f play symmetricaloles. For instance, phase shifting the low pass by � isquivalent to phase shifting the high pass by −�. Usingq. (18), one can check immediately that �R�2 also satisfies

he second-degree Eq. (23). This means that, by properlyluing the local functions C±, one can construct also theodule of the attenuated high-pass �R� /k. Consider now

he particular case of a perfectly flat mirror. Then f van-shes outside the field stop and is the unity function in-ide the field stop. The results are that within the fieldtop not only is S smooth but R= f−S is also smooth.aturally, the same holds for the square of the modules,nd we have finally exhibited a wave f such that twomooth functions can be constructed out of C± inside theeld stop [see Fig. 4(d)].

PPENDIX B: APPROXIMATING THEONVOLUTANT g BY A WIDER CLASSF FUNCTIONS THAN THEOLYNOMIAL FUNCTIONS

n Subsection 5.D it was claimed that the polynomial ap-roximation used to reconstruct the phase of the low passan be modified so that merely the same accuracy is kepthile one works with polynomials of significantly loweregree. We have claimed in Subsection 5.B.3 that if g cane accurately approximated by a polynomial of low de-ree, then the same holds for the convolution

S�x,y� =� f�s,t� � g�x − s,y − t�dsdt. �B1�

he polynomial approximation of S is obtained by con-olving f with the polynomial approximation of g. That is,he function g�x−s ,y− t� in the integral (B1) is replaced by

polynomial function in the four variables x ,y ,s , t. Theey point is that it is enough to require that the function�x−s ,y− t� be a polynomial in the �x ,y� variables only, inrder to ensure that S�x ,y� is a polynomial. In otherords, if the function g�x−s ,y− t� can be accurately ap-roximated by a function of the form

gn�x,y,s,t� = �i+j�n

gi,j�s,t�xiyj, �B2�

here the functions gi,j�s , t� are general functions (andot only polynomials), then S�x ,y� can be accurately ap-roximated by a polynomial of degree n. Classical math-matical techniques provide decompositions of the typeB2). Clearly, we have relaxed the approximating condi-ion: For a given polynomial degree n in �x ,y�, the func-

ion g�x−s ,y− t� can be more accurately approximated byfunction gn�x ,y ,s , t� of the form (B2) than by a polyno-ial in the four variables �x ,y ,s , t�. When one replaces

�x−s ,y− t� with gn�x ,y ,s , t� and sets S=�i+j�nsi,jxiyj, theystem S= ��S��gn reduces again to a linear system of di-ension ��n+1��n+2�� /2.

PPENDIX C: SENSITIVITY TO NOISE OFHE LEAST-SQUARES PROCEDURENVOLVED FOR THE RECONSTRUCTIONF THE LOCAL FUNCTIONS �0 ,�c ,�s

ecall the notations of Subsection 6.B. For the sake oflarity, we introduce for this appendix only the notationsj=cos �j and sj=sin �j. Using these notations, one hasrom Eq. (34):

U � PTP = � N � cj � sj

� cj � cj2 � cjsj

� sj � cjsj � sj2 � . �C1�

ince U is definite positive, the eigenvalues of U aretrictly positive. We will prove that the condition number�U� satisfies ��U��2. If �j are chosen equidistributed,amely, �j= �2��j−1�� /N+�0 for 1� i�N, then ��U�=2.ndeed, in such case the matrix U is diagonal, and the el-ments on the diagonal are N, N /2, and N /2.

We prove now that ��U��2. Let us denote by �j thehree eigenvalues of U. By definition, �j are the roots ofhe characteristic polynomials Q��det�U−�Id�, whereet represents the determinant of the matrix and Id is theiagonal unit matrix. From direct computation and fromhe fact that ��i=tr�U�=2N, one deduces that

Q�N� = �� cj�2 � cj2 + �� sj�2 � sj

2 − 2 � cj � sj � cjsj,

�C2�

hich by Cauchy–Schwarz implies that

Q�N� � �� cj�� cj2 − � sj�� sj

2�2� 0. �C3�

ince Q��→−� when �→�, we deduce that at least one ofhe eigenvalues �i is greater than N. Since ��i=2N, weonclude that at least one of the two remaining eigenval-es is �N /2, resulting in ��U��2.

CKNOWLEDGMENTShis research as well as the spatial phase-shift opticaletup was carried out in Nano-Or Technologies, and theuthors thank Y. Saban and G. Feigin, and more gener-lly all the Nano-Or Technologies team, for their helphroughout the research.

Corresponding author S. Wolfling can be reached by-mail at [email protected].

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Documents

Spatial phase-shift interferometry—a wavefront analysis technique for three-dimensional topometry