2010 Using Orientation Zernike Polynomials to predict the imaging performance of optical systems with birefringent and partly polarizing components.pdf

8/10/2019 2010 Using Orientation Zernike Polynomials to predict the imaging performance of optical systems with birefringe

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Using Orientation Zernike Polynomials to predict the imagingperformance of optical systems with birefringent and partly

polarizing components

Johannes Ruoff a , Michael Totzeck b

a Carl Zeiss SMT AG, D-73446 Oberkochen, Germanyb Carl Zeiss AG, D-73446 Oberkochen, Germany

ABSTRACT

Orientation Zernike Polynomials have been shown to provide a complete and systematic description of polarizedimaging using the polar decomposition of the Jones pupil. We use this concept to predict the polarizationperformance of high NA lithography lenses.

Keywords: polarization, hyper NA, Jones pupils, polar decomposition, wave front aberrations, lithography,projection systems

1. INTRODUCTION

A well controlled wavefront is a prerequisite for a high resolution optical imaging system. In particular opticalsystems for microlithography have to provide an extremely stable and constant performance on customer side,with regard to specication, manipulation, set-up and tool-to-tool matching. These conditions have to be fullledalso for polarized imaging, which is in particular relevant for the hyper-NA immersion scanners. In order toprovide the maximum performance a set of values is needed that is tightly connected to the origin of imagingdegradation in the optical system but has nevertheless a simple relationship to the polarized imaging performance.

Typically, the polarization properties of a non-depolarizing imaging system are encoded in a complex elec-tromagnetic transfer function, such as the Jones pupil. Following the seminal work of Lu and Chipman 1, Gehet al. 2 showed that, in current lithography lenses, these rather unintuitive Jones pupils can be decomposed intopupil maps corresponding to the basic physical effects of wavefront, apodization, diattenuation, and retardation.Apodization, which describes the transmission variation across the pupil, and wavefront aberrations are well-known quantities from scalar imaging that are sufficient to describe imaging at moderate NA. Diattenuation,which is the polarization-induced transmission splitting, and retardation, which is the polarization-induced phasesplitting, start to become important when polarized light or high NA values are used in the imaging process.Having this particular Jones-pupil decomposition at hand, we can use the well known scalar Zernike polyno-mials to describe the wavefront and apodization maps. However, in order to quantify the diattenuation andretardation maps, we have to construct a new set of Zernike-like base functions, the so-called orientation Zernikepolynomials, 3, 4 which are adapted to the vector-like nature of these pupils.

In Section 2 we shall briey recapitulate the basic ideas behind the above mentioned Jones pupil decompositionas presented in Geh et al. 2 In Section 3 we present the concept of an orientator, which is the mathematicalobject that describes the orientation of a polarization state or the principal axes of a linear retarder, and whichserves as a basis for the derivation of the proposed set of orientation Zernike polynomials. Section 4 discusses

the relations between retardation and polarized wavefronts and Section 5 is devoted to some applications.E-mail: [email protected]

International Optical Design Conference 2010, edited by Julie Bentley, Anurag Gupta,Richard N. Youngworth, Proc. of SPIE-OSA Vol. 7652, 76521T 2010 SPIE

CCC code: 0277-786X/10/$18 doi: 10.1117/12.871896

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2. JONES PUPIL AND ITS POLAR DECOMPOSITION

The propagation of a plane wave through any nondepolarizing optical element can be completely characterized bya change in the propagation direction, and changes both in amplitude and phase of the electric eld components.Hence, after passing through the optical element the new Jones vector E = ( E 1 , E 2 )T can be computed fromthe original state E = ( E 1, E 2)T by matrix multiplication E = J E with a complex valued 2 2 matrix J , the so-called Jones matrix, which contains the full information about the polarization properties of the optical element

under consideration. For optical systems the Jones matrix usually is a function of pupil and eld coordinates.For a given eld point, the collection of Jones matrices for all pupil coordinates is called a Jones pupil.

Starting point for the above mentioned decomposition is the so-called polar decomposition theorem fromlinear algebra, which states that any complex n n-matrix M can be written as a product of a Hermitian matrixH (or H ) and a unitary matrix U . Applying this theorem to our Jones matrix J , it reads

J = HU = UH . (1)

Noting that any unitary matrix U can be written in the form

U = eiP , (2)

with another Hermitian matrix P . Hence we can write Eq.(1) as

J =

He

iP

= eiP H

. (3)Since Hermitian matrices have real eigenvalues, this representation is similar to the polar representation of acomplex number c = r exp(i) with modulus r > 0 and phase . Hence the representation H exp(iP ) can beconsidered as a generalized polar representation of a complex matrix J with an amplitude matrix H (or H ) anda phase matrix P . Now it is obvious that the matrix H is responsible for the polarization dependent apodizationpart of J and P is responsible for the polarization dependent phase changes, viz. retardation effects. However,this separation is not so clean, since, in general, H can have elliptical eigenvectors and therefore also may containphase changing elements. Nevertheless, we can consider H to represent a generalized partial polarizer with realeigenvalues and elliptical eigenstates and U to represent a generalized retarder with pure phase eigenvalues anddifferent elliptical eigenstates.

Another difference with respect to the polar representation of a complex number is that, in general, thematrices H and U do not commute, and therefore H and H are different. However, when polarization effects are

weak, H and U do commute to a sufficient approximation, and the ordering does not matter.As is well known, a Hermitian matrix always has real eigenvalues and orthogonal eigenvectors, thus both H

and P can be diagonalized according to

H = V H D H V H , (4)P = V P D P V P , (5)

with unitary transformation matrices V H and V P containing the eigenvectors and the diagonal matrices D H andD R containing the real valued eigenvalues. The unitary retardation matrix U can then be diagonalized accordingto

U = V P D U V P = V P eiD P V P . (6)

It further can be shown that an arbitrary unitary 2 2-matrix can be parameterized by four independentparameters. However, in the combination VDV occurring in (4) through (6), the matrix V can be written usingonly two independent parameters:

V = cos sin e i

sin e i cos . (7)

Hence, the eigenvectors, which are given by the two columns of the transformation matrix V are

E 1 = cossin e i ,

E 2 = sin e i

cos , (8)

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representing mutually orthogonal elliptical polarization states. Writing the diagonal transmission matrix D H as

D H = t1 + d 0

0 1 d , (9)

with t denoting an overall transmission amplitude and d representing the polarization dependent amplitude split,and the phase matric D U as

D U = ei e i 0

0 ei , (10)with denoting an overall phase and representing the polarization dependent phase split, we can nally writethe polar decomposition of an arbitrary Jones matrix J as

J = te i J pol (d, p , p) J ret (, r , r ) . (11)

Herein, p and r denote the orientation angles of the corresponding polarization eigenstates of the partialpolarizer J pol ( H ) and generalized retarder J ret ( U ) and p and r are related to their respective ellipticities.In Geh at al 2 it has been shown that the elliptical retarder can, by means of a Poincare decomposition, berepresented as combination of a linear retarder and a rotator. Similiarly the elliptical partial polarizer canfuther be decomposed into a combination of two linear retarders and a linear polarizer. Now, in lithographicalprojection systems, the ellipticy is usually very small and can therefore be safely neglected. But even in cases,where the ellipticity cannot be neglected it can always be separated out and treated independently from the

linear retardation and diattenuation maps. In what follows we therefore will no longer pay attention to apossible ellipticity of the eigenstates and focus, besides apodization and wavefront, only on the linear retardationand diattenuation maps.

Setting p = r = 0, the matrices J pol and J ret represent a linear partial polarizer and retarder with respectiveorientation angles p and r with the functional forms given by

J pol (d, p) =cos p sin psin p cos p

1 + d 00 1 d

cos p sin p sin p cos p

= 1 + d cos2 p d sin2 pd sin2 p 1 d cos2 p , (12)

J ret (, r ) =cos r sin rsin r cosr

e i 00 ei

cosr sin r sin r cosr

= cos isin cos2r isin sin2r isin sin2r cos + i sin cos2r . (13)

Note that the angles p and r enter the Jones matrix of the rotated elements always together with the factor 2,which means the Jones matrix is -periodic in these variables. In other words a 180 -rotation leaves the retarderor polarizer invariant. This property is responsible for the fact that one cannot use ordinary vector Zernikepolynomials to describe retardation or polarization maps, since the latter are no true vector elds, which wouldbe invariant under 360 rotation.

We should also mention that when talking about transmission and diattenuation, usually the intensity basedquantities T and D are considered, which are related to the amplitude based parameters t and d from Eqn. (9)by

T = t2 1 + d2 , (14)

D = 2d1 + d2 . (15)

Using the above described decomposition, we can generate from a Jones pupil its respective apodization, wave-front, diattenuation and retardation pupils. The apodization, i. e. the scalar pupil transmission, as well as thephase can be expanded into the well known scalar Zernike polynomials. As already mentioned the retardationand diattenuation maps are vector-like quantities as they possess magnitude and orientation. They therefore haveto represented by an appropriate vector-Zernike description, which takes care of their rotational -invariance.This is accomplished by the so-called orientation Zernike polynomials, which shall be dened in the next section.

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3. ORIENTATORS AND ORIENTATION ZERNIKE POLYNOMIALS

In order to properly dene the proposed set of orientation Zernike polynomials is is very convenient to use theconcept of orientators. Orientators have been rst introduced by Heil et al. 3 and then put on a more rigorousmathematical basis by Ruoff and Totzeck. 4 Here, we briey review their basic properties.

An orientation is a direction modulo 180 . Orientations appear in polarization optics as the direction of thepolarization ellipse as well as the direction of diattenuation and retardance for instance as the orientation of the bright and fast axis, respectively. An orientation with orientation angle can be represented by a vectorwith doubled angle 2 . By including a scalar quantity a (representing, i. e. , an amplitude, phase or ellipticity)we obtain an orientator O that is denoted by

O (a, ) = a cos2sin2 . (16)

The case a = 1 shall be abbreviated by O (). Let us repeat the basic properties, which are easily proved usingrelation (16):

Two orientators enclosing an angle of 45 are orthogonal to each other.

An orientator and its negative (inverse) element enclose an angle of 90 .

These relations immediately follow from the representation of orientators with orientation angle as vectors withcorresponding doubled angle 2 . This vector is then invariant under 360 rotation and its inverse is obtainedby 180 rotation, which are the required mathematical properties of true vectors. Since two orthogonal vectorsenclose an angle of 90 , two orthogonal orientators must enclose an angle of 45 .

Coming back to the linear partial polarizer and retarder, we nd that their respective Jones matrices as givenin Eqns. (12) and (13) can be written as

J pol (d, ) = I + d O () (17)

for the partial polarizer and

J ret (, ) = cos I isin O () (18)

for the retarder with I denoting the unit matrix and O () a matrix formed by two orthogonal orientators

O () = cos2 sin2sin2 cos2 = O (), O 4

. (19)

If the diattenuation is weak then the product of two partial polarizers J 1 and J 2 is given by the weighted sum of their corresponding orientators:

J pol (d1 , 1 )J pol (d2 , 2 ) I + d1 O (1 ) + d2 O (2 ) . (20)

Similarly one obtains for a sequence of two weak retarders

J ret (1 , 1 )J ret (2 , 2 ) cos 1 cos2 I isin 1 O (1 ) isin 2 O (2 ) . (21)

Since the retardation is assumed to be small such that quadratic terms in i can be neglected, we can go onestep further and expand cos and sin to obtain an expression similar to the diattenuation case:

J ret (1 , 1 )J ret (2 , 2 ) I i 1 O (1 ) i 2 O (2 ) . (22)

These relations are very useful and important, since they assure that the total polarization properties of anoptical system with weak polarization properties of each of its optical components can be found by just summing

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OZ 5 OZ 5 OZ 6 OZ 6

Figure 1. Correspondence between the orientation Zernike polynomials OZ 5 and OZ 6 and their representation in matrix

form.

up the individual contributions. In particular, this also holds for the representation of these quantities in termsof orientation Zernike polynomials.

To fully appreciate the advantages of this formalism, we now nally introduce the orientation Zernike poly-nomials, by directly relating them to the vector Zernike polynomials. Again, we only briey present the mainformulae, the more elaborate derivation can be found in Ruoff and Totzeck. 4

Let an arbitrary orientator eld O be given as a function of pupil coordinates r and restricted onto theunit disc. Choosing the vector representation we can expand it into vector Zernike polynomials according to

O (a(r, ), (r, )) = a(r, ) cos2(r, )sin2(r, ) =

n =1

n

m = n

1

=0

cmn, Rmn (r )

m () , (23)

with coefficients cmn, , the radial dependence Rmn (r ) known from the scalar Zernike polynomials, and the twomutually orthogonal vector components

m0 =cos msin m , (24)

m1 = sin mcos m =

cos(m + / 2)sin(m + / 2) . (25)

These two vectors can be considered as the double angle vector representation of two corresponding orientatorsO m :

O m0 () := Om

2 = m0 () , (26)

O m1 () := O m2 +

4 = m1 () . (27)

Finally, the orientation Zernike polynomials (OZP) can be dened by including the radial function R mn :

OZ mn, (r, ) := Rmn (r )O m () . (28)

Like the vector Zernike polynomials, OZPs possess three independent indices, which can be used to order themaccording to their symmetry properties. However, as for the scalar Zernike polynomials, which possess two

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indices, it is more convenient to use a linear numbering system. The labeling system, we will use is chosen tobe as close as possible to the Fringe numbering system, which nowadays is the most common numbering schemeused in optical lithography.

Loosely spoken, we label a given OZ mn, by OZ j , if, when considered as a retardation pupil, it will give riseto a polarized wavefront represented by the scalar Zernike polynomial Z j in Fringe notation, when the inputpolarization state is x-polarized. However, this correspondence is not unique, as there are two distinct OZPs,

which lead to the same wavefront, therefore we will differentiate between these two cases by introducing a sign inthe OZP labelling. The difference between OZ j and OZ j will become obvious when looking at the Jones-matrixrepresentation, which can be schematically be written as:

OZ j= Z j Z j +1Z j +1 Z j

, (29)

OZ j= Z j Z j +1 Z j +1 Z j

. (30)

Hence, both OZ j and OZ j have the same diagonal elements, but differ in sign in the off-diagonal elements.These relationships are exemplarily depicted in Fig. 1 for OZ 5 and OZ 6 . For the OZP which are related tothe rotationally symmetric Zernike polynomials Z 1 , Z 4 , Z 9 , . . . , the Jones-matrix representations contain eitheronly diagonal or only off-diagonal matrix elements:

OZ j= Z j 00 Z j

, (31)

OZ j= 0 Z jZ j 0

, j = 1 , 4, 9, 16, . . . (32)

Finally in Fig. 2, we show the rst nine OZPs, arranged in a table according to their symmetry propertieswith respect to rotation, denoted by the M -quantum number. OZPs with M = 0 are invariant under rotation,whereas OZPs with M = 1 exhibit no rotational symmetry at all, hence they return to the same image onlyafter rotation by 360 . OZPs with symmetry class M = 0 are invariant under rotations by integer multiples of 360 /M . It worthwile noting that although OZ j and OZ j just differ by the sign in the off-diagonal componentsof the Jones-matrix representations, they belong to different symmetry classes. Whereas the scalar Zernikepolynomials with square indices Z 1 , Z 4 , Z 9 , . . . are rotationally symmetric, the corresponding OZPs belong theM = 2 class, that is they are invariant under 180 rotation. Instead, the rotationally symmetric OZPs aregiven by OZ 5 / 6 , OZ 12 / 13 , . . . , which correspond to the 2-wave or astigmatic scalar Zernike polynomials. But thisust reects the well-known fact that a rotationally symmetric retardation distribution leads to an astigmatic

wavefront when illuminated with linearly x or y-polarized light.

4. CORRESPONDENCE TO WAVEFRONTS

The effect of linear retardation or diattenuation on incoming light depends its polarization state. If the lightis linearly polarized along the fast/slow axis of the retarder or the bright/dark axis of the partial polarizer,then the only effect on the light ray is a change in phase or amplitude, whereas the polarization state remainsunaltered. Things change, however, when the polarization state and the fast/slow or bright/dark axis enclose anangle which is different from 0 and 90 . In this case, the output polarization state will be altered. In particular,a linear polarization state becomes rotated by a certain amount when passing a partial polarizer, and elliptical,

when passing a linear retarder.These effects have to be kept in mind whenever one wants to study the impact of diattenuation or retardation

on the imaging behavior of a lithography lens. If the polarization of the illumination and all the diffractionorders due to the mask structures are nearly parallel to the lens retardation, then its impact is dominated bypure wavefront effects, and the imaging effects are similar to those created by scalar aberrations for unpolarizedlight. For the same conditions, the diattenuation can be treated as an additional scalar apodization. If, however,some diffraction orders pass through regions, where the orientation of the retardation or diattenuation encloses

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M =0

M =1

M =2

M =3

M =4

Figure 2. OZP table arranged according to the M -symmetry properties. The colors code the radial dependence, which isgiven by the polynomial R mn .

a signicant angle with the polarization (in the worst case 45 ), then the change of the polarization state canhave an additional impact on the imaging and has to be considered as well.

Figure 3 shows the resulting wavefronts, when polarized light passes through a lens with the retardationgiven by 10 nm OZ j with j = 5, 6. For purely x-polarized light the resulting wavefronts are described bythe scalar Zernike polynomials Z | j | , since that is how the orientation Zernike polynomial OZ j was dened. Forinstance, 10 nm OZ 5 lead to 5 nm Z 5 and 10nm OZ 6 lead to 5nm Z 6 wavefront. Switching to y-polarizedillumination only changes the sign of the resulting wavefront. In addition to the retardation induced wavefronts,the resulting IPS (intensity in preferred state) is plotted, as well. In this analysis the preferred state is alwaysthe input polarization state. As already mentioned, in the case, where the orientation of the retardation axiscomprises an angle to the polarization state of the incoming light, the polarization state will be changed, anda previously purely x-polarized light wave will now contain a small y component. Hence, the intensity in thepreferred state, which is the x polarization, will be reduced by the amount which is transferred into the orthogonaly-polarization state. For OZ 5 , the orientations of the fast axes are parallel or orthogonal to the x polarization

along the coordinate axes, hence the IPS along these axes is 100%. In the diagonals the orientation of the fastaxis is tilted by 45 or 135 with respect to the x-polarization state, therefore the IPS is smallest along theselines. For high-NA imaging, IPS loss usually corresponds to contrast loss in the aerial image, since the preferredpolarization is usually chosen such that it leads to maximum contrast. The fraction of the light, which, due topolarization effects, is shifted into the orthogonal polarization state, cannot fully interfere and therefore will leadto an additional background, resulting in contrast loss.

Turning now to tangential (TE) or radial (TM) polarization, it turns out that the resulting wavefront causedby a single OZP can no longer be described by a single scalar Zernike coefficient. However, the wavefront willpossess the same M -fold symmetry as the underlying retardation. Hence, the wavefront caused by the rotationallysymmetric OZ 5 for TE or TM polarized light is given by Z 4 (plus an irrelevant Z 1 ) leading to a defocus of theimage. Note that in this case IPS = 100% across the whole pupil, since the TE-polarized illumination pupil isin the retardation eigenstate for each pupil point.

Now the difference between OZ 5 and OZ 5 , which were indistinguishable for linear x or y polarization,becomes obvious. In contrast to OZ 5 , which leads to a rotationally symmetric wavefront, OZ 5 possesses afour-fold symmetry and gives rise to a wavefront containing four-wave Zernike polynomials such as Z 17 and Z 28and some higher orders. The corresponding IPS exhibits even an eight-fold symmetry. As can be seen fromFig. 3, when the retardation is proportional to OZ 6 , it does not create any wavefront aberrations, since at eachpupil point the orientation of the input polarization and the principal retardation axes enclose an angle of 45 .Hence, the angle of the input polarization is orthogonal to the retardation orientation, and therefore there is nonet wavefront effect. The only impact is a change in the polarization state, which is apparent from the fact that

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OZ5

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

OZ5

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

OZ6

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

1 0 11

0.5

0

0.5

1

OZ6

1 0 11

0.5

0

0.5

1

0

2

4

6

8

10

1 0 1

1

0.5

0

0.5

1

5

0

5

1 0 1

1

0.5

0

0.5

1

97.5

98

98.5

99

99.5

100

1 0 1

1

0.5

0

0.5

1

5

0

5

1 0 1

1

0.5

0

0.5

1

97.5

98

98.5

99

99.5

100

Figure 3. Resulting wavefronts and intensity distributions for different retardation pupils when illuminated with polarizedlight. Row a) Retardation plots corresponding to 10nm OZ 5 and OZ 6 . Row b) Induced wavefronts when illuminated byx -polarized light. Row c) Corresponding intensity distributions of the x -polarized component. Row d) Induced wavefrontswhen illuminated by tangentially (TE) polarized light. Row e) Corresponding intensity distributions of the TE-polarizedcomponent.

a)

b)

c)

d)

e)

retardation

x-pol.wavefront

IPS

TE-pol.wavefront

IPS

the IPS is everywhere less than 100% (except the origin) with a Z 4 -like ngerprint. For OZ 6 the situation issimilar to the OZ 5 case, although instead of Z 17 and Z 28 , we now nd Z 18 and Z 29 .

From these considerations it becomes apparent that the imaging impact of retardation OZP strongly dependson the input polarization. Only for x or y-polarized input states, a one-to-one correspondence with scalar Zernikepolynomials can be immediately established and the imaging impact of the polarized wavefront is essentially thesame as that of a scalar wavefront on unpolarized light. However, this is strictly true only if the retardation axesare parallel to the input polarization state, otherwise IPS loss will occur, leading to an additional contrast loss,which has no counterpart in unpolarized low-NA imaging.

As a nal remark we should note that retardation in general also leads to contrast loss for unpolarized

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Retardation [nm]

1 0 1

1

0.5

0

0.5

1

0

2

4

6

8

10

1 1 2 2 3 3 4 4 5 55

0

5

OZ #

r e t a r d a

t i o n

[ n m

]

x = 0.6

1 0 11

0

1x = 0.3

1 0 11

0

1x = 0

1 0 11

0

1x = 0.3

1 0 11

0

1x = 0.6

1 0 1

1

0

1

0

5

10

Figure 4. a) Retardation distribution of the clear aperture together with the positions of ve different subapertures. b)Retardation plots corresponding to the chosen subapertures. c) Corresponding OZP spectra. The different colors denotedifferent eld positions for each OZP coefficient. In order to illustrate the eld dependence more clearly, nine eld positionshave been sampled.

a)

b)

c)

imaging. Due to the effect of birefringence it creates two different images for the two orthogonal polarizationstates, which are typically laterally or vertically displaced with respect to each other. The resulting image isthen the incoherent superposition of these two polarized displaced images.

5. APPLICATIONS

5.1 OZP in subapertures and eld dependence

As a simple but quite instructive example we consider an optical system, wherein a certain lens element possessessome rotationally symmetric birefringence given by 10nm OZ 5 for the clear aperture of the full lens. Such aretardation map can, e.g., be induced by antireecting coatings on the lens elements surfaces, although 10nmmight seem a little exaggerated.

If the lens element is situated close to a pupil plane, the corresponding pupil retardation map will be essentiallybe domintated by OZ 5 . If, however, the lens element is closer to a eld plane, only a small circular part of thelens element will be intersected by a ray bundle emanating from a certain eld point. Depending on the eldposition of the bundle under consideration, different parts of the lens element will be intersected. This is depictedin Fig. 4a, where the subapertures of the ray bundles for ve different eld locations are plotted.

It is clear that in these subapertures the retardation will no longer be described by OZ 5 alone, since, in general,the retardation distribution will no longer be rotationally symmetric, as can be seen from the corresponding pupilsmaps shown in Fig. 4b together with their respective expansions into OZPs (Fig. 4c), which contain various otherOZP coefficients.

Let us try to understand the origin of these OZP coefficients. To this end, suppose rst, that the centerof the subaperture is located at the origin. Then the only contributing OZP would be OZ 5 since the pupil isperfectly rotationally symmetric. It amplitude only depends on the radius r of the subaperture and is given by10r 2 nm, with r is given in normalized pupil coordinates. In our example, we chose r = 0 .4, hence we obtain1.6nm OZ 5 . Next, shift the subaperture slightly along the positive y axis. This case is depicted in the middlegraph of Fig. 4b. The rotationally part will now be superposed by an additional tilt in y-direction, which isrepresented by OZ 3 , and a small offset given by OZ 1 . These are the only additional contributions, as can be

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Retardation [nm]

1 0 1

1

0.5

0

0.5

1

0

2

4

6

8

10

1 1 2 2 3 3 4 4 5 55

0

5

10

15

OZ #

r e

t a r d a

t i o n

[ n m

]

x = 0

1 0 11

0

1x = 0.16137

1 0 11

0

1x = 0.32275

1 0 11

0

1x = 0.48412

1 0 11

0

1x = 0.6455

1 0 1

1

0

1

0

5

10

Figure 5. a) Retardation distribution of the clear aperture together with the position of ve different subapertures withshrinking aperture radius. b) Retardation plots corresponding to the chosen subapertures. c) Corresponding OZP spectra.The different colors denote different eld positions for each OZP coefficient. In order to illustrate the eld dependencemore clearly, nine eld positions have been sampled.

(a)

(b)

(c)

seen from the bar graph in Fig. 4c. Moving now along the x-axis leaves OZ 3 unaltered, but gives rise to theappearance of OZ 2 , representing the x-tilt with a linear dependence on the x coordinate. The value of OZ 1shows a quadratic behavior with x, which comes from the fact that the amplitude of the retardation distributioninside the clear aperture, given by OZ 5 , is quadratic in the pupil radius r . In addition to the offset OZ 1 , we alsosee the appearance of OZ 1 , which is the constant offset with fast axis along the diagonal. It is needed to changethe orientation of the fast axis, making it to point into the direction, which is given by the vector pointing fromthe origin of the full aperture to the origin of the subaperture.

Figure 5 shows the behavior of the OZP coefficients when shrinking the subaperture radius and movingit towards the pupil edge. As already mentioned the value of OZ 5 depends quadratically on the subapertureradius, hence we see it quadratically disappear, when shrinking the subaperture down to zero radius. Thevanishing of OZ 5 goes hand in hand with a quadratic growth of OZ 1 , since for shrinking subaperture size, theretardation distribution within this subaperture becomes more and more constant, as can be seen in Fig. 5b.Hence, eld lenses with small subaperture sizes mostly induce the low order OZP coefficients OZ 1 as well asOZ 2 / 3 , whereas lenses situated close to the pupil plane with large subapertures show the full ngerprint of theretardation distribution within the lens. Moreover, the eld dependence of the latter is usually small, since thesubapertures do not move signicantly across the clear aperture. An exception might be when the retardationwithin the lens strongly depends on the ray direction as is the case for the intrinsic birefringence of certaincrystalline materials such as CaF 2 . The eld lenses, in contrast, can exhibit a strong eld dependence, when theretardation strongly varies across the lens elements.

With these preliminary examples in mind, we can now turn to a realistic lithography projection lens anddemonstrate how the OZPs provide a convenient tool to analyze optical designs during a concept study in termsof their polarization impact. To this end we examine the retardation and diattenuation maps induced by theAR and HR coatings on the lens elements and mirrors of a catadioptric RCR design with 1.3 NA taken fromWO 2005/111689. We have chosen an RCR design because it provides a characteristic polarization ngerprintwhich is useful for illustration purposes. Figure 6a shows a sketch of an RCR lens design, Fig. 6b shows thediattenuation and retardation pupils for ve eld points, and Fig. 6c shows the corresponding OZP coefficients.Both the retardation and diattenuation induced by the coatings on the lens elements and the pupil mirror arerotationally symmetric. Hence, for a eld points located on the optical axis, the only induced OZP coefficients

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Figure 6. a) Sketch of an RCR lens design. b) Diattenuation pupils (upper row) and retardation pupils (lower row) for veeld points across the scanner slit. c) Corresponding eld dependent OZP spectra. The different colors denote differenteld positions for each OZP coefficient. In order to illustrate the eld dependence more clearly, nine eld positions have

been sampled.

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are the rotationally symmetric ones, namely OZ 5 and OZ 12 and maybe some higher order coefficients as well.However, since the eld is off-axis, we also expect the presence of the constant OZPs OZ 1 and the tilts OZ 2 / 3as well as the higher orders OZ 7 / 8 . As can be seen from Fig. 6c OZ 1 , OZ 5 and OZ 12 are clearly present, whereasthe tilt like OZP have very small values. This is due the fact, that for projection lenses, most lens elements

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Figure 7. Lens surface resolved OZP spectra of the RCR design for OZ 5 and OZ 5 .

are located close to pupil positions and therefore have large subapertures and little eld variation. In additionto the above mentioned OZP, we also nd OZ 4 and OZ 5 as dominant OPZs as well as some other higherorder coefficients, which are entirely due to the presence of the two folding mirrors, which obviously break therotational symmetry of the system.

5.2 Budget breakdown

In order to guarantee optimal imaging performance of the optical lens, the polarization impact has to be accu-rately controled. From imaging consideration one can deduce the maximum level of retardation and diattenuation

that can be tolerated without having signicant image deterioration. For wavefront errors, the commonly usedspecication framework is given by the Zernike polynomials and the total wavefront errors are specied by xingmaximal values for each of a certain set of Zernike polynomials, say from Z 2 to Z 36 and a value for the residualhigh frequency errors. In the same way, we can specify the maximally allowed levels of retardation and diatten-uation of the total projection optics by specifying the single OZPs, ranging from OZ 1 up to some OZ max and, if necessary, some value for the residual high frequency errors.

Having specied the total amount of allowed polarization impact in terms of OZPs, we can then try to use theadditivity of single OZP coefficients to create subbudgets for each lens elements. The maximally allowed levelof polarization impact of each lens element then, for instance, puts constraints on the levels of stress inducedbirefringence allowed for a particular lens element, or the maximally allowed coating variations.

To be able to perform such kinds of budget breakdown, it is required that the polarization effects are weakenough so that the linearity is assured and Eqns. (20) and (22) hold, which means that the total polarization

impact of all lens elements can be obtained by just summing up the OZP coefficients of each lens element.To check the validity of this approach we rst decompose the polarization pupils of each lens element sur-

face into its OZPs. In Fig. 7 the values of OZ 5 and OZ 5 are shown for each lens element surface, both fordiattenuation and retardation. Herein, it is assumed that the polarization effects are entirely due to the ARcoatings on the lens surfaces and the HR coatings on the three mirror surfaces. So, no bulk material inducedeffects are taken into account. From these plots, it is dicernable that almost all surfaces have OZ 5 contributions,although of quite different levels, whereas only two surfaces have an OZ 5 contribution. These stem from the

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Figure 8. Comparison of OZP spectra resulting from adding up the individual coefficients of each surface with thecoefficients which are obtained by expanding the polarization pupils of the whole system.

two surfaces of the folding mirror, which break the rotational symmetry of the system. As one can see, theycompletely dominate the overall retardation of the system, which therefore has no rotational symmetry. For thediattenuation, the impact of the folding mirror is of the same order as of the most critical surfaces, and thereforethe total diattenuation distribution still bears some rotational symmetry.

Finally, in Fig. 8 we compare the OZP specta resulting from adding up the individual coefficients of eachsurface with the coefficients which are obtained by expanding the polarization pupils of the whole system, shownin Fig. 6. As can be seen, the differences are negligible, which clearly justies the above described approach of

breaking down the total polarization budget into individual subbudgets in terms of OZPs. It should be mentionedthat this approach will certainly start to fail, when strongly polarizing elements are part of the optical system.However, our purpose here is to quantify the polarization properties of lithographic systems, which in generaldo have weak polarizing properties, as is the case for many other optical systems.

6. CONCLUSIONS

We have demonstrated that the concept of orientation Zernike polynomials is a natural extension of the wellknown scalar Zernike polynomials to describe the polarization performance of optical imaging systems. Thecomplete Jones pupil of a moderately polarizing optical system can be parametrized by two sets of scalar Zernikepolynomials and two sets of orientation Zernike polynomials. In general, a very limited set of Zernike coefficients isnecessary to completely describe the polarization performance of an optical system. Moreover, orientation Zernikepolynomials provide a verypowerful means to characterize and quantify the imaging impact of polarization effects.

This can be used for budget breakdowns and to monitor the polarization performance of both single lens elementsand the whole optical lens. Hence, in the same way as scalar Zernike polynomials are used to assess the wavefrontquality, OZP can be used to control the polarization aberrations, which, when taken together, serves to create aholistic picture of the imaging performance of any optical imaging system.

ACKNOWLEDGMENTS

We would like to thank Paul Gr aupner and Thomas Schicketanz for valuable input.

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REFERENCES

1. S.-Y. Lu and R. A. Chipman, Homogeneous and inhomogeneous Jones matrices, J. Opt. Soc. Am. A 11 ,pp. 766773, 1993.

2. B. Geh, J. Ruoff, J. Zimmermann, P. Gr aupner, M. Totzeck, M. Mengel, U. Hempelmann, and E. Schmitt-Weaver, The impact of projection lens polarization properties on lithographic process at hyper-NA, Proc.SPIE 6520 , p. 65200F, 2007.

3. T. Heil, J. Ruoff, J. T. Neumann, M. Totzeck, D. Krahmer, B. Geh, and P. Gr aupner, Orientation ZernikePolynomials: a systematic description of polarized imaging using high NA lithography lenses, Proc. SPIE 7140 , p. 714018, 2008.

4. J. Ruoff and M. Totzeck, Orientation Zernike polynomials: a useful way to describe the polarization effectsof optical imaging systems, J. Micro/Nanolith. MEMS MOEMS 8, p. 0314104, 2009.

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2010 Using Orientation Zernike Polynomials to predict the imaging performance of optical systems with birefringent and partly polarizing components.pdf