light distribution close to focus in biaxially ... mypapers/josaa_2004_2.pdf · light distribution...

Sjoerd Stallinga Vol. 21, No. 9 /September 2004 /J. Opt. Soc. Am. A 1785

Light distribution close to focus in biaxiallybirefringent media

Sjoerd Stallinga

Philips Research Laboratories, Professor Holstlaan 4, 5656 AA Eindhoven, The Netherlands

Received October 15, 2003; revised manuscript received February 18, 2004; accepted April 2, 2004

The effect of focusing into a biaxially birefringent medium on the light distribution in the focal region of ahigh-NA optical system is investigated with the Debye approach to vectorial diffraction theory. Attention islimited to media with small birefringence. The electric field in the focal region is the sum of the field of thetwo polarization eigenmodes of the biaxially birefringent medium. Both modes are generally astigmaticallyaberrated, are defocused with respect to each other, and have a polarization field that is nonuniform over thepupil. The diffraction integrals are calculated numerically on the basis of an expansion of the field close tofocus in terms of partial waves. © 2004 Optical Society of America

OCIS codes: 260.1440, 260.1960, 210.4770, 180.1790.

1. INTRODUCTIONThe light distribution close to focus in various circum-stances is relevant to the imaging properties of scanningmicroscopes and to the readout of optical disks. Focusingthrough or into a birefringent layer results in blurring ofthe focal spot that deteriorates the image of a scanningmicroscope and the readout of significant bits from an op-tical disk. These birefringence effects can occur in thesubstrate–cover layer of optical disks as a result of me-chanical stresses that are introduced into the plastic dur-ing manufacturing. As the beam is focused onto the datalayer through the substrate–cover layer, birefringence inthe latter layer can be particularly harmful. The aim ofthis paper is to study the effect on the focal spot of focus-ing into a biaxially birefringent medium. Such a me-dium has three different refractive indices associatedwith three mutually orthogonal axes, the principal axes.This paper generalizes the results obtained in a previouspaper1 on the effect of focusing into a uniaxially birefrin-gent layer with principal axis parallel to the optical axis.

The focal light distribution can be studied within theframework of vectorial diffraction theory. Focusing in asingle isotropic medium was originally considered by Wolfand coworkers,2–5 and later by Mansuripur,6–8 Kant,9,10

and Sheppard and Torok.11 Focusing through an inter-face between two dielectric media was first investigatedby Ling and Lee,12 followed by a number of otherauthors.13–19 A general framework for describing focus-ing into uniaxial and biaxial anisotropic media was devel-oped by Stamnes and coworkers.20–24 In Ref. 24 the for-malism is applied to the paraxial limit of small NAs.

This paper deals mainly with biaxially birefringent me-dia with small birefringence. This means that the differ-ences between the three refractive indices are taken to bemuch smaller than 1. For stress-induced birefringencein the plastic substrate–cover layer of optical disks thesedifferences are of the order of 1025 –1023, which makesthe approximation quite appropriate for this case. Thereare also many crystals for which the small-birefringence

1084-7529/2004/091785-14$15.00 ©

approximation is justified. For example, the short list ofbiaxial crystals mentioned in Ref. 25 includes gypsum,feldspar, mica, topaz, and YAlO3 , for which the birefrin-gence parameters are of the order of 1023 or less. Themain advantage of the small-birefringence approximationis that it allows for a considerable simplification of theformalism needed to calculate the light distribution closeto focus. Such a simplified formalism clarifies the physi-cal effects governing the deformation of the focal spot bythe effects of birefringence. For example, in Ref. 1 it isshown that the spot distortion is determined largely by arelative defocus of the ordinary and extraordinary polar-ization modes. Results that are similar in character canbe obtained for the case of biaxial birefringence.

The content of this paper is as follows. The existingresults in the literature based on the Debye approach aresummarized in Section 2. These results are generalizedto biaxially birefringent media in Section 3. Numericalresults are discussed in Section 4. The paper concludeswith a summary of the main results in Section 5.

2. DEBYE APPROACHA. Light Distribution Close to FocusWe consider a collimated beam of light of wavelength lpropagating in the z direction that is focused by a lensinto a biaxially birefringent medium with refractive indi-ces that deviate slightly from an average refractive indexn. The cone of light converging to focus has a top angle2b making NA 5 n sin b. The distance between the in-terface and the focal point is d. In the Debye approachthe electromagnetic field close to the focal point is ex-pressed as a superposition of plane waves. Each of theseplane waves corresponds to a point on the referencesphere of radius R touching the exit pupil, as shown inFig. 1. The propagation direction of the plane wave isalong the line between the corresponding point on the ref-erence sphere and the (geometrical) focal point of the lens.Clearly, plane waves with wave vectors k 5 nkk, with k5 2p/l, and

2004 Optical Society of America

1786 J. Opt. Soc. Am. A/Vol. 21, No. 9 /September 2004 Sjoerd Stallinga

k 5 ~sin u cos f, sin u sin f, cos u!, (1)

contribute to focus if the polar angle u < b. This sharpcutoff in the angular spectrum of plane waves is the basicassumption of the Debye approach and is valid providedthe Fresnel number N is much larger than 1. TheFresnel number is typically the ratio of the pupil radiusa 5 R sin b and the diffraction length l/NA. The pupilradius is taken to be of the order of 1 mm and the diffrac-tion length is of the order of 1 mm for visible light andmoderate to high NA, giving a Fresnel number of the or-der of 103. It follows that the Debye approach may besafely applied. Even for small NA values, well within theparaxial regime, the Fresnel number is still sufficientlylarge that the Debye approach can be applied. The lowerlimit for NA is '1022 for visible light and a fixed pupil ra-dius of the order of 1 mm. A more extensive discussion ofthe validity of the Debye aproach can be found in the textsby Stamnes.26,27

It is often convenient to map this part of the referencesphere to the unit circle. The position vector of a point inthe unit circle

r 5 ~r cos f, r sin f ! (2)

is associated with a direction parametrized by u and f forthe following definition of the radial pupil coordinate r:

r 5 sin u/sin b. (3)

It follows that r is a measure of the distance between theoptical axis and the point on the reference sphere associ-ated with the plane wave with polar angle u.

Associated with each plane wave are the p- ands-polarization vectors (see Fig. 1):

p 5 ~cos u cos f, cos u sin f, 2sin u!, (4)

s 5 ~2sin f, cos f, 0!. (5)

The vectors p, s, and k make an orthonormal set with p3 s 5 k. Both the electric and magnetic fields in the farfield R @ l are in the plane spanned by p and s.

The goal of our work is to calculate the electromagneticfield in the proximity of the geometrical focus. This pointis taken to be the center of our coordinate frame. A pointrp in the proximity of focus can be parameterized inspherical coordinates as

Fig. 1. Reference sphere centered at the focal point O. The ra-dius is R and the maximum polar angle, defined by the apertureof the lens, is b. A point Q on the reference sphere correspondsto a plane wave contributing to the field at focus. The unit vec-tor k along the wave vector is parametrized by the polar angle uand the azimuthal angle f, where u is always smaller than b.The electric field is a linear combination of the polarization vec-tors p and s.

rp 5 ~rp sin up cos fp , rp sin up sin fp , rp cos up!. (6)

A different parametrization is in (normalized) cylindricalcoordinates as

rp 5 S l

NAv cos fp ,

l

NAv sin fp ,

nl

NA2u D , (7)

where u and v are the normalized axial and radial coor-dinates, respectively (see Fig. 2).

The literature on vectorial diffraction based on the De-bye approach can be summarized by the following for-mula for the components of the electric field E at a pointrp in the proximity of focus:

Ea~rp! 5E0pN

i (j51,2

Faj~rp!Aj , (8)

for a 5 x, y, z. Here E0 is the amplitude in the entrancepupil; A1 and A2 are the x and y components of the polar-ization vector in the entrance pupil, respectively; N5 R sin2 b/l is the Fresnel number; and the functions Fajare defined by the integrals over the reference sphere (theangular range 0 < u < b, 0 < f < 2p) as

Faj~rp! 51

p sin2 bE

RdV cos u (

l51,2v laJlj exp~iF!

51

pE

Rd2r (

l51,2v laJlj exp~iF!, (9)

where dV 5 sin u dudf, F is the phase, v l are the polar-ization vectors, and J is a 2 3 2 matrix, the Jones matrix.All quantities depend on the pupil coordinates r and f, orequivalently, u and f. The polarization vectors are a lin-ear combination of p and s:

v1 5 cos fp 2 sin f s

5 ~cos u cos2 f 1 sin2 f, 2~1

2 cos u!sin f cos f, 2sin u cos f !, (10)

v2 5 sin fp 1 cos f s

5 ~2~1 2 cos u!sin f cos f, cos2 f

1 cos u sin2 f, 2sin u sin f !, (11)

and do not depend on the image position rp . The Jonesmatrix does not depend on the image position either.The phase is the only quantity that depends on the imageposition rp , and can be written as

Fig. 2. Parametrization of the wavevector k by the polar andazimuthal angles u and f, respectively, and of the position vectorrp by spherical coordinates rp , up , and fp and by dimensionlesscylindrical coordinates u, v, and fp .


F 5 k–rp

5 nkrp~sin u sin up cos~f 2 fp! 1 cos u cos up!

5 2pvr cos~f 2 fp! 22pur2

1 1 ~1 2 r2 sin2 b!1/2

12pu

sin2 b. (12)

The second line gives the expression in spherical coordi-nates, the third in dimensionless cylindrical coordinates.The third term of the third line, a phase term not depend-ing on the pupil coordinates, is usually left out. It isnoted that the magnetic field can be expressed similarlyto the electric field, but with polarization vectors wl thatare found from the electric polarization vectors v l by thesubstitutions p → s and s → 2p.

B. Jones Matrix in Known CasesThe Jones matrix can take different forms. In case theamplitude and phase are constant over the exit pupil ofthe focusing lens, it is equal to the unit matrix

J 5 F1 0

0 1G . (13)

When the amplitude and phase vary over the exit pupilthe Jones matrix is given by

J 5 B exp~iW !F1 0

0 1G , (14)

with B and W (which are functions of r) the apodizationand aberration function, respectively. The lens is oftenconsidered to be aplanatic. Assuming a uniform ampli-tude in the entrance pupil, the apodization function isthen given by

B 51

Acos u5 ~1 2 r2 sin2 b!21/4. (15)

This is the original case considered by Richards andWolf.3 A more complicated case is that of focusingthrough an interface between a medium with refractiveindex n1 and a medium with n2 5 n. It is often the casethat the spherical aberration introduced in this manner iscorrected for by the lens. For example, in the practice ofoptical disk readout the scanning objective lens is de-signed such that the spherical aberration induced by asubstrate–cover layer of nominal thickness is compen-sated for, leading to a diffraction-limited scanning spot.Residual spherical aberration is then related to a devia-tion of the thickness from the nominal thickness or to de-viations in the refractive index from the assumed nominalvalue of the plastic. For such corrected systems thephase factor corresponding to spherical aberration can beleft out. Furthermore, the distortion of the focal spot byspherical aberration can overshadow the more subtle ef-fects of the interface itself, or of the birefringence of themedium. As this paper deals with the latter effect, thespherical aberation is ignored in our treatment. Thisleads to a Jones matrix:

J 5 R~f !F tp 0

0 tsGR~2f !

5 F tp cos2 f 1 ts sin2 f ~tp 2 ts!sin f cos f

~tp 2 ts!sin f cos f ts cos2 f 1 tp sin2 fG .

(16)

Here R(f ) is the rotation matrix,

R~f ! 5 F cos f 2 sin f

sin f cos fG , (17)

and tp and ts are the p and s Fresnel coefficients,

tp 52n1 cos u1

n1 cos u2 1 n2 cos u1, (18)

ts 52n1 cos u1

n1 cos u1 1 n2 cos u2, (19)

where u1 and u2 5 u are the angles of incidence in me-dium 1 and 2, respectively, related by Snell’s law. Thematrix R(2f )JR(f ) is the Jones matrix on the basisformed by the p- and s-polarization vectors, i.e., it relatesthe complex p and s amplitudes on the exit side to thecomplex p and s amplitudes on the entrance side. Theapodization factor related to the aplanatic condition isnow B 5 1/Acos u1. The case of a single interface can begeneralized to the case of multilayer interference stacks,which may include evanescent wave coupling throughlayers of subwavelength thickness.

A somewhat more complicated case is that of a uniaxi-ally birefringent medium with its principal axis along theoptical axis of the system (the z axis). This is the case ofaxial birefringence treated in Ref. 1. The conventionaldesignation ‘‘optic axis’’ for the principal or symmetry axisof the uniaxial medium is avoided to prevent confusionwith the term ‘‘optical axis’’ (the line through the center ofthe pupil and the geometrical focal point). Furthermore,similar nomenclature is used for uniaxial and biaxial me-dia. For biaxial media the three principal axes (the axesthat define a frame in which the dielectric tensor is diag-onal) do not coincide with the optic axes (the axes alongwhich the two orthogonal polarization modes have thesame refractive index).

The ordinary and extraordinary refractive indices are nand n 1 Dn, respectively, and the birefringence Dn is as-sumed to be small compared to 1. The ordinary modecorresponds to the field component that is tangentially ors polarized in the exit pupil of the lens, whereas the ex-traordinary mode corresponds to the field that is radiallyor p polarized in the exit pupil of the lens. This impliesthat the Jones matrix on the basis of the p and s polariza-tion vectors is diagonal. The diagonal terms of this ma-trix depend on the p and s aberration functions, which aregiven by

Wp 5 W 1 DW 5 W 1kdDn sin2 u8

cos u8, (20)

Ws 5 W. (21)

Here W is the classical isotropic aberration function, d isthe distance between the entrance interface and the geo-


metrical focus, and u8, the polar angle of the wave vectorinside the medium, is related to the dimensionless pupilcoordinate r by sin u8 5 rNA/n.

When focusing into the birefringent layer is considered,the effect of the interface must also be taken into account.This may be done with an interface matrix that gives thecoupling between the s and p amplitudes of the field inthe isotropic media in front of the birefringent mediumand the e and o amplitudes of the field inside the medium.Such interface matrices can be considerably simplified inthe limit of small birefringence.25 Then the interfacecoupling may be approximated by neglecting the birefrin-gence of the medium. The effect of birefringence then ap-pears only in the phases accumulated by the polarizationeigenmodes as the wave traverses the medium to the focalpoint; the effect of the birefringence on the amplitudes ofthe polarization eigenmodes is neglected. It turns out1

that for focusing into the birefringent medium the overallJones matrix is JbirJen , where Jen is the interface Jonesmatrix as defined in Eq. (16) and Jbir is the birefringenceJones matrix, defined by

For focusing through a birefringent slab the overall Jonesmatrix is JexJbirJen , where Jex is the interface matrix cor-responding to the exit side of the slab. The factorizationof Jones matrices for focusing through the birefringentslab is accurate only if multiple reflections can be ne-glected.

C. Electromagnetic Energy DensityThe energy density of the electric field is U5 1/2«0«abEaEb* , where the dielectric tensor «ab is as-sumed to be real. This may be approximated by the sumover the energy densities of the x, y, and z components ofthe electric field, where the energy density of the a com-ponent of the electric field is Ua 5 1/2«0n2uEau2, becausethe difference between the refractive indices is assumedto be much smaller than the average refractive index n.By introducing the scaling factor

U0 512 «0n2p2E0

2N2, (23)

it is found that

Ua 5 U0 (j,l51,2

FajFal* AjAl*

5 U0 (m50,3

IamMm , (24)

with the quantities

Ia0 512 ~ uFa1u2 1 uFa2u2!, (25)

Jbir 5 R~f !Fexp~iWp! 0

0 exp~iWs!GR~2f !

5 F exp~iWp!cos2 f 1 exp~iWs!sin2 f @exp~iWp! 2 ex

@exp~iWp! 2 exp~iWs!#sin f cos f exp~iWs!cos2 f

Ia1 512 ~ uFa1u2 2 uFa2u2!, (26)

Ia2 5 Re$Fa1Fa2* %, (27)

Ia3 5 Im$Fa1Fa2* %, (28)

and the Stokes parameters Mj , which describe the stateof polarization in the entrance pupil:

M0 5 uA1u2 1 uA2u2 5 1, (29)

M1 5 uA1u2 2 uA2u2 5 cos 2e cos 2j,(30)

M2 5 2 Re$A1* A2% 5 cos 2e sin 2j,(31)

M3 5 2 Im$A1* A2% 5 sin 2e. (32)

Here e is the ellipticity angle and j is the angle betweenthe long axis of the polarization ellipse and the x axis.Eq. (24) is valid for arbitrary states of polarization in theentrance pupil. Finally, it is noted that expressions forthe magnetic energy density 1/2m0H2 and for the Poyn-

ting vector S 5 Re$E 3 H* % can be derived along similarlines.

3. JONES MATRIX IN THE CASE OFBIAXIAL BIREFRINGENCEA. General Expression for the Jones MatrixIn this section we consider the generalization of the Jonesmatrix for the case of axial birefringence to the case of bi-axial birefringence. The biaxially birefringent layer hasrefractive indices n1 , n2 , and n3 , and principal axes

a1 5 ~cos g, sin g, 0!, (33)

a2 5 ~2sin g, cos g, 0!, (34)

a3 5 ~0, 0, 1 !; (35)

i.e., one of the principal axes is taken parallel to the opti-cal axis. The three refractive indices are parameterizedas

n1 5 n/~1 2 Dn i /n !1/2, (36)

n2 5 n/~1 1 Dn i /n !1/2, (37)

n3 5 n/~1 2 2Dn' /n !1/2, (38)

where n is an average refractive index, Dn i is the lateralbirefringence, and Dn' is the axial birefringence. Thisparametrization is somewhat arbitrary and is motivatedprimarily by future convenience. In the limit of small bi-refringence these expressions may be approximated by

n1 5 n 1 Dn i/2, (39)

n2 5 n 2 Dn i/2, (40)

n3 5 n 1 Dn' . (41)

Ws!#sin f cos f

exp~iWp!sin2 fG . (22)

p~i

1


There are two polarization eigenmodes with aberrationfunctions W1 and W2 . They correspond to two orthogo-nal polarization fields that are a linear sum of the p- ands-polarization fields. The angle between the 1mode po-larization vector and p (which is equal to the angle be-tween the 2mode polarization vector and s) is x 2 f (seeFig. 3). With R(2f )JR(f ) as the Jones matrix on thebasis formed by the p- and s-polarization vectors, theJones matrix J then follows as

The aberration functions turn out to be

W6 5kd

2 cos u8@a 6 ~b2 1 c2!1/2#, (43)

and the angle x follows from

cos~2x 2 2f ! 5 b/~b2 1 c2!1/2, (44)

sin~2x 2 2f ! 5 c/~b2 1 c2!1/2, (45)

where the quantities a, b, and c are defined as

a 5 Dn' sin2 u8 212 Dn i sin2 u8 cos~2g 2 2f !, (46)

b 5 Dn' sin2 u8 112 Dn i~1 1 cos2 u8!cos~2g 2 2f !,

(47)

c 5 Dn i cos u8 sin~2g 2 2f !, (48)

and where u8 is given by sin u8 5 rNA/n. A formal deri-vation of these results is presented in Subsection 3.B. Itis remarked that the overall Jones matrix must also takeinto account the interfaces between the birefringent layerand the isotropic entrance medium. In the limit of smallbirefringence this interface can be described by the isotro-pic Fresnel coefficients, just as discussed in Subsection2.B for the special case of axial birefringence. The over-all Jones matrix for focusing into the birefringent me-dium is then JbirJen , where Jbir is the birefringence Jonesmatrix of Eq. (42) and Jen is the interface Jones matrix ofEq. (16). For focusing through a birefringent slab the

J 5 R~x!Fexp~iW1! 0

0 exp~iW2GR~2x!

5 F exp~iW1!cos2 x 1 exp~iW2!sin2 x @exp~iW1! 2 exp

@exp~iW1! 2 exp~iW2!#sin x cos x exp~iW1!sin2 x 1

Fig. 3. Polarization fields of the two eigenmodes E1 and E2

make an angle x 2 f with p and s, respectively. The angle be-tween the eigenpolarizations and the Cartesian x and y axes istherefore x. This angle x can vary with the pupil point Q.

overall Jones matrix is JexJbirJen , where Jex is the Jonesmatrix of the exit surface, and where multiple reflectionsare neglected.

In the special case of uniaxial birefringence with prin-cipal axis along the optical axis, so-called axial birefrin-gence (Dn i 5 0), the aberration functions (taking Dn'

. 0) are

W1 52pdDn' sin2 u8

l cos u8, (49)

W2 5 0, (50)

and the eigenpolarization angle is x 5 f, in agreementwith my previous paper.1 The two polarization eigen-modes are now the purely radially polarized field (ex-traordinary mode) and the purely azimuthally polarizedfield (ordinary mode). The difference aberration functionW1 2 W2 is proportional to r2 for small u8, meaning thatthe two focal spots are relatively defocused. The spheri-cal aberration terms of all orders that appear as well areless important. Figure 4 shows the aberration functions(with the average defocus subtracted) and polarization

fields of the two eigenmodes for Dn' 5 0.016 (and Dn i

5 0). The defocus difference between the two modesand the radial and azimuthal eigenpolarizations areclearly visible. The parameters that are used are n5 1.58, NA 5 0.85, d 5 100 mm, and l 5 405 nm andpertain to the readout of so-called Blu-ray disks.28–30

This new type of optical disk has a significantly higherstorage capacity ('25 Gbyte) than existing compact disks(CDs, 650 Mbyte) and digital versatile disks (DVDs, 4.7Gbyte) as a result of the use of a shorter wavelength (785nm for CD and 655 nm for DVD) and a larger NA (0.45–0.50 for CD and 0.60–0.65 for DVD).

In the special case of uniaxial birefringence with prin-cipal axis perpendicular to the optical axis—so-called lat-eral birefringence, (Dn' 5 2Dn i/2, i.e., n3 5 n2)—theaberration functions (taking Dn i . 0) are

W1 5pdDn i

l cos u8@1 2 2 sin2 u8 cos2~g 2 f !#, (51)

W2 5 2pdDn i

l cos u8. (52)

The aberration function of the extraordinary mode con-tains terms r2 cos(2g 2 2f ) and r2 for small u8, meaningthat this mode suffers from astigmatism and defocus.The aberration function of the ordinary mode suffers fromdefocus alone. The amount of defocus in the two modesis the same and can therefore be eliminated by a shift of

2!#sin x cos x

xp~iW2!cos2 xG . (42)

~iW

e


the image plane. The eigenpolarization angle x is givenby

cos~2x 2 2f ! 5cos~2g 2 2f ! 2 sin2 u8 cos2~g 2 f !

1 2 sin2 u8 cos2~g 2 f !,

(53)

sin~2x 2 2f ! 5cos u8 sin~2g 2 2f !

1 2 sin2 u8 cos2~g 2 f !. (54)

For small polar angles u8 it follows that x 5 g. For largepolar angles u8, deviations occur that are of second orderin u8. Figure 5 shows the aberration functions (with theaverage defocus subtracted) and polarization fields forDn i 5 22Dn' 5 0.016 (and the same values for n, NA, d,and l as for the axial birefringence case). The saddleshape of the extraordinary aberration function and the

near flatness of the ordinary aberration function implythat there is no defocus difference and that the extraordi-nary mode is astigmatically aberrated. The eigenpolar-izations turn out to be indeed nearly uniformly linear.

For a general biaxial layer a crossover between the sa-lient features of the two uniaxial cases takes place as theratio Dn i /Dn' varies. The pupil point on the optical axiswhere W1 5 W2 for Dn i 5 0 splits into two points whereW1 5 W2 when Dn i Þ 0. These zero-retardation pointsin the pupil correspond to the optic axes of the biaxial me-dium. These two distinct optic axes of the biaxial me-dium are at the heart of the phenomenon of conicalrefraction.31 The absence of retardation for propagationalong the optic axes has been used by Goodman andMansuripur32 to measure the birefringence parameters.The polar angle of incidence (in the birefringent layer) ucis given by (take Dn i . 0 and Dn' . 0)

Fig. 4. Aberration functions of the (a) extraordinary and (b) ordinary modes, and polarization fields of the (c) extraordinary and (d)ordinary modes as a function of pupil coordinates for a uniaxially birefringent layer with optic axis parallel to the optical axis. Thecircle in (c) and (d) indicates the pupil rim. We have taken Dn' 5 0.016, Dn i 5 0, n 5 1.58, NA 5 0.85, d 5 100 mm, and l5 0.405 mm. The aberration functions are scaled by 2p, and the average defocus has been subtracted. The arrows have been addedfor the sake of clarity; their direction is arbitrary from a physical point of view.


Fig. 5. Aberration functions of the (a) extraordinary and (b) ordinary modes, and polarization fields of the (c) extraordinary and (d)ordinary modes as a function of pupil coordinates for a uniaxially birefringent layer with optic axis perpendicular to the optical axis.The circle in the bottom panels indicates the pupil rim. We have taken Dn i 5 22Dn' 5 0.016, n 5 1.58, NA 5 0.85, d 5 100 mm, andl 5 0.405 mm. The aberration functions are scaled by 2p, and the average defocus has been subtracted. The arrows have been addedfor the sake of clarity; their direction is arbitrary from a physical point of view.

tan2 uc 52Dn i

Dn i 1 2Dn'

. (55)

For polar angles smaller than uc the aberration functionshave a saddle shape of opposite sign (representing astig-matism); for polar angles larger than uc the aberrationfunctions have a parabolic shape of opposite sign (repre-senting defocus). As Dn i /Dn' is increased the zero-retardation points at uc move out of the pupil, giving twolargely astigmatic aberration functions. When Dn' ap-proaches zero one of the two aberration functions flattensout. The eigenpolarizations reflect the same crossoverbehavior. For polar angles smaller than uc the eigenpo-larizations are close to the symmetry axes at angles g andg 1 p/2, similar to the case of lateral birefringence,whereas for polar angles larger than uc the eigenpolariza-

tions are close to the radial and azimuthal directions,similar to the case of axial birefringence. Along the op-tical axes the eigenpolarizations are ill defined. Whenthe eigenpolarizations are parallel transported over a looparound this singularity they experience a geometric phasep, i.e., they change sign.33,34 Figure 6 shows the aberra-tion functions and polarization fields for Dn' 5 0.016 andDn i 5 0.002 (and the same values for n, NA, d, and l asfor the uniaxial birefringence cases). The differences be-tween the part of the pupil close to the optical axis, whichis similar to the lateral birefringence case, and the part ofthe pupil close to the rim, which is similar to the axial bi-refringence case, are as described above.

Stamnes and coworkers24 have considered the limitingcase of a small NA in some detail. When NA ! n sin ucthe intricate behavior of the eigenmodes close to the optic


axes does not play a role. The eigenpolarizations arethen substantially uniformly linear across the pupil atangles g and g 1 p/2, and both eigenmodes are astig-matically aberrated. The astigmatic distances obtainedfrom Eq. (43) in the limit of small NA are d(Dn i/22 Dn') for the 1mode and d(Dn i/2 1 Dn') for the2mode. This is in agreement in the limit of small bire-fringence with the expressions (n1

2 2 n32)/n1n3

2(z12 z0)/n (1) and (n3

2 2 n22)/n2n3

2(z1 2 z0)/n (1) obtainedin Ref. 24 @n (1) is the refractive index of the isotropic me-dium in front of the medium and z1 2 z0 is the distancebetween the front surface and the focal point before in-serting the medium into the converging beam of light, andd 5 (z1 2 z0)n/n (1)].

B. Derivation of Aberration Functions andEigenpolarization VectorsIn this subsection the expressions for W6 and x are for-mally derived. This subsection may be skipped by theless mathematically inclined readers.

The starting point is a formal plane-wave solution ofMaxwell’s equations in the biaxially birefringent medium.Consider a plane wave with wave vector k propagating inthe medium. The components of k are given by

kx 5 nk sin u8 cos f, (56)

ky 5 nk sin u8 sin f, (57)

kz 5 nk cos u8, (58)

where n is the refractive index of the medium, to be de-termined later on. Orthogonal to k are polarization vec-tors p and s:

p 5 ~cos u8 cos f, cos u8 sin f, 2sin u8!, (59)

s 5 ~2sin f, cos f, 0!. (60)

The magnetic field and induction can be eliminated fromMaxwell’s equations for a plane wave, giving the followingequation for the electric field E and the dielectric dis-placement D:

Fig. 6. Aberration functions of the (a) extraordinary and (b) ordinary modes and polarization fields of the (c) extraordinary and (d)ordinary modes as a function of pupil coordinates for a biaxially birefringent layer. The circle in (c) and (d) indicates the pupil rim. Wehave taken Dn' 5 0.016, Dn i 5 0.002, n 5 1.58, NA 5 0.85, d 5 100 mm, l 5 0.4059 mm. The aberration functions are scaled by 2p,and the average defocus has been subtracted. The arrows have been added for the sake of clarity; their direction is arbitrary from aphysical point of view. This is indicated by the opposing arrows on the line connecting the zero-retardation points.


1

«0

v2

c2Da 5 @k2dab 2 kakb#Eb . (61)

Clearly, D is perpendicular to k. As a consequence, Dmay be written as

Da 5 Dppa 1 Dssa . (62)

The electric field may be eliminated by using the consti-tutive relation

Da 5 «0«abEb , (63)

with the dielectric tensor

«ab 5 F cos g 2 sin g 0

sin g cos g 0

0 0 1GF n1

2 0 0

0 n22 0

0 0 n32G

3 F cos g sin g 0

2sin g cos g 0

0 0 1G , (64)

giving

Ea 51

«0~Dp«ab

21pb 1 Ds«ab21sb!. (65)

Substitution results in the following equations for Dp andDs :

FM11 2 1/n2 M12

M21 M22 2 1/n2G FDp

DsG 5 F00 G , (66)

with the matrix elements

M11 5 pa«ab21pb 5

cos2 u8 cos2~g 2 f !

n12

1cos2 u8 sin2~g 2 f !

n22

1sin2 u8

n32

,

(67)

M12 5 pa«ab21sb 5 2S 1

n22

21

n12D cos u8

3 sin~g 2 f !cos~g 2 f !, (68)

M21 5 sa«ab21pb 5 2S 1

n22

21

n12D cos u8

3 sin~g 2 f !cos~g 2 f !, (69)

M22 5 sa«ab21sb 5

sin2~g 2 f !

n12

1cos2~g 2 f !

n22

. (70)

Nontrivial solutions of the characteristic equation Eq.(66) are found when the determinant of this symmetric2 3 2 matrix equals zero, or

@M11 2 1/n2#@M22 2 1/n2# 2 M122 5 0. (71)

The two solutions corresponding to the two eigenmodesare

1

n62

5M11 1 M22

27 F S M11 2 M22

2 D 2

1 M122G1/2

.

(72)

The eigenvectors D6 of the two modes are orthogonal be-cause of the symmetry of the 2 3 2 matrix in Eq. (66). Itfollows that the eigenvectors can be expressed in the fol-lowing form,

D1a 5 «0A1n12@cos~x 2 f !pa 1 sin~x 2 f !sa#,

(73)

D2a 5 «0A2n22@2sin~x 2 f !pa 1 cos~x 2 f !sa#,

(74)

where A1 and A2 are constants, to be fixed by the bound-ary conditions, and where x is an angle to be determinedfrom the characteristic equation (66). The correspondingelectric fields are

E1a 5 A1n12@cos~x 2 f !«ab

21pb 1 sin~x 2 f !«ab21sb#

5 A1@cos~x 2 f !pa 1 sin~x 2 f !sa 1 e1ka#,

(75)

E2a 5 A2n22@2sin~x 2 f !«ab

21pb 1 cos~x 2 f !«ab21sb#

5 A2@2sin~x 2 f !pa 1 cos~x 2 f !sa 1 e2ka#,

(76)

where e1 and e2 are proportional to the birefringence pa-rameters and are therefore small compared to one.These small electric field components are responsible forthe rays’ not being perpendicular to the wave fronts, i.e.,for the different refraction of the two eigenmodes. By us-ing the expressions for the refractive indices of the eigen-modes the angle x turns out to be given by

cos~2x 2 2f ! 5 2~M11 2 M22!/2

$@~M11 2 M22!/2#2 1 M122%1/2

,

(77)

sin~2x 2 2f ! 5 2M12

$@~M11 2 M22!/2#2 1 M122%1/2

.

(78)

The angle f is present in these equations for future con-venience. This completes the general solution.

The biaxially birefringent medium has an interfacewith air perpendicular to the z axis. A plane wave propa-gating in air along

kair 5 k~sin u cos f, sin u sin f, cos u!, (79)

will be refracted into two waves corresponding to the twomodes with polar angles u68 given by Snell’s law:

sin u 5 n6 sin u68 . (80)

As the refractive indices n6 depend on u68 this is an equa-tion for this angle. In general, angles u18 and u28 are dif-ferent. The resulting wave vectors of the two modes canbe expressed as

k6 5 k~sin u cos f, sin u sin f, ~n62 2 sin2 u!1/2!,

(81)

where Snell’s law and uk6u2 5 k2n62 are used.


The mode amplitudes A1 and A2 can be expressed interms of the p and s amplitudes of the isotropic mediumin front of the biaxially birefringent medium as

FA1

A2G 5 FT1p T1s

T2p T2sG FAp

AsG , (82)

where the numbers T6p and T6s describe the interfacecoupling. These numbers must be solved from theboundary conditions for the electromagnetic field at theinterface.

To this point no approximations have been made.From here on the small-birefringence approximation willbe used. The matrix elements Mjk may be rewritten interms of the birefringence parameters as

M11 1 M22

25

1

n22

1

n3 FDn' sin2 u68

21

2Dn i sin2 u68 cos~2g 2 2f !G , (83)

M11 2 M22

25 2

1

n3 FDn' sin2 u68 11

2Dn i~1

1 cos2 u68 !cos~2g 2 2f !G , (84)

M12 5 21

n3@Dn i cos u68 sin~2g 2 2f !#.

(85)

The angles u68 can be taken equal to the angle of refrac-tion in the isotropic limit of zero birefringence, i.e.,sin u68 5 sin u8 5 sin u/n. Equations (44) and (45) for theangle x now follow directly. Also, the electric field may betaken in the plane spanned by the polarization vectors pand s, as the component along k is proportional to thesmall-birefringence parameters. The resulting polariza-tion vectors of the 1 and 2 modes make an angle x 2 fwith p and s, respectively.

The refractive indices of the eigenmodes are given by

1

n62

51

n22

a 6 ~b2 1 c2!1/2

n3, (86)

where the parameters a, b, and c, defined in Eqs. (46),(47), and (48), are proportional to the birefringence pa-rameters. In the small-birefingence limit the refractiveindices of the two eigenmodes then follow as

n6 5 n 1 a 6 ~b2 1 c2!1/2, (87)

and the z component of the wave vector is

kz6 5 k~n62 2 sin2 u!1/2

5 kn cos u8 1k

2 cos u8@a 7 ~b2 1 c2!1/2#. (88)

In the limit of zero birefringence the z component is sim-ply kn cos u8. The lateral wave vector components arethen still the same as they are fixed by Snell’s law for re-fraction at the isotropic–birefringent interface. Conse-

quently, the phase difference between the nonzero bire-fringence case and the zero birefringence case is

W6 5 ~kz6 2 kn cos u8!d, (89)

which gives the aberration functions of the two eigen-modes according to Eq. (43).

The mode coupling at the interface can be simplified aswell. Effects of the small-birefringence parameters onthe interface matrix elements are only small amplitudeeffects and can be safely neglected. The interface canthus be described by the Fresnel coefficients tp and ts ofthe interface between air and an isotropic medium withrefractive index n. This is entirely similar to the ap-proach described in the text of Yeh25 for uniaxial media.From Eqs. (75) and (76) the following relations betweenthe mode amplitudes and the entrance pupil amplitudesA1 and A2 can now be obtained:

cos~x 2 f !A1 2 sin~x 2 f !A2

5 tp~cos fA1 1 sin fA2!, (90)

2sin~x 2 f !A1 1 cos~x 2 f !A2

5 ts~2sin fA1 1 cos fA2!, (91)

leading to

FA1

A2G 5 R~2x!JenFA1

A2G , (92)

where Jen is the interface Jones matrix as given in Eq.(16). The overall Jones matrix for focusing into the biax-ially birefringent medium then follows as JbirJen , whereJbir is the birefringence Jones matrix as defined in Eq.(42).

4. NUMERICAL RESULTSA. Multipole Expansion of the Diffraction IntegralsThe diffraction integrals are evaluated in terms of a mul-tipole expansion.35 The method used is a variation of themethod proposed by Sheppard and Torok.11 The diffrac-tion integrals can be cast in the form

Faj~rp! 51

4pE

0

b

du sin uE0

2p

dfVaj~u, f !exp~ik • rp!,

(93)

where

Vaj~u, f ! 54 cos u

sin2 b(

s51,2vasJsj , (94)

is the pupil function, and where the phase may be writtenas

k • rp 5 nkrp cos z

5 nkrp@cos u cos up 1 sin u sin up cos~f 2 fp!#,

(95)

where the parametrization of rp in spherical coordinatesis used. Using Bauer’s formula the phase term may berewritten as


Fig. 7. Intensity distribution in the meridional planes parallel (left) and perpendicular (right) to the linear entrance polarization for thecase of axial birefringence. Parameters used are Dn' 5 0.016, n 5 1.58, NA 5 0.85, d 5 100 mm, and l 5 0.405 mm. The intensitydistribution is plotted logarithmically to show relatively fine details.

Fig. 8. Intensity distribution in the meridional planes parallel (left) and perpendicular (right) to the linear entrance polarization for thecase of lateral birefringence with the entrance polarization perpendicular to the principal axis (ordinary mode). Parameters used areDn i 5 2Dn' 5 0.016, n 5 1.58, NA 5 0.85, d 5 100 mm, l 5 0.405 mm. The intensity distribution is plotted logarithmically to showrelatively fine details.

`

l Flm 5 ~2i !lb

du sin u2p

dfY* ~u, f !V ~u, f !.
exp~inkrp cos z! 5 (l50
i ~2l 1 1 !jl~nkrp!Pl~cos z!,

(96)

with jl(x) the spherical Bessel function of order l andPl(x) the Legendre polynomial of order l. This Legendrepolynomial may be rewritten in terms of spherical har-monics by using the addition theorem,

Pl~cos z! 54p

2l 1 1 (m52l

l

Ylm* ~u, f !Ylm~up , fp!,

(97)

leading to

exp~ik • rp!

5 4p(l50

`

(m52l

l

i lYlm* ~u, f !Ylm~up , fp!jl~nkrp!. (98)

Substitution in the diffraction integral gives the final re-sult,

Faj~rp! 5 (l50

`

(m52l

l

FajlmYlm~up , fp!jl~nkrp!, (99)

with the coefficients

aj E0

E0

lm aj

(100)

Clearly, we have expanded the field close to focus in termsof partial waves or multipoles, which are elementary so-lutions of the scalar wave equation. As each electric fieldcomponent satisfies the wave equation it follows directlythat such an expansion is possible. The surprising ad-vantage of the Debye approach is that the coefficients ofthe expansion Faj

lm can be easily expressed in terms of thepupil function. This approach is similar in spirit to thatof Kant,9,10 later followed in Ref. 14, the difference beingthat Kant’s expansion is based on the Gegenbauer poly-nomials instead of the spherical harmonics of the presentmethod. The use of the multipole expansion was firstproposed by Sheppard and Torok.11 In the simple casewithout aberrations or birefringence they even showedthat the three components of the full vectorial field can bederived from a single set of multipole coefficients. Fi-nally, it is mentioned that the multipole expansion of theimage field of Eqs. (99) and (100) can also be derived fromthe eigenfunction expansion of the Green’s function,35 thefar-field behavior of spherical Hankel functions, and the(second) Rayleigh–Sommerfeld form of the diffraction in-tegral.


The multipole expansion works best for moderate tohigh NA values. Then the spatial extent of the focal spotis relatively small, meaning that the argument of thespherical Bessel function is relatively small. For smallarguments the terms in the series decrease rapidly andthe cutoff order lmax may be taken to be relatively low.For small NA, large birefringence, large aberrations, orfar from focus, more terms in the expansion are needed toproduce accurate results. The method is implemented asfollows. First the pupil function is sampled, typically 16azimuthal values and 129 radial values. The fast-Fourier-transform (FFT) method is applied to the azi-muthal dependence and Romberg integration to the ra-dial dependence.36 For NA 5 0.85 the cutoff at l 5 32and m 5 8 gives accurate results. Calculating the ex-pansion coefficients and the image field in '4000 pointstakes '1 s on present-day workstations.

A different eigenfunction method has recently been pro-posed by Janssen,37 and by Braat, Dirksen, andJanssen.38 Their eigenfunctions can be expressed as afast-converging Taylor series in a defocus parameter ofwhich the coefficients can be analytically expressed interms of Bessel functions of integer order. The main ad-vantage of eigenfunction expansions compared with bruteforce, FFT methods is that the same expansion coeffi-cients can be used for all values of the axial position–defocus value. The FFT method requires a new calcula-tion for each axial position–defocus value. On the otherhand, efficient ways to evaluate FFTs are available in theliterature.39 It is an open question which method is themost efficient one for computing diffraction integrals.

B. IsophotesFigure 7 shows contour plots of the intensity in the me-ridional planes parallel and perpendicular to the linearentrance polarization for focusing by an aplanatic lensinto an axially birefringent layer with Dn' 5 0.016, d5 100 mm, l 5 0.405 mm, and NA 5 0.85. These pa-

rameters correspond to the aberration functions and po-larization fields of Fig. 4. The lines of constant intensityare called isophotes. The plots clearly show the two focalspots, corresponding to the ordinary and extraordinarymodes, with different defocus. The uniform linear en-trance polarization couples to both eigenmodes, as theeigenpolarizations are radially and azimuthally polar-ized. The entrance polarization couples differently to theeigenmodes in the planes parallel and perpendicular tothe entrance polarization, leading to the asymmetry be-tween the two focal spots in each meridional plane and tothe asymmetry between the two meridional planes. An-other consequence is the appearance of a secondary focalstructure, namely, the ring in the original focal plane.

Figures 8 and 9 show the same plots for a laterally bi-refringent layer with Dn i 5 2Dn' 5 0.016 for the linearentrance polarization perpendicular (ordinary mode) tothe principal axis and parallel (extraordinary mode) tothe principal axis. These parameters correspond to theaberration functions and polarization fields of Fig. 5.The ordinary mode is hardly affected by the birefringence,whereas the extraordinary mode is significantly aber-rated by astigmatism, as can be clearly seen from the twofocal ‘‘lines’’ that are orthogonal. When the entrance po-larization is such that both modes are excited (e.g., for acircular polarization) the intensity distribution in the fo-cal region of both modes becomes mixed.

Figure 10 shows intensity contour plots for a biaxiallayer with Dn i 5 0.002 and Dn' 5 0.016 for linear en-trance polarizations along the x axis and along the y axis.The birefringence parameters correspond to the aberra-tion functions and polarization fields of Fig. 6. Both en-trance polarizations show that one of the two axially dis-placed focal spots of the axially birefringent case hasbecome slightly astigmatic, whereas the other mixes withthe secondary focal ring in three distinct parts. Whenthe ratio Dn i /Dn' increases these structures reorganizeinto a hardly aberrated ordinary focal spot and an astig-

Fig. 9. Intensity distribution in the meridional planes parallel (left) and perpendicular (right) to the linear entrance polarization for thecase of lateral birefringence with the entrance polarization parallel to the principal axis (extraordinary mode). Parameters used areDn i 5 2Dn' 5 0.016, n 5 1.58, NA 5 0.85, d 5 100 mm, and l 5 0.405 mm. The intensity distribution is plotted logarithmically toshow relatively fine details.


Fig. 10. Intensity distribution in the meridional xz (left) and yz (right) planes for a linear entrance polarization along the x axis (top)and along the y axis (bottom) for the case of biaxial birefringence with Dn' 5 0.016, Dn i 5 0.002 (so n3 . n1 . n2), g 5 0, n 5 1.58,NA 5 0.85, d 5 100 mm, and l 5 0.405 mm. The intensity distribution is plotted logarithmically to show relatively fine details.

matically aberrated extraordinary focal spot. This reor-ganization is most likely accompanied by creation and an-nihilation of singular structures of the kinds described inRef. 40. This subject is not considered further here, butwould certainly be of interest for a separate study.

5. SUMMARY AND CONCLUSIONThe influence of focusing into a biaxially birefringentlayer on the distribution of light close to focus is investi-gated by means of vectorial diffraction theory. The dif-fraction integrals are evaluated within the framework ofthe Debye approach and for small birefringence values.The field close to focus is the superposition of the field ofthe two polarization eigenmodes of the birefringent layer.In the limiting case of uniaxial birefringence with princi-pal axis parallel to the optical axis, the most important ef-fect is the relative defocus of the two polarization eigen-modes; when the principal axis is perpendicular to theoptical axis the dominant effect is astigmatism in the ex-traordinary polarization eigenmode. For the biaxial casea crossover between the two cases occurs, as a result ofwhich, for small angles of incidence, both modes are as-tigmatic, while for large angles of incidence both modesare relatively defocused. For the intermediate case thebehavior is determined by the mode structure close to theoptic axes of the biaxial medium.

The resulting blurring of the focal spot can be describedby the deviation of the Strehl ratio from the ideal value of1. An analytical calculation of the Strehl ratio based on ageneralization of the ordinary scalar aberration theory ispresented in a companion paper.

ACKNOWLEDGMENTSJean Schleipen and Paul Urbach are thanked for a criticalreading of the manuscript.

Sjoerd Stallinga can be reached by e-mail [email protected].

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