Flagello et al. Vol. 13, No. 1 /January 1996 /J. Opt. Soc. Am. A 53

Theory of high-NA imaging in homogeneous thin films

Donis G. Flagello

ASM Lithography, 1110 de Run, 5503 LA Veldhoven, The Netherlands

Tom Milster

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Alan E. Rosenbluth

IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598

Received May 23, 1994; accepted July 26, 1995; revised manuscript received July 28, 1995

A description is given of a modeling technique that is used to explore three-dimensional image distributionsformed by high numerical aperture sNA . 0.6d lenses in homogeneous, isotropic, linear, and source-free thinfilms. The approach is based on a plane-wave decomposition in the exit pupil. Factors that are due topolarization, aberration, object transmittance, propagation, and phase terms are associated with each plane-wave component. These are combined with a modified thin-film matrix technique in a derivation of thetotal field amplitude at each point in the film by a coherent vector sum over all plane waves. One thencalculates the image distribution by squaring the electric-field amplitude. The model is used to show howasymmetries present in the polarized image change with the influence of a thin film. Extensions of the modelto magneto-optic thin films are discussed. 1996 Optical Society of America

1. INTRODUCTION

In the field of microphotolithography the demand forhighly integrated electronic circuits has motivated in-vestigation into better lens resolution. Traditionally, ahighly corrected optical system is used to record imagesin a thin photoresist layer sø1 mmd that is coated overelectronic thin-film layers and a substrate such as silicon.The developed photoresist forms a stencil pattern that istransferred into underlying films. Since resolution im-proves as the numerical aperture (NA) of the optical sys-tem increases, high-NA systems sNA . 0.6d are desired.In this paper we discuss a theory for estimating the ir-radiance distribution that results from high-NA imagingin thin films.

Since photolithographic optical systems are highly cor-rected, the achievable density and resolution of printedpatterns are limited by diffraction. For low-NA systemssNA , 0.6d the Fresnel approximation1 can be used to pre-dict accurately the aerial image. The initial work in thisarea was done with a crude model of periodic structures,based on a modulation transfer function approximation.2

The irradiance distribution throughout the photoresistwas assumed to be proportional to the aerial image.Subsequent enhancements to this simple theory werebased on the Fresnel approximation and the incorpora-tion of partial coherence theory.3 Although the use of theFresnel approximation and paraxial propagation of theaerial image into the photoresist does not adequately de-scribe high-NA imaging, it has proven invaluable for thestudy of related photoresist chemical reactions.

There is much literature on the subject of vector high-NA diffraction.4 – 8 These works tend to discuss aerial im-age formation in terms of the point spread function (PSF),or Airy disk. Also, the description of a focused electro-

0740-3232/96/010053-12$06.00

magnetic distribution within a medium has been exam-ined extensively by Ling and Lee,9 based on methods byGasper et al.10 However, the description of high-NA vol-ume imaging into layered media with the use of extendedobjects, such as is the case in microlithography, is notdiscussed at any length. It was not until Yeung that ahigh-NA vector model for photoresist emerged.11 Yeungbased his imaging theory on the work of Richards andWolf 4 and used a numerical solution to derive the three-dimensional (3-D) image as a weighted sum of planewaves propagating into the photoresist. Yeung simpli-fied his model by assuming that the object is a periodicgrating, thus reducing the problem to a two-dimensional(2-D) solution for S or P polarization. He treated propa-gation into a bleaching photoresist as a one-dimensional(1-D) electromagnetic grating problem, using differentialmethods to calculate the fields. After realizing that thebleaching effects do not substantially alter the image,Yeung presented a further simplification12 by assuminghomogeneous and linear thin films. This allowed the useof standard thin-film matrix techniques for calculation ofthe electromagnetic fields within the photoresist. Yuanderived a 1-D vector imaging model with a 2-D wave-guide scattering model to simulate high-NA images thatcan accommodate nonplanar substrates.13 However, likeYeung’s work, it is limited to polarization in either S or Pand cannot describe the more general 3-D imaging case.

The intent of this work is to present a model for vectorimage formation within the volume of the first film of ageneral thin-film stack.14 – 18 The primary film of inter-est is photoresist, but the theory is sufficiently general toaccommodate other applications. The films are consid-ered to be homogeneous, isotropic, linear, and source free.The model is based on the generalized Debye approach19

in which the vector image field is characterized as a plane-

1996 Optical Society of America

54 J. Opt. Soc. Am. A/Vol. 13, No. 1 /January 1996 Flagello et al.

wave decomposition of each component of the electricfield. This approach is similar to Yeung’s in that the vec-tor diffraction from the exit pupil of the imaging system isbased on the concepts of Richards and Wolf; however, theanalysis here extends the development into a matrix for-malism of the imaging within a Cartesian coordinate sys-tem. This formalism lends itself to easy implementationof the numerical computations for 3-D simulation of pho-tolithographic applications. Also, we use assumptions ofisoplanarity for object and image fields, which gives usa broader scope of validity. This allows us to constructthe imaging from the object using plane-wave decomposi-tion as described by Hopkins.20 Since the first film of thefilm stack of interest is assumed to be located at or nearfocus, the amplitude and the phase of each plane wave,weighted by factors that are due to polarization, aberra-tion, and diffraction, can be used as input into a thin-filmmatrix routine that calculates the local field within thefilm, with the final image distribution being proportionalto jEj2. The result is a matrix formalism that views high-NA imaging within a thin film as the output of a linearsystem.

In Sections 2 and 3 we discuss the basic methodologyand the assumptions required for the derivation. In Sec-tion 4 we proceed to derive the general scalar form ofhigh-NA imaging in free space. The transition to vectortheory in thin films is shown in Section 5. The use of thevector theory for gyro-optic media is outlined in Section 6.In Section 7 we present the results of the vector theoryfor a PSF and an extended object imaged into a film. Weconclude in Section 8 with a summary.

2. METHODOLOGYOur model simulates a Kohler optical projection sys-tem with polarized coherent illumination. The projectorforms an image of the object in the vicinity of a thin-filmassembly, as shown in Fig. 1. The polarization just be-fore the object is decomposed into components along thex axis and the y axis. The imaging lens is assumed tobe lossless, with an image-side NA typically $0.6 but anobject-side NA , ,0.2. The imaging lens is representedby an entrance pupil reference sphere whose center ofcurvature is at the object and by an exit pupil referencesphere whose center of curvature is at the geometricalfocus. Aberrations are defined by small deviations fromthe exit pupil sphere in terms of wave-front phase errors.

The imaging is stationary (or shift invariant),21,22 whichimplies that an x–y translation of the object field resultsin only a likewise translation of the image. This assump-tion results in isoplanatic conjugate fields, such that theaberrations across the fields are constant. Isoplanaticimaging imposes strong constraints on the image-formingproperties of the system, allowing one to model it withoutconsidering in detail the propagation of light between theentrance and exit pupil surfaces. Hopkins used this as-sumption to show that the entrance pupil must exist inthe Fraunhofer region of the object, whereas the imageexists in the Fraunhofer region of the exit pupil.

The propagation direction of a general plane wavethrough the optical system is described by a geometricalray with a direction given by the propagation vector k, asshown in Fig. 2. The polarization direction of the elec-

tric field remains perpendicular to the geometrical rayand is represented by a polarization vector. The reduc-tion ratio is assumed large enough that the diffractionfrom the object to the entrance pupil can be modeledas a scalar process. The polarization vectors for object-side propagation rays are then approximately parallelto the object plane. Calculation of the exact polariza-tion mapping from the entrance pupil to the exit pupilby means of polarization ray tracing requires an exactlens prescription23,24; however, in the absence of a spe-cific lens prescription, further simplifying assumptionscan be made that are suitable for investigating generalfeatures of high-NA imaging. If the isoplanatic patch islarge enough to include the axial field position, all rayscan be assumed meridional without loss of generality. Insuch circumstances the angle that the polarization vectormakes with the meridional plane (the plane containingthe propagation vector and the z axis) will remain un-changed during propagation through the lens if polarizingeffects from the lens coatings and the like are assumednegligible. A general meridional plane is illustrated inFig. 3. The S-polarization state is assumed perpendicu-lar to the meridional plane, and the P-polarization state isassumed parallel to this plane. The polarization vectorof each plane wave has global Cartesian components thatare decomposed into local projections on the S and P axes.

Debye has shown with a stationary phase argumentthat one can decompose the electric-field amplitude con-verging on the geometrical focus into a simple spectrum ofplane waves.25 Subsequently, Luneburg26 showed that aseparate expansion can be made for each Cartesian com-ponent of the electric field, E . For this paper we as-sume that the cone of light emerging from the exit pupilof an optical system defines the angular subtense of thespectrum.

The conditions for the validity of the Debye approach27

require that the image region be small compared with thepupil diameter and that the Fresnel number be large, i.e.,

Fig. 1. Kohler illuminated projection system focusing to athin-film stack.

Fig. 2. Propagation vector k .

Flagello et al. Vol. 13, No. 1 /January 1996 /J. Opt. Soc. Am. A 55

Fig. 3. Meridional plane through a high-NA optical system.

the NA must be large compared withp

lyr0, where r0 isthe distance from the exit pupil to the image. In thiswork a local field size less than 10 mm about the axis isused, with r0 . 500 mm and l ­ 0.442 mm. This givesp

lyr0 , 0.03, which satisfies the conditionp

lyr0 , NA.We calculate the E fields in the volume of the film by

summing forward and backward plane waves within thefilm and satisfying boundary-value conditions at the in-terfaces. The E fields are derived through a modificationof the thin-film matrix technique outlined by Macleod,28

which defines additional reflection and transmission co-efficients with respect to the interface between the firstfilm and the second film. Since the intent here is to cal-culate the total vector fields inside a film, the solution ofthe axial szd electric-field component is required, in ad-dition to the transverse components on which traditionalthin-film matrices operate. Finally, the image distribu-tion within the first film of a thin-film stack is given byJoule’s heat term, Q, defined by Poynting’s theorem andis proportional to jEj2 for an isotropic medium.

3. PRELIMINARY DEFINITIONSAND NOTATIONSA plane wave with a propagation vector k at a positionr is given by

U sx, y, zd ­ U0sx, y, zdexpsik ? rd

­ U0sx, y, zdexpf2i2pNsax 1 by 1 gzdg , (1)

where

r ­ l0sxx 1 yy 1 zzd ,

k ­2pN

l0

√kx

kx 1

ky

ky 1

kz

kz

!

­2pN

l0fscos f sin udx 1 ssin f sin udy 1 scos udzg

­2pN

l0sax 1 by 1 gzd . (2)

Here and throughout this paper, scalar distances x, y,z, and r are normalized to the vacuum wavelength l0.U0sx, y, zd is defined as the complex amplitude [the har-monic time dependence of the form expsivtd is implicitlyassumed], and N is the complex index of refraction givenby N ­ n 2 ik. k defines the length of the propagationvector, and a, b, and g are the direction cosines, relatedby

a2 1 b2 1 g2 ­ 1 .

Traditionally, the magnitude of the time-averagedPoynting vector has been used to calculate the imageirradiance; however, it is more appropriate to use theabsorbed energy, i.e., local Joule heating per unit volume,in calculations of the image field within a thin film ofphotoresist. We estimate the Joule heating by consider-ing the conservation of energy using Poynting’s theorem,where the time-averaged divergence of the Poynting vec-tor is shown to represent the power absorbed within thevolume. In this paper we define

Q ­ jk= ? sE 3 Hdlj ­12 sjEj2 ­ k0YnkjEj2, (3)

where s is the conductivity, Y is the free-space admit-tance, and k0 ­ 2pyl0. If we assume that only Ohmiclosses are being considered, Q represents the form of therecorded image distribution within an absorbing mediumwith units of power per volume. It has been interpretedby Stratton29 as the power dissipated by thermochemicalactivity. In general, for photosensitive films, a fractionof the energy absorbed is used in a photochemical reactionwith the remainder being dissipated as heat. Thereforethe recorded image within a film is proportional to jEj2.

4. SCALAR HIGH-NA IMAGINGIN FREE SPACEThis section describes general image formation throughan optical system in the absence of a thin-film stack,where the object and image spaces are defined in air.The derivation is based on the scalar imaging conceptsintroduced by Goodman,30 where the image distribution isformed by propagation of plane waves sequentially fromobject to entrance pupil, entrance pupil to exit pupil, andexit pupil to image.

Harvey31 has shown that a direction cosine spectrumof plane waves can be used to describe the propagationof a scalar field in free space. His geometry consisted ofa diffracting aperture illuminated by a converging beam.The diffracted electric-field amplitude is observed on ahemisphere in the far field, i.e., the Fraunhofer region.Harvey’s results may be applied to the propagation be-tween the object and the entrance pupil reference sur-face. Figure 4 displays the diffraction geometry, wherer is the distance from the object to the entrance pupil.The entrance pupil surface is described in direction co-sine space, where locations on the surface are referencedby coordinates sa, bd. The limit of the pupil is a circularring defined by the NA, that is,

NA ­ sin umax ­p

amax2 1 bmax

2 ,

Fig. 4. Diffraction geometry from object to entrance pupil.

56 J. Opt. Soc. Am. A/Vol. 13, No. 1 /January 1996 Flagello et al.

Fig. 5. Circular pupil defined in direction cosine space.

where umax is the maximum marginal ray angle definedin object space. The direction cosine coordinates in theentrance pupil reference surface are shown in Fig. 5.

The diffraction to the entrance pupil, based on the workof Hopkins, uses the Huygens–Fresnel principle. It iswritten with the use of a Fourier transform relationshipand is given by

Osa, bd ­ igexps2i2prd

r

Z `

2`

Z `

2`

Osx, yd

3 expfi2psax 1 bydgdxdy

­ igexps2i2prd

rF hOsx, ydj , (4)

where the tilde is used to define a transformed object.Without any loss of generality the object is placed atz ­ 0. The distribution of the electric-field amplitude onthe entrance pupil is represented by

Esa, bd ­ Osa, bdT sa, bd , (5)

where a transmission function T sa, bd specifies the sizeand the shape of the entrance pupil and hence umax.T sa, bd has a value of 1 on the entrance pupil and 0 other-wise. Equations (4) and (5) are rearranged to give

rEsa, bd ­ ig exps2i2prdT sa, bdF hOsx, ydj , (6)

which is an important relationship that will be used inthe development below.

Again, from the work of Hopkins, we calculate theimage field by recognizing that it is proportional to theinverse Fourier transform of the spherical exit pupil sur-face S and thus is a weighted integration of plane waves.The solution is consistent with Wolf ’s generalized Debyeapproximation and is given by

E 0sx0, y 0; z0 d

­ 2iexpsi2pr0d

r0

ZZS 0

E 0sa0, b0dexps2ik0 ? r0 ddS 0

­ 2i expsi2pr0dF 21

8<:r0E 0sa0, b0dexps2i2pg0z0dg0

9=; , (7)

where an element of wave front dS 0 is defined by

dS 0 ­ r02 dV0 ­ r02 da0db0

g0

and the primed notation is used to represent the image-side geometry, as shown in Fig. 6. r0 has been written in-side the transform for reasons that will be obvious shortly.E 0sa0, b0d is the scalar complex amplitude distribution inthe exit pupil. Note that the exponential term in z0 rep-resents an image focus term. The geometrical focus, lo-cated at z0 ­ 0, is at the vertex of the solid angle sub-tended by the exit pupil.

A simple magnification scaling is assumed between en-trance and exit pupils. Any aberrations within the lensare treated with an aberration term that modifies thephase of the wave front at the exit pupil reference sur-face. The transverse magnification is defined as

m ­h

0

h,

where h and h0 refer to normalized object and image size.From the Abbe sine condition,

hNa ­ h0N 0a0, hNb ­ h0N 0b0,

and since N ­ N 0 ­ 1, the transverse magnification isgiven as

m ­h0

h­

a

a0­

b

b0.

The field distribution in the exit pupil is derived fromscaling Eq. (5) and is written as

E 0sa0, b0 d ­1

m2Esma0, mb0 d

­1

m2Osma0, mb0 dT sma0, mb0 d . (8)

The relationship between the magnitudes of the en-trance pupil fields and the exit pupil fields must be takeninto account. Since there is a physical difference in size,i.e.,

T 0sa0, b0d ­ T sma0, mb0 d , (9)

and T 0sa0, b0 d is the exit pupil transmission function, con-servation of energy must be ensured. Figure 7 illustratesthe differential areas on the entrance and exit pupil sur-

Fig. 6. Diffraction geometry from exit pupil to image focus.

Flagello et al. Vol. 13, No. 1 /January 1996 /J. Opt. Soc. Am. A 57

Fig. 7. Mapping geometry between entrance and exit pupils.

faces. An energy balance yields

jEsa, bdj2da ­ jE 0sa0, b0dj2da0,

where the differential areas on each pupil are given by

da ­ r2 dV ­ r2 dadb

g,

da0 ­ r02 dV0 ­ r02 da0db0

g0.

The electric fields must satisfy the conservation equation

r0jE 0sa0, b0 dj ­ rjEsa, bdj

sg0

g

pdadbpda0db0

­ rjEsa, bdj

sg0

gm . (10)

As stated above, lens aberrations are given by a wave-front phase error W sa0, b d and are represented by a scalarterm, expf2i2pW sa0, b0 dg. In the absence of any filmstack the scalar electric-field distribution, given by thecombination of Eqs. (7), (8), and (10), is

E 0sx0, y 0; z0 d

­ 2i expsi2pr0d

3 F 21

8<:r0E 0sa0, b0 dexps2i2pg0z0dexpf2i2pW sa0, b0dgg0

9=;­ 2i

expsi2pr0dm

F 21

(rEsma0, mb0 dexps2i2pg0z0d

3 expf2i2pW sa0, b0dg1

pgg0

). (11)

Substitution of Eq. (6) and the application of Eq. (8) yield

E 0sx0, y 0; z0 d ­expfi2psr0 2 rdg

m

3 F 21hOsma0, mb0 dCsa0, b0; z0 dj , (12)

where Csa0, b0; z0 d is a function that contains only thescalar elements of the optical system and is defined by

Csa0, b0; z0 d ­ T 0sa0, b0 dexps2i2pg0z0 d

3 expf2i2pW sa0, b0 dgr

g

g0. (13)

Osma0, mb0 dCsa0, b0; z0 d represents the complex ampli-tude distribution of plane waves for a given z0. The im-age irradiance is proportional to jE 0sx0, y 0; z0 dj2.

5. VECTOR HIGH-NA IMAGINGIN THIN FILMSThe transition to a vector field is first accomplished byestablishing the initial state of polarization that illumi-nates the entrance pupil. A polarization state matrix Mi

is defined such that multiplication with Osa, bd gives thedescription of polarization in the entrance pupil, i.e.,

Osa, bd ­ Osa, bdMi ,

where Osa, bd and Mi are 2 3 1 matrices. Mi has ele-ments given by Sl that may be complex, i.e.,

Mi ­

√Sx

Sy

!.

For example, x-polarized illumination results in

Osa, bd ­

√Ox

0

!­ Osa, bd

√10

!­ Osa, bdMi .

Similarly,

Osma0, mb0 d ­ Osma0, mb0 dMi .

A vector analogy to Eq. (12) is derived by first weight-ing the plane-wave amplitudes, Osma0, mb0 dCsa0, b0; z0 d,by the polarization amplitudes for each Cartesian compo-nent of the electric field. Figure 8 illustrates a propaga-tion vector, k0, with a direction sa0, b0d emerging from theexit pupil. The polarization vector remains at a constantorientation with respect to k0 and the meridional planedefined by the plane of incidence with the film stack, andthe component amplitudes are projected onto the local Sor P coordinate system. The component amplitudes havebeen defined by Mansuripur8 in terms of the propagationvector direction and the initial x or y polarizations fromthe object. It is given here as a 5 3 2 matrix MP sa0, b0 dwith matrix elements Plmn, where l gives the initial objectpolarization, m refers to the global sx, y, zd coordinates,and n is the local S or P coordinate. Hence

MP sa0, b0 d ­

266666664

PxxS PyxS

PxxP PyxP

PxyS PyyS

PxyP PyyP

PxzP PyzP

377777775 ­

26666666666666666664

b02

1 2 g02

2a0b0

1 2 g02

g0a02

1 2 g02

a0b0g0

1 2 g02

2a0b0

1 2 g02

a02

1 2 g02

a0b0g0

1 2 g02

g0b02

1 2 g02

2a0 2b0

37777777777777777775

.

(14)

Fig. 8. Rotation of the polarization vector emerging from exitpupil.

58 J. Opt. Soc. Am. A/Vol. 13, No. 1 /January 1996 Flagello et al.

By simple orthogonal decomposition any polarizationstate can be defined by Eq. (14). As an example, con-sider a propagation vector direction given by sa0, b0, g0 d ­s0.5, 0.5, 0.707d. The corresponding matrix is

MP ­

266666640.5 20.50.35 0.35

20.5 0.50.35 0.35

20.5 20.5

37777775 .

The resultant amplitude projections onto the x, y, z axisare simply the sum of the S and P contributions. Ifthe initial object polarization is along the x axis, theamplitudes are Pxx ­ 0.854, Pxy ­ 20.147, and Pxz ­ 20.5.

The complex amplitudes of the plane-wave componentslocated at the top surface (interface I) of a thin-film stackat z0 ­ z0 are now written in matrix form as

AiIsa0, b0; z0d ­ MP sa0, b0dU sa0, b0; z0d

­ MP sa0, b0dOsma0, mb0 dCsa0, b0; z0d ,

(15)

where AiI is a 5 3 1 matrix with elements given bysAiIdmn. The distance z0 is an offset term that deter-mines the initial phase for each plane wave incident oninterface I. Variation of z0 is equivalent to moving thetop film surface in and out of focus.

The derivation of the electric-field image amplitudewithin the first layer of a thin-film stack is based onthe thin-film matrix techniques presented by Macleod.28

Figure 9 shows the geometry used for the calculation.The vector field for any point within the first film is foundby summation of the downward and upward plane wavesfor each S and P contribution, n, of the Cartesian compo-nent, m. The derivation presented here departs from thetypical thin-film method by deriving this field in termsof the field at interface II, the bottom of the film, andthen relating the result to the incident field at interface Ithrough the matrix formalism. If the first film has a nor-malized thickness of d1, the incident and reflected fieldsat interface II are

A1sa1, b1d ­ AiII expfi2pN1g1sd1 2 z0 1 z0dg3 expf2i2pN1sa1x0 1 b1y 0dg , (16)

A2sa1, b1d ­ ArII expf2i2pN1g1sd1 2 z0 1 z0dg3 expf2i2pN1sa1x0 1 b1y 0 dg , (17)

where A1 is the downward field and A2 is the upwardfield. It is understood in Eqs. (16) and (17) that the com-plex amplitudes AiII and ArII represent mn matrix ele-ments. The numerical subscripts denote the underlyingmedia of interest in image space, and the quantity z0 2 z0

ranges from 0 at interface I to d1 at interface II.The following relationships are used from Snell’s law

in direction cosine notation (assume an incident mediumof N 0 ­ 1):

a0 ­ Nj aj , b0 ­ Nj bj .

Since for any film j the z-direction cosine is referenced to

the incident medium with

gj ­

"1 2

sin2scos21 g0dNj

2

# 1/2

,

the substitution

Fsz0 d ­ 2pN1g1sd1 2 z0 1 z0d

­ 2psd1 2 z0 1 z0dfN12 2 sin2scos21 g0dg1/2 (18)

yields the total field in the first film as

A1sa0, b0; z0 d ­ expf2i2psa0x0 1 b0y 0 dg

3 fAiII exp iF 1 ArII exps2iFdg , (19)

where it is understood that F is a function of z0.The following derivation shows that AiII and ArII in

Eq. (19) are related to the incident field through the trans-verse sx, yd and axial szd transmission and reflection co-efficients. Since the fundamentals of thin-film matrixtechniques are well known, only the departures from thestandard formalism will be presented. It is assumed thatthe film in question is the first film on an arbitrary filmstack composed of homogeneous, linear films. A matrixcan be defined for each film by

Mj ­

"cos dj i sin djyhj

ihj sin dj cos dj

#, (20)

where hj is defined as the tilted optical admittance for Sand P polarization in free-space units and dj is the phasefor a normalized film thickness d1. Hence

hj ­

(Nj gj S polarizationNjygj P polarization

,

dj ­ 2pNjgj dj .

The characteristic matrix for a film assembly with q lay-ers is defined by√

BC

!­

0B@ qYj­1

Mj

1CA√1

hm

!, (21)

where hm is the substrate admittance. A subassemblycharacteristic matrix is defined without the first film as√

Bs

Cs

!­

0B@ qYj­2

Mj

1CA√1

hm

!. (22)

With reference to Fig. 9, transverse and axial reflec-tion and transmission coefficients are defined for the full

Fig. 9. Thin-film stack with incident plane waves and electric-field amplitudes.

Flagello et al. Vol. 13, No. 1 /January 1996 /J. Opt. Soc. Am. A 59

Table 1. Reflection and Transmission CoefficientsComponent Full Assembly Subassembly

Transverse reflection r ­ArI

AiIrII ­

ArII

AiII

Transverse transmission t ­Atm

AiItII ­

Atm

AiII

Axial reflection rz ­ArIz

AiIzrIIz ­

ArIIz

AiIIz

Axial transmission tz ­Atmz

AiIztIIz ­

Atmz

AiIIz

film assembly and subassembly in Table 1. The originalassumption of film homogeneity and linearity has been in-voked to allow the equality of x and y coefficients. Sincethe transverse components of E and H are required tobe continuous across all interfaces, the coefficients can beexpanded in B and C terms for the total assembly and inBs and Cs terms for interface II as

t ­2h0

Bh0 1 C, r ­

Bh0 2 CBh0 1 C

, (23)

tII ­2h1

Bsh1 1 Cs

, rII ­Bsh1 2 Cs

Bsh1 1 Cs

, (24)

where h0 is the incident admittance. The axial szd coeffi-cients are not continuous across the boundaries. We de-rive them here by requiring that propagating plane wavesin the first film be solutions to Maxwell’s equations ina source-free medium. In particular, the divergence ofthe field must be 0 in the incident medium and the film;hence, for a general plane wave,

= ? E ­ k ? E ­ 0 .

Solving this equation for the electric field in the incidentand reflected media results in the following relationshipsbetween the transverse and axial reflection and transmis-sion coefficients:

tz

rIIz­

t

tII

N 0g0

N1g1

, rIIz ­ 2rII . (25)

Equations (23)–(25) are now used to express generalplane-wave amplitudes in the film in matrix notation as

A1sa0, b0; z0 d ­ MF sa0, b0; z0 dAiIsa0, b0; z0d , (26)

where MF sa0, b0; z0 d is called the film function matrix. Ithas dimensions 3 3 5 with elements given by Fn such that

MF sa0, b0; z0 d ­

264FS FP 0 0 00 0 FS FP 00 0 0 0 FzP

375 , s27d

Fn ­

√t

tII

!n

fexpsiFd 1 srIIdn exps2iFdg , s28d

FzP ­N 0g0

N1g1

√t

tII

!P

fexpsiFd 2 srIIdP exps2iFdg .

(29)

The matrix amplitude function A1sa0, b0; z0 d has dimen-sions 3 3 1 pertaining to the Cartesian coordinates. The

total vector electric image field within the film is calcu-lated by the summation of all plane waves. Therefore

E1sx0, y 0; z0 d ­ c0F21hA1sa0, b0; z0 dj

­ c0F21 hMF sa0, b0; z0 dAiIsa0, b0; z0dj

­ c0F 21 hMF sa0, b0; z0 dMP sa0, b0 d

3 Osa0, b0 dCsa0, b0; z0dj . (30)

Equation (30) suggests that image formation within a thinfilm can be regarded as the output of a linear system.The polarization, film, and scalar lens terms behave astransfer functions. The film stack in this context mustthen be treated as an integral part of the image formationsystem. If the object is a delta function, A1sa0, b0; z0 d be-comes the vector version of the system transfer function,and the vector point spread function (PSF) is given by itsFourier transform.

Alternatively, E1sx0, y 0; z0 d can be written in terms of aconvolution of the transformed elements:

E1sx0, y 0; z0 d ­ c0MF sx0, y 0; z0 d ≠ MP sx0, y 0 d

≠ O

√x0

m, y 0

m

!≠ Csx0, y 0; z0d . (31)

This formalism further implies that the image shape isa function of the film as well as of the polarization andscalar terms.

Finally, the image distribution within the first film isgiven by

Qsx0, y 0; z0 d ­ k0Yn1k1jE1sx0, y 0; z0 dj2. (32)

The advantage of Eq. (30) is that the field within thefilm is represented in a concise format that lends it-self to computer programming. The interaction of thedownward- and upward-traveling waves in the film is suc-cinctly represented by one film function per component inthe matrix MF sa0, b0; z0 d.

6. GYRO-OPTIC MEDIAUnfortunately, Eq. (30) is not easily used with gyro-opticmedia, for which it is traditional to use 2 3 2 matriceswith strictly S- and P-polarization decomposition. In thissection we derive an alternative form of Eq. (30) by main-taining the local S and P coordinate system through thefilm, then using an xyz projection matrix to map to aglobal coordinate system.

As explained in Section 2, the polarization vectorremains perpendicular to the propagation vector withcomponents parallel (P) and perpendicular (S) to themeridional plane. It is easily shown that the amplitudesof the S and P components in the entrance pupil aregiven in matrix form by

MSP O ­

"sin f 2 cos f

cos f sin f

#√Ox

Oy

!. (33)

Since the direction cosine relationships give

cos f ­a0p

1 2 g02, sin f ­

b0p1 2 g02

,

60 J. Opt. Soc. Am. A/Vol. 13, No. 1 /January 1996 Flagello et al.

Fig. 10. PSF for NA0 ­ 0.95 given by values of normalized jEj2:(a) isoimage contours, (b) profiles along x and y.

Table 2. Parameters for PSF Simulation

Film Assembly Optics

N 0 ­ 1 NA0 ­ 0.95, l ­ 0.442 mmN1 ­ 1.656 2 i0.004, d1 ­ 1 mm NA ,, 1

x-polarized

Nm ­ 1.656 2 i0.004 Osx, yd ­ dsx, ydW ­ 0, z0 ­ 0

MSP is written as

MSP ­

266664b0p

1 2 g02

2a0p1 2 g02

a0p1 2 g02

b0p1 2 g02

377775 . (34)

The downward field for each local S and P wave throughthe film is now written as

A1sa0, b0 d ­ M1F sa0, b0; z0 dMSP sa0, b0 dOsa0, b0d

3 Csa0, b0; z0d , (35)

where A1sa0, b0 d is a 2 3 1 matrix corresponding to theS and P fields and M1

F sa0, b0; z0 d is defined as

M1F sa0, b0; z0 d ­ expsiFd

"cos c sin c

2 sin c cos c

#

3

2666664√

t

tII

!S

0

0

√t

tII

!P

3777775 , (36)

where c ­ pNqsd1 2 z0 1 z0d and q is the off-diagonalpermittivity matrix coefficient given by32

e ­

264 e 2iqe 0iqe e 00 0 e

375 .

c relates to the Faraday rotation in the material, andq ­ 0 for pure dielectrics.

Likewise, the upward field is

A2sa0, b0 d ­ M2F sa0, b0; z0 dMSP sa0, b0 dOsa0, b0 d

3 Csa0, b0; z0d , (37)

where

M2F sa0, b0; z0 d ­ exps2iFd

"cos c 2 sin c

sin c cos c

#

3

2666664√

t

tII

!S

rIIS 0

0

√trII

tII

!P

rIIP

3777775 . (38)

Fig. 11. Q distributions of PSF at interface I with NA0 ­ 0.95:total Q distribution on top surface with x, y, and z components.

Fig. 12. Normalized PSF distribution in film: (a) meridionalslices, (b) comparison of x and y polarizations at interface I andinterface II.

Flagello et al. Vol. 13, No. 1 /January 1996 /J. Opt. Soc. Am. A 61

Fig. 13. Magnitudes of normalized system transfer functionA1sa0, b0; z0 ­ 0d for x, y, and z components given in exit pupilcoordinates for NA0 ­ 0.95. The magnitudes are normalized tothe maximum x-component value.

After they emerge from the exit pupil, it can be shownthat the projections of the S and P components onto thex, y, and z axes are given by a matrix Mxyz in terms ofthe direction cosines of the propagation vector:

M6xyz ­

266666666664

b0p1 2 g02

a0g0p1 2 g02

2a0p1 2 g02

b0g0q1 2 g

02

0 6N 0 g0

N1g1

p1 2 g02

377777777775, (39)

where the rows represent the x, y, and z components andthe columns represent the S and P components. The nec-essary correction to the z component has been incorpo-rated into downward and upward projection matrices such

that the total field through the film is now written as

E1sx0, y 0; z0d ­ c0F21hM1

xyzsa0, b0 dA1sa1, b1d

1 M2xyzsa0, b0 dA2sa1, b1dj . (40)

7. RESULTS

A. Point Spread FunctionIt is useful to examine first the normalized image of adelta function object that produces a constant electric-field distribution across the entrance pupil referencesurface. The resultant image in the film, Qsx0, y 0; z0 d,becomes the point spread response function (PSF). Sub-sequently, we call A1sa0, b0; z0 d the system transfer func-tion within the exit pupil domain. Note that the film isincorporated in this function.

Table 3. Parameters for Tribar Simulation

Film Assembly Optics

N 0 ­ 1 NA0 ­ 0.95, l ­ 0.442 mmN1 ­ 1.656 2 i0.004, d1 ­ 1 mm NA ­ 0.048

Osx, yd ­ tribar, 3 barsof unit amplitude onan opaque field

Nm ­ 1.656 2 i0.004 W ­ 0, z0 ­ 0, m ­ 0.05

Fig. 14. (a) Tribar object with scaled image dimensions,(b) entrance pupil electric-field distribution normalized to unitmagnitude.

62 J. Opt. Soc. Am. A/Vol. 13, No. 1 /January 1996 Flagello et al.

Foa

piiarsmcaipl

wtnrtcapaiim

ig. 15. Magnitude of A1sa0, b0; z0 ­ 0d components, with thebject, scalar lens, polarization, and film terms at interface I fortribar object and NA0 ­ 0.95.

Figure 10 illustrates the effect of the film stack by com-aring aerial PSF images (no film stack) to the PSF atnterface I of a thin-film stack, where jEj2 is normalized tots maximum value. The parameters set for this examplere given in Table 2. The index of the first thin film rep-esents a mildly absorbing photoresist. The substrate isilicon. Clearly, interaction with the film reduces asym-etry in the PSF. This is caused by the reduction in the

one angle of the transmitted beam through Snell’s laws well as the angular dependence of reflectivity. Themage width decreases along the direction parallel to theolarization and slightly increases along the perpendicu-ar direction.

The image at interface I is further illustrated in Fig. 11,here the PSF is shown for each xyz component and

he image, which is the component sum. The x compo-ent contains the most power and is similar to a scalaresponse. The z component contains substantial powerhat is distributed along the x axis. It is the primaryontributor to image asymmetry. The y component hasn interesting quadrupole symmetry but contains littleower and has an insignificant effect on the overall im-ge. We further analyze PSF film images by consideringmage contours in the film shown in Fig. 12(a) for merid-onal slices along the x and y axes. A slight image asym-

etry exists along the film depth. Figure 12(b) shows

that the x slice is slightly wider with fewer interferenceeffects, such as ringing, than the y slice. The differencesare more apparent 1 mm from the focal plane.

Figure 13 shows the magnitude of the normalized sys-tem transfer function, A1sa0, b0; z0 ­ 0d, at interface I.The maximum value of the x component is used for nor-malization. Since the initial polarization is x polarized,the x component transfers the bulk of the power. Theobliquity terms tend to counter the polarization term, pro-ducing a top hat shape. The y and z transfer functionsare mostly dominated by the polarization terms. Thequadrupole symmetry for y does not give a significantmagnitude as compared with that for z, where the great-est effect is at the edge of the pupil and along the direc-tion of polarization. The interaction of the film reducesthe overall magnitude of the z transfer function relative

Fig. 16. Simulation of Q distribution of tribar object at inter-face I with NA0 ­ 0.95: Q distribution at top surface with x, y,and z components.

Fig. 17. Q simulation of tribar image with NA0 ­ 0.95 andz0 ­ 0: (a) x –z meridional plane, (b) profiles comparing x andy polarizations at interface I and interface II.

Flagello et al. Vol. 13, No. 1 /January 1996 /J. Opt. Soc. Am. A 63

Fig. 18. Q simulation of tribar image with NA0 ­ 0.95 andz0 ­ 45 mm: (a) x –z meridional plane, (b) profiles comparingx and y polarizations at interface I and interface II.

to x, thus reducing the image asymmetry of the PSF ascompared with the aerial image.

B. Extended ObjectThis study simulates the image of an object with finitedimensions using a system magnification of 0.05. Theobject-side NA is 0.048, so that g ø 1 with an image-sideNA0 ­ 0.95. Other parameters are given in Table 3. Aclear tribar object is used with scaled image dimensions of0.25 mm lines and 0.25 mm spaces parallel to the x axis,as shown in Fig. 14(a). The tribar object has clear barswith unit amplitude on an opaque background field. Thenormalized entrance pupil distribution is presented inFig. 14(b). Since there is substantial magnitude at theedge of the pupil, the object dimensions are approximatelyat the limit of resolution.

The magnitudes of the A1sa0, b0; z0 ­ 0d components foran initial x polarization are shown in Fig. 15, where thefunctions are normalized to the maximum magnitude ofthe x component. These functions now include the am-plitude contributions of the object. Most of the energy iscontained in the x component. The y and z componentshave a small amount of energy concentrated at the ex-tremes of the pupil. The z component has a maximumof ,30% compared with that of the x component, whereasthe y component has a maximum of ,2%. The resultingQ distributions just after interface I are shown in Fig. 16.Most of the energy is in the x component. The maximumvalue of the z component is approximately 3.7% of themaximum image magnitude. The contribution of the ycomponent is negligible, at approximately 0.01% of themaximum image magnitude.

The asymmetry that is due to polarization is shownin Fig. 17(a) as iso-Q contours, where the x–z merid-ional plane is shown for initial x and y polarizationswith z0 ­ 0. A comparison of the two polarizations isshown for two film depths in Fig. 17(b). The x-polarizedslice has slightly more power at the top surface for x ­60.25 mm and is broader than the y-polarized slice.

High-NA effects are clearer at the bottom surface. Thephenomenon of spurious resolution is evidenced by thethree maxima at interface I becoming four maxima atinterface II. Although the differences between polariza-tions appear small at the top surface, they are muchlarger at the bottom. This increase of the polarizationdifferences is again seen in Fig. 18, with z0 ­ 0.45 mm.The focus offset is chosen to reproduce the tribar imageat interface II. The respective profile difference at thebottom of the film is much larger than that in Fig. 17.

8. SUMMARYThis work presents a vector description of high-NA imag-ing in homogeneous thin films. The resultant derivationgives a form for the image distribution within the vol-ume of the first film of a thin-film stack. It is based on ageneralized Debye plane-wave decomposition of the radia-tion that propagates from the pupil. Each plane waveis weighted by polarization, aberration, input amplitude,and phase terms. We combine this decomposition witha thin-film matrix technique to derive the electric fieldswithin the film.

The model is limited to imaging systems that haveisoplanatic object and image fields. In general, this re-quires that the optical system be well corrected, i.e., thewave-front aberrations are below 0.25l. This require-ment further implies that the electric-field polarizationis not substantially altered upon propagation throughthe optical elements of the imaging lens. The polariza-tion maintains a constant orientation with respect to thepropagation vector and the meridional plane.

We also present a vector formalism of image formationas a linear, shift-invariant system. This gives rise to avector representation of transfer functions that are due tothe lens, the polarization, and the thin-film stack. Wefurther show a method to incorporate gyro-optic mediainto the imaging.

Finally, examples were presented showing asymmetriesthat are due to vector imaging components. The modelalso accurately predicts classic spurious resolution as thefocal depth is increased.

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