Electromagnetic Diffraction in Optical Systems. I. an Integral Representation of the Image Field

$Page 1: Electromagnetic Diffraction in Optical Systems. I. an Integral Representation of the Image Field$
Torok et al. Vol. 12, No. 2 /February 1995 /J. Opt. Soc. Am. A 325

Electromagnetic diffraction of light focused througha planar interface between materials of mismatched

refractive indices: an integral representation

P. Torok

Department of Materials, University of Oxford, Oxford OX1 3PH, UK

P. Varga

Central Research Institute for Physics of the Hungarian Academy of Sciences,Research Institute for Materials Science, Budapest P.O. Box 49, H-1525, Hungary

Z. Laczik and G. R. Booker

Department of Materials, University of Oxford, Oxford OX1 3PH, UK

Received June 1, 1994; revised manuscript received August 29, 1994; accepted August 29, 1994

The diffraction of electromagnetic waves for light focused by a high numerical aperture lens from a firstmaterial into a second material is treated. The second material has a different refractive index from thatof the first material and introduces spherical aberration. We solve the diffraction problem for the case of aplanar interface between two isotropic and homogeneous materials with this interface perpendicular to theoptical axis. The solution is obtained in a rigorous mathematical manner, and it satisfies the homogeneouswave equation. The electric and magnetic strength vectors are determined in the second material. Thesolution is in a simple form that can be readily used for numerical computation. A physical interpretationof the results is given, and the paraxial approximation of the solution is derived.

1. INTRODUCTION

The structure of focused electromagnetic waves has beenstudied by a number of authors. Of particular interestfrom both the theoretical and practical points of viewis the diffracted-light distribution in the region of focuswhen light is focused by a glass lens. The results of suchinvestigations have helped in the design of better objectivelenses, especially those with high numerical apertures.

Focusing through materials of mismatched refrac-tive indices has attracted particular interest because ofmicroscopy applications in the biological and materialsciences. Microscope objective lenses have long beenmanufactured with aberration correction for certain butfixed penetration depths, e.g., lenses to be used with acover glass for biological applications. Microscope ob-jective lenses have recently been produced that providecontinuously variable correction for spherical aberrationintroduced by off-axis propagation of light through ma-terials of different refractive indices. Nevertheless, theimportance of investigations directed at a better under-standing of high aperture focusing is still great, and thisdemand is increasing, especially with new material sci-ence applications.

The literature that deals with the general focusingproblems of electromagnetic waves is well established.In an early paper Wolf1 treated high aperture focusing ofelectromagnetic waves in a single homogeneous material.The starting point of his elegant theory was the represen-tation of the angular spectrum of plane waves, from which

0740-3232/95/020325-08$06.00

integral formulas similar to the Debye integral2 wereobtained. Wolf and Li3 subsequently showed that thisapproach was valid for systems that satisfied the high-aperture condition. Although, during the derivation ofhis formulas, Wolf1 made a number of approximations,his transformed integral formulas were formally identi-cal to Luneburg’s4 representation of the Debye integral,which Luneburg had previously shown to be an exact so-lution of the homogeneous wave equation. Luneburg’sformulation is valid for an idealized problem, i.e., whenthe probe function and the field to be determined havethe same boundary values at infinity as their geometricaloptics solution. Kant5 recently showed how Wolf’s inte-gral formulas could be used when (Seidel) lens aberrationswere present in the optical system.

The first paper that dealt with the focusing of electro-magnetic waves into mismatched refractive index mate-rials was by Gasper et al.,6 who considered an arbitraryelectromagnetic wave as it traversed a planar interface.They also used the representation of the angular spec-trum of plane waves and considered the focusing prob-lem for isotropic homogeneous media. For a treatment ofdiffraction in anisotropic materials the reader is referredto Stamnes and Sherman.7,8 Gasper et al.6 derived arigorous solution of the problem and its asymptotic ap-proximation and gave expressions for the electric andmagnetic fields. However, because of the complexity oftheir theory, the use of their formulas for the computationof the electromagnetic field near focus was not practical inmost cases. Ling and Lee9 treated the focusing of electro-

1995 Optical Society of America

326 J. Opt. Soc. Am. A/Vol. 12, No. 2 /February 1995 Torok et al.

magnetic waves through an interface. A boundary condi-tion in the form of a current distribution was used as thestarting point, and the representation of the angular spec-trum of plane waves was applied. In a semigeometricalapproach, with the use of the stationary-phase method,expressions were obtained for the electric and magneticfields. However, Ling and Lee used approximations toobtain their integral formulas. Ji and Hongo10 treatedthe different problem of a point source and a spheri-cal dielectric interface, using Maslov’s method to obtainthe electric field in the focal region. Although the finalequations showed the same general form as those of thepresent paper, they could not be applied to the presentproblem. A comprehensive treatment of different focus-ing theories was later given by Stamnes.11

In a recent paper Hell et al.12 considered the focusingproblem for mismatched refractive index materials usingthe Fresnel–Kirchhoff integral. They decomposed theincident electric vector into s- and p-polarized parts andalso calculated the effect of spherical aberration on theimage formation for a confocal fluorescence microscope.However, the integral formula that they used is derivedfrom Green’s theorem, which requires the continuity ofthe field and its first and second derivatives within andon the boundary of the area of integration; this is not thecase for the normal component of the electric field andthe tangential component of the magnetic field, and sothe final integral formulas obtained may not be rigorouslycorrect.

The purpose of this paper is to extend Wolf’s treatmentof the diffraction problem for the case when an electro-magnetic wave is focused in a single medium of propaga-tion to the case when light is focused by a high-aperturesystem into a bulk specimen that has a refractive indexdifferent from that of the medium of propagation and in-troduces a considerable amount of spherical aberration.The starting point of our treatment is Wolf’s integral for-mulas, with which we derive the electromagnetic field justbefore the interface between the two media. The field isthen traversed across the interface by application of theFresnel refraction law to the individual plane waves inci-dent upon the interface. The field so derived just afterthe interface is used as the boundary condition for a sec-ond set of integral formulas corresponding to a superposi-tion of plane waves, which represent the field inside thesecond medium. In this way the diffraction problem issolved in a rigorous mathematical manner, and the solu-tion satisfies the homogeneous wave equation. We alsoobtain formulas for the strength vectors in the second ma-terial in a form more generally applicable than that ofthose previously published.

In our calculations the spherical aberration is intro-duced by the specimen rather than by an aberrationfunction related to the lens. The reason is that mod-ern high-performance lenses are generally well correctedfor spherical aberration. Therefore, in considering a realoptical system, we feel justified in the assumption of aperfect spherical wave emerging from the lens and con-verging toward the Gaussian focus point (with no spheri-cal aberration). Although strong spherical aberrationscan be treated by geometrical optics successfully describ-ing the main features of the focal region (e.g., caustic andcircle of least confusion), the fine structure and the elec-

tromagnetic properties (e.g., polarization and energy flow)of the field can be described only by a full electromagnetictreatment. This becomes increasingly important whenthe field so calculated is to be applied to wave scatter-ing by small objects situated close to the focus and whosesize is comparable with that of the wavelength, e.g., inbiological and material science applications.

The presentation of our paper is as follows: InSubsection 2.A we derive the integral representationof the electromagnetic field in the image space. InSubsection 2.B we decompose the electric and mag-netic vectors as each traverses the optical system. InSubsection 2.C we make use of the above results and givesimplified formulas for the electric and magnetic fields inthe second material. Finally, in Subsections 2.D and 2.Ewe consider the physical interpretation of our formulasand give a paraxial approximation of our solution.

2. PROBLEM, FORMULATION,AND SOLUTION

A. Integral RepresentationConsider an optical system of revolution with an opticalaxis z as shown in Fig. 1. This system images a pointsource that is situated in the object space at z 2` andradiates a linearly polarized monochromatic and coherentelectromagnetic wave. This wave is incident upon a lensof aperture S that produces a convergent spherical wavein the image space. The origin O of the sx, y, zd coor-dinate system is positioned at the Gaussian focus. Theelectric and magnetic fields are determined at the arbi-trary point P in the focal region. The aperture size andthe distance of P from the aperture was taken to be largecompared with the wavelength. In Fig. 1 and what fol-lows, s ssx, sy , szd is the unit vector along a typical rayin the image space and rp sx, y, zd is the position vec-tor pointing from O to P.

Let EsP , td indicate the time-dependent electric fieldand HsP , td indicate the time-dependent magnetic fieldat P at time t, so that

EsP , td RefEsP dexps2ivtdg ,

HsP , td RefHsP dexps2ivtdg , (1)

where Re indicates the real part.

Fig. 1. Diagram showing light focused by a lens into a singlemedium.


In a homogeneous image space the time-independentelectric and magnetic fields can be represented as a super-position of plane waves,2 and we use the form developedby Wolf1:

EsP d 2ik2p

ZZV

assx, sy dsz

exphikfFssx, sy d 1 s ? rpgj

3 dsxdsy , (2)

HsP d 2ik2p

ZZV

bssx, sy dsz

exphikfFssx, sy d 1 s ? rpgj

3 dsxdsy , (3)

where Fssx, sy d is the aberration function (describing theoptical path difference between the aberrated and thespherical wave front along s), a and b are the electricand magnetic strength vectors, respectively, of the unper-turbed electric and magnetic fields in the exit aperture S,k is the wave number, and V is the solid angle formed byall the geometrical optics rays (and which is therefore alimit for all the s unit ray vectors).

Before we proceed with the formulation, it is worthmentioning that Eqs. (2) and (3) represent a summation ofthe plane waves (unlike the Fresnel–Kirchhoff integral)that are leaving the aperture. It is also evident that theelectric and magnetic fields, as represented by Eqs. (2)and (3), do not depend on the particular wave front withinthe solid angle V over which the integration is performed.This statement can be proved in a rigorous mathemati-cal manner.4 Equations (2) and (3) also show that thephase factor (apart from the aberration function) is thescalar product of the vectors s and rp. It follows fromthe above that the phase factor expresses the optical pathdifference between wave fronts going through the point Pand the Gaussian focus O, unlike the Fresnel–Kirchhoffintegral, for which the phase factor is proportional to thefull optical path from the aperture to P.

In this section henceforth we shall not present thederivation of the formulas corresponding to the magneticfield because, apart from the strength vectors, Eqs. (2)and (3) are in the same form.

The case corresponding to Fig. 1, but for an image spaceconsisting of materials 1 and 2 with refractive indicesn1 and n2, respectively, separated by a planar interfaceperpendicular to the optical axis, is shown in Fig. 2. Theorigin O is again positioned at the Gaussian focus. Wereformulate Eq. (2) as follows. In material 1 and at theinterface sz 2dd the incident electric field is given by

E1sx, y, 2dd 2ik1

2p

ZZV1

W sedss1dexpfik1ss1xx

1 s1yy 2 s1zddgds1xds1y , (4)

where subscripts 1 and 2 denote values corresponding toregions with materials 1 and 2, respectively, the objectivelens is taken to be aberration free fFss1x, s1y d 0g, and

W sedss1d ass1x, s1y d

s1z

. (5)

To describe the field in the second material, we assumethat each plane wave component refracting at the inter-face obeys Fresnel’s refraction law, and the resulting field

is constructed as a superposition of refracted plane waves.If the amplitude of the plane waves incident upon the in-terface is described by W sed, then the amplitude of thetransmitted plane waves in the second material is a lin-ear function of W sed, i.e.,

T s2dW sed , (6)

where the operator T s2d is a function of the angle of inci-dence and n1 and n2. The transmitted field in the secondmaterial at the close vicinity sz 2d 1 dd of the inter-face is given by

E2sx, y, 2dd 2ik1

2p

ZZV1

T sedW sedss1dexpfik1ss1xx

1 s1yy 2 s1zddgds1xds1y (7)

when d ! 0. We represent the field inside the secondmaterial again as a superposition of plane waves. Thisrepresentation is a solution of the time-independent waveequation and can be written as

E2srpd 2ik2

2p

ZZV2

F sedss2dexpsik2s2 ? rpdds2xds2y . (8)

We must determine the function F sedss2d, and for this weshall make use of Eq. (7), which represents a boundarycondition for Eq. (8). Furthermore, we shall establishthe relationship between s1 and s2.

It is evident from the vectorial law of refraction,

k2s2 2 k1s1 sk2 cos f2 2 k1 cos f1du , (9)

where u represents the normal of the interface, f1 ss1, ud, and f2 ss2, ud, that

k2s2x k1s1x, k2s2y k1s1y . (10)

In what follows, we shall give a solution for Eq. (8) withthe boundary condition [Eq. (7)].

Fig. 2. Diagram showing light focused by a lens into two mediaseparated by a planar interface.


As a result of a coordinate transformation and settings2 f ss1d, Eq. (8) yields

E2srpd 2ik2

2p

ZZV1

F sedss2dexpsik2s2 ? rpd

3 JJJ 0ss1x, s1y ; s2x, s2ydds1xds1y , (11)

where JJJ 0 is the Jacobian of the coordinate transformation:

JJJ 0

√k1

k2

! 2

,

obtained by the use of Eqs. (10). Equation (11) satisfiesthe boundary condition represented by Eq. (7) when

F sedss1, s2d

√k2

k1

!T sed ass1x, s1y d

s1zexpf2idsk1s1z 2 k2s2zdg .

(12)

On substituting from Eq. (12) into Eq. (11), we obtain theelectric field in the second material:

E2sx, y, zd 2ik2

2

2pk1

ZZV1

T sed ass1x, s1y ds1z

3 expf2idsk1s1z 2 k2s2zdgexpsik2s2zzd

3 expfik1ss1xx 1 s1yydgds1xds1y . (13)

The same procedure yields a similar expression for themagnetic vector:

H2sx, y, zd 2ik2

2

2pk1

ZZV1

T smd bss1x, s1y ds1z

3 expf2idsk1s1z 2 k2s2zdgexpsik2s2zzd

3 expfik1ss1xx 1 s1yydgds1xds1y . (14)

From the above it also follows that in Eqs. (13) and (14)we have

s2z

√1 2

n12

n22

1n1

2

n22

s1z2

!1/2

"1 2

n12

n22

ss1x2 1 s1y

2d

#1/2

.

(15)

It is important to emphasize that, since both the bound-ary condition represented by Eq. (7) and the integralrepresentation [Eq. (8)] are exact solutions of the homo-geneous wave equation, our formulas for the electric vec-tor [Eq. (13)] and the magnetic vector [Eq. (14)] in thesecond material also satisfy the homogeneous wave equa-tion. Therefore we have successfully obtained a consis-tent extension of Wolf ’s solution in the second material.We shall show below in Subsection 2.E that the parax-ial approximation of Eq. (13) yields the well-known ax-ial distribution13 for one material, and it also correctlypredicts the axial position of the Gaussian focus for twomaterials.

B. Electric and Magnetic Strength VectorsThe determination of the electric and magnetic strengthvectors for a single medium of propagation was describedpreviously.14 Now we obtain these vectors for a plane po-larized wave incident upon the lens, and a material whose

refractive index is different from that of the medium ofpropagation is placed into the image space. The mate-rial is taken to be isotropic and homogeneous and to havean optically smooth planar surface that is perpendicu-lar to the optical axis. For our decomposition the usualassumptions are made, namely, that the electric vectormaintains its direction with respect to a meridional planeand the electric vector remains on the same side of ameridional plane on passing through the system.

For the optical system under consideration we denotethe angle of incidence on the interface by f1 and the angleof refraction by f2. The unit vectors s1 and s2 and thevector rp (Fig. 2) are given in spherical polar coordinatesby

s1 ssin f1dscos udi 1 ssin f1dssin udj 1 scos f1dk , (16)

s2 ssin f2dscos udi 1 ssin f2dssin udj 1 scos f2dk , (17)

rp rpfssin fpdscos updi 1 ssin fpdssin updj 1 scos fpdkg ,

(18)

where i, j , and k are the unit base vectors of the sx, y, zdorthogonal system and the spherical polar coordinatesr, f, and u are defined so that r . 0, 0 # f , p, and0 # u , 2p. The coordinate system is chosen so that they component of the incident electric vector is zero. Forthe incident electric vector Es0d in front of the lens we have

Es0d

0BB@E0

00

1CCA .

For a treatment of the refraction that occurs at the in-terface of the mismatched materials it is convenient todecompose the electric vector into s- and p-polarized vec-tor components Es and Ep, respectively, and to rotate thecoordinate system so that the new coordinate system willcontain components in the s p, s, z d system. This coordi-nate system is defined in such a way that Ez 0. Theelectric vector components Es1d

s p,s,z d after the lens are thenin the form

Es1ds p,s,z d Asf1dP s1dL R Es0d

sx,y,zd , (19)

where Asf1d is an amplitude function (defined below),the matrix R describes the coordinate transformation forrotation around the z axis,

R

264 cos u sin u 02sin u cos u 0

0 0 1

375 , (20)

the matrix L describes the changes in the electric field asit traverses the lens,

L

264 cos f1 0 sin f1

0 1 02sin f1 0 cos f1

375 , (21)

and the matrix P s1d describes the coordinate system rota-tion that generates Es and Ep components with Ez 0,

P s1d

264cos f1 0 2sin f1

0 1 0sin f1 0 cos f1

375 . (22)


It is interesting to note that in our case, when E s0dz 0,

the operations by L and P s1d cancel out.The electric field in the second material, Es2d, is given by

Es2dsx,y,zd R21fP s2dg21I Es1d

sp,s,z d , (23)

where the matrix I describes the effect of the interface,

I

2664tp 0 00 ts 00 0 tp

3775 , (24)

in which tp and ts are the Fresnel coefficients,

ts 2 sin f2 cos f1

sinsf1 1 f2d, tp

2 sin f2 cos f1

sinsf1 1 f2dcossf1 2 f2d,

(25)

the matrix fP s2dg21 describes the rotation of the coordinatesystem required to return it to the s p, s, zd system,

fP s2dg21

264 cos f2 0 sin f2

0 1 02sin f2 0 cos f2

375 , (26)

and the matrix R21 describes the inverse rotation aroundthe z axis.

From Eqs. (19)–(21) and (23) we obtain the componentsof the electric vector in the second material on settingE0 1:

Es2dsx,y.zd Asf1d

0BBB@tp cos f2 cos2 u 1 ts sin2 u

tp cos f2 sin u cos u 2 ts sin u cos u

2tp sin f2 cos u

1CCCA .

(27)

The function Asf1d can be regarded as an apodizationfunction that depends on the lens used in imaging.Richards and Wolf14 showed that when the system obeysAbbe’s sine condition, i.e., is aplanatic, then

Asf1d fl0 cos1/2 f1 , (28)

where f is the focal length of the lens in vacuo and l0 isan amplitude factor. Otherwise, this function can takedifferent forms.11

Having derived the electric vector Es2dsx,y,zd, we can write

the electric strength vector c as

c T seda Es2d . (29)

The magnetic strength vector d is then given by

d

√e2

m2

!1/2

s2 3 c , (30)

where m2 is the permeability and e2 is the permittivity ofthe second material. The magnetic strength vector fromEqs. (17), (27), (29), and (30) is given by

d

√e2

m2

!1/2

Asf1d

3

0BB@2tp cos u sin u 1 ts sin u cos u cos f2

tp cos2 u 1 ts sin2 u cos f2

2ts sin u cos f2

1CCA . (31)

We note that, for the special case f1 f2, e1 e2 1, and

m1 m2 1 (in the CGS unit system), Eqs. (27) and (31)reduce to the electric strength vector given by Richardsand Wolf,14 as they should.

C. Electric and Magnetic Field VectorsFirst we formulate the expressions needed to simplifyEqs. (13) and (14). We transform the integral variablesds1x and ds1y to the spherical polar coordinate system.The Jacobian of the transformation is

JJJp sin f1 cos f1 ;

thus

ds1xds1y sin f1 cos f1df1du . (32)

We define

k n1 sin f1 sin fp cossu 2 upd

1 n2 cos f2 cos fp , (33)

Csf1, f2, 2dd 2dsn1 cos f1 2 n2 cos f2d . (34)

So far we have constructed the quantities that we requireto determine the electric and magnetic vectors. With thehelp of the above results we can rewrite Eqs. (13) and(14) to express the electric sE2d and magnetic sH2d vectorsinside the second material:

E2srp, 2dd 2ik2

2

2pk1

ZZV1

csf1, f2, ud

3 exphik0frpk 1 Csf1, f2, 2ddgj

3 sin f1df1du , (35)

H2srp, 2dd 2ik2

2

2pk1

ZZV1

dsf1, f2, ud

3 exphik0frpk 1 Csf1, f2, 2ddgj

3 sin f1df1du , (36)

where k0 indicates the wave number in vacuo. On sub-stituting from Eqs. (27), (29), (31), and (32) into Eqs. (35)and (36), changing the integration limits, and assumingthat the system obeys the sine condition [Eq. (28)], weobtain the following expressions for the electric-field com-ponents:

e2x 1iK2p

Z a

0

Z 2p

0scos f1d1/2ssin f1d

3 fstp cos f2 1 tsd 1 scos 2udstp cos f2 2 tsdg

3 exphik0frpk 1 Csf1, f2, 2ddgjdf1du ,

e2y 1iK2p

Z a

0

Z 2p

0scos f1d1/2ssin f1dssin 2ud

3 stp cos f2 2 tsd


e2z 2iKp

Z a

0

Z 2p

0scos f1d1/2ssin f1dtp sin f2 cos u

3 exphik0frpk 1 Csf1, f2, 2ddgjdf1du , (37)

where

K k2

2fl0

2k1

pn22fl0

ln1

.


We also obtain the following expressions for the magnetic-field components:

h2x 1iKn2

2p

Z a

0

Z 2p

0scos f1d1/2ssin f1dssin 2ud

3 sts cos f2 2 tpd


h2y 1iKn2

2p

Z a

0

Z 2p

0scos f1d1/2ssin f1d

3 fstp 1 ts cos f2d 1 stp 1 ts cos f2dscos 2udg


h2z 2iKn2

p

Z a

0

Z 2p

0scos f1d1/2ssin f1dts sin f2 sin u

3 exphik0frpk 1 Csf1, f2, 2ddgjdf1du , (38)

where we assumed that m2 1 and set a to be the angularsemiaperture of the lens. The integration in Eqs. (37)and (38) can be carried out14 with respect to u, and fromthe result the electric- and magnetic-field componentscan be expressed as the combination of two sets of threeintegrals, I0, I1, and I2:

e2x 2iKfI sed0 1 I sed

2 coss2updg ,

e2y 2iKI sed2 sins2upd ,

e2z 22KI sed1 cos up , (39)

h2x 2iKn2I shd2 sins2upd ,

h2y 2iKn2fI shd0 2 I shd

2 coss2updg ,

h2z 22Kn2I shd1 sin up , (40)

where up is defined in Eq. (18) and we have put n2 p

e2.After we substitute the normalized optical coordinates

v k1sx2 1 y2d1/2 sin a k1rp sin fp sin a ,

u k2z sin2 a k2rp cos fp sin2 a (41)

into Eqs. (37) and (38), the integrals I0, I1, and I2 aregiven by

I sed0

Z a

0scos f1d1/2ssin f1dexpfik0Csf1, f2, 2ddg

3 sts 1 tp cos f2dJ0

√v sin f1

sin a

!

3 exp

√iu cos f2

sin2 a

!df1 ,

I sed1

Z a


3 tpssin f2dJ1

√v sin f1

sin a

!exp

√iu cos f2

sin2 a

!df1 ,

I sed2

Z a


3 sts 2 tp cos f2dJ2

√v sin f1

sin a

!

3 exp

√iu cos f2

sin2 a

!df1 ; (42)

I shd0

Z a


3 stp 1 ts cos f2dJ0

√v sin f1

sin a

!

3 exp

√iu cos f2

sin2 a

!df1 ,

I shd1

Z a


3 tsssin f2dJ1

√v sin f1

sin a

!exp

√iu cos f2

sin2 a

!df1 ,

I shd2

Z a


3 stp 2 ts cos f2dJ2

√v sin f1

sin a

!

3 exp

√iu cos f2

sin2 a

!df1 , (43)

where Jn is the Bessel function of the first kind, of order n.Equations (39), (40), (42), and (43) conclude our solutionof the problem. It should be noted that, for n1 n2 1(therefore f2 f1 and d 0), Eqs. (42) and (43) reduce toI sed

n I shdn In, which are then identical to the correspond-

ing equations of Richards and Wolf.14 Therefore our for-mulation of the vector diffraction theory with sphericalaberration introduced by an interface between two mate-rials of mismatched refractive indices is consistent withprevious results derived for the nonaberrated case.

We can see from Eqs. (42) and (43) that the time-independent electric and magnetic fields can be describednot by only one set of integral functions but rather bytwo such sets. The reason is that the electric and mag-netic vectors are orthogonal with respect to each otherand the polarization-dependent refraction at the interfaceacts differently on different electric and magnetic vectordirections. It is emphasized, however, that all the lawsthat describe the transition of these two vectors throughthe system are essentially the same, and the difference be-tween the two sets of integrals does not originate from thenature of the wave (whether it is an electric or a magneticwave) but rather depends on whether the wave incidentupon the interface possesses p or s polarization.

D. Physical InterpretationAlthough the formal solution of our problem has beenconcluded by Eqs. (39) and (40), we recall here Eq. (13)for an analysis of our results. This latter equation canbe rewritten in the form

E2sx, y, zd K 0ZZ

V1

W0ss1x0, s1y

0 d

3 expfi2pss1x0x, s1y

0ydgds1x0ds1y

0 , (44)

where

s1x0

n1s1x

l, s1y

0 n1s1y

l, (45)

W0 css1x, s1y d

s1zexpf2isk1s1z 2 k2s2zddgexpsik2s2zzd

css1x, s1y d

s1zexps2ik1s1zddexpfik2s2zsz 1 ddg , (46)


and K 0 is constant. If we extend the limit of integrationin Eq. (44) to the range f2`, `g, the latter equation be-comes a Fourier integral. The physical meaning of theabove extension is that the integration is performed overthe full p solid angle and not only over the area of theaperture. In other words, the integral operation is beingperformed not only in the range

√k1

k2

!2

ss1x2 1 s1y

2d , 1 (47)

but also outside this region. Relation (47) representsthe condition for homogeneous waves. Thus, when theintegration limits are extended to f2`, `g, the integralrepresentation also includes evanescent waves into theresulting field. These evanescent waves propagate per-pendicular to the optical axis, i.e., very close and par-allel to the plane of the aperture. These waves decayexponentially at larger z distances from the aperture andtherefore do not contribute to the far-field distribution.

We can conclude that the electromagnetic field of theoptical system under consideration can be represented inthe second material as the Fourier transform of a functionW0, which consists of an amplitude factor and two phasefactors as shown by Eq. (46). These two phase factorsrepresent the phase at the point of observation (P) withrespect to the Gaussian focus (O); i.e., one of these phasefactors represents the phase expfik2s2zsz 1 ddg as thewave propagates in the second material, and the other onerepresents the phase exps2ik1s1zdd that the wave wouldhave had in the absence of the second material. The signof the latter phase component is negative, and so the totalof the two phases gives the phase difference with respectto the Gaussian focus.

There is an alternative way to write Eq. (44):

E2sx, y, zd K 0ZZ

V1

W00ss1x

0, s1y0 dexpsik2s2zzd

3 expfi2pss1x0x 1 s1y

0ydgds1x0ds1y

0 , (48)

where

W00

W0

expsik2s2zzd.

Equation (48) is formally identical to the angular spec-trum representation given by Goodman.15 Thus we canconclude that our extension of the Debye integral for theoptical system under consideration can also be regardedas the angular spectrum representation. The spectrumis given as an amplitude factor describing the electric (andalso the magnetic) field as it traverses the system anda phase factor corresponding to the phase difference be-tween the ray vectors along the z direction (which latterphase factor, apart from the distance d, is often referredto as the astigmatic constant).

It should also be mentioned that Eq. (35) is in a formidentical to that of the corresponding equation of Wolf,1

i.e., the aberration function is represented in an exactform by Csf1, 2dd, as defined by Eq. (34). Since thisfunction is dependent only on the azimuthal angle f1, itdescribes spherical aberration.

E. Paraxial ApproximationEquation (13) can be approximated paraxially so as to cor-respond to the axial light distribution for a low-apertureobjective lens. For this we recall the expression for theelectric field:

E2sx, y, zd 2ik2

2

2pk1

ZZV1

T sed ass1x, s1y ds1z

3 expf2idsk1s1z 2 k2s2zdg

3 expsik2s2zzdexpfik1ss1xx 1 s1yydgds1xds1y .

(49)

In the paraxial approximation the strength vector T sedacan be considered as constant, and therefore it is in-dependent of the integration. Furthermore, we can set1ys1z constant and x y 0. The z components ofthe unit ray vectors can be approximated with the firstand second terms of their expansion:

s1z f1 2 ss1x2 1 s1y

2dg1/2 ø 1 2s1x

2 1 s1y2

2 1 2

s2

2,

s2z f1 2 ss2x2 1 s2y

2dg1/2 ø 1 2s2x

2 1 s2y2

2

1 2

√n1

n2

!2s2

2. (50)

After we rearrange Eq. (49) and perform further calcu-lations, the paraxial approximation of the electric field,E

spard2 , is derived as

Espard2 s0, 0, zd 2

ik22

k1B exp

8><>:2k1

24d

√1 2

n2

n1

!1

n2

n1z

359†=†;

3Z b

0exp

√2

12

ik1Ds12

!sds , (51)

where b is the paraxial semiangle of the aperture, B isa constant, and

D

√1 2

n1

n2

!d 1

n1

n2z . (52)

After we perform the integration with respect to s, theintensity is given by

I s0, 0, zd jEspard2 s0, 0, zdj2

I0

ØØØØØØØØØØexp

µ2

12

ik1Db2

∂2 1

14

k1Db2

ØØØØØØØØØØ

2

4I0

sin2µ

14

k1Db2

∂µ

14

k1Db2

∂ 2. (53)

By putting n1 n2 in Eqs. (52) and (53), we obtain thewell-known axial intensity distribution for a single mate-rial. If we use the integral formalism I sed

1 I sed2 0 in


the paraxial approximation, I sed0 yields a form identical to

that of Eq. (51). By putting D 0 in Eq. (52), we obtainthe location of the paraxial focus, for the case of two ma-terials, that corresponds to that obtained by geometricaloptics.

In a subsequent paper we will show how our integralformulas can be solved analytically, and we will presentresults for the numerically computed time-averaged elec-tric energy density as a function of the numerical aper-ture of the lens and the depth of focusing into the bulkspecimen.

It is emphasized that our solution for the extension ofthe Debye integral is an exact solution of the homoge-neous wave equation and is therefore valid everywhereexcept in the immediate vicinity of the aperture.

3. CONCLUSIONSWe have presented a new solution of electromagneticdiffraction for the problem of light focused into anisotropic and homogeneous material with refractive indexdifferent from that of the medium of propagation andsituated in the image space of a high aperture coherentoptical system. Focusing into such materials results inspherical aberration.

We have solved the above diffraction problem for thecase of a planar interface in a rigorous mathematicalmanner, and the solution satisfies the homogeneous waveequation and is therefore valid everywhere except in theimmediate vicinity of the aperture. The solution can beregarded as an extension of Wolf ’s integral formulas, forfocusing in a single medium, and is valid for high aperturefocusing into materials of mismatched refractive indices.We have shown that for our case the aberration function isin a closed analytical form. We have obtained the electricand magnetic strength vectors in the second material.We have given a physical interpretation of our results andhave obtained the paraxial approximation of our solution.

The main advantages of our method are as follows.The solution is in a simple form that can be directlyused for numerical computation. The method for obtain-ing the strength vectors is generally applicable. Incidentelectric vectors, with directions other than perpendicularto the optical axis, can readily be treated. The solutionis consistent, and, for the special case of a single mediumof propagation, it reduces to the results published previ-ously. Apart from the initial assumption of Wolf ’s inte-gral formulas, no approximations are used.

ACKNOWLEDGMENTSThe authors gratefully acknowledge E. Wolf for manystimulating discussions. Thanks are also due to

T. Wilson for comments, J. J. Stamnes for providinginformation concerning his research, R. Falster for dis-cussions, and D. G. Pettifor for provision of computingfacilities in the Materials Modelling Laboratory of theDepartment of Materials, Oxford University, which waspartially funded by Science and Engineering ResearchCouncil (UK) grant GRyH58278. P. Torok thanksMEMC Electronic Materials SPA, Novara, Italy, forsupport, and P. Varga thanks the Hungarian ScienceResearch Council for partial support through NationalScience Research Fund (Hungary) grant 2957. P. Torokand Z. Laczik are on leave from the Central ResearchInstitute for Physics, Hungarian Academy of Sciences.

REFERENCES1. E. Wolf, “Electromagnetic diffraction in optical systems. I.

An integral representation of the image field,” Proc. R. Soc.London Ser. A 253, 349–357 (1959).

2. P. Debye, “Das Verhalten von Lichtwellen in der Naha einesBrennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig)30, 755–776 (1909).

3. E. Wolf and Y. Li, “Conditions for the validity of the Debyeintegral representation of focused fields,” Opt. Commun. 39,205–210 (1981).

4. R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U.of California Press, Berkeley, Calif., 1966).

5. R. Kant, “Analytical solution of vector diffraction for focus-ing optical systems with Seidel aberrations. I. Sphericalaberration, curvature of field, and distortion,” J. Mod. Opt.40, 2293–2310 (1993).

6. J. Gasper, G. C. Sherman, and J. J. Stamnes, “Reflection andrefraction of an arbitrary electromagnetic wave at a planeinterface,” J. Opt. Soc. Am. 66, 955–961 (1976).

7. J. J. Stamnes and G. C. Sherman, “Radiation of electromag-netic fields in uniaxially anisotropic media,” J. Opt. Soc. Am.66, 780–788 (1976).

8. J. J. Stamnes and G. C. Sherman, “Radiation of electromag-netic fields in biaxially anisotropic media,” J. Opt. Soc. Am.68, 502–508 (1978).

9. H. Ling and S.-W. Lee, “Focusing of electromagnetic wavesthrough a dielectric interface,” J. Opt. Soc. Am. A 1,965–973(1984).

10. Y. Ji and K. Hongo, “Analysis of electromagnetic waves re-fracted by a spherical dielectric interface,” J. Opt. Soc. Am.A 8, 541–548 (1991).

11. J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger,Bristol, UK, 1986).

12. S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aber-rations in confocal fluorescence microscopy induced bymismatches in refractive index,” J. Microsc. (Oxford) 169,391–405 (1993).

13. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon,Oxford, UK, 1970).

14. B. Richards and E. Wolf, “Electromagnetic diffraction inoptical systems. II. Structure of the image field in anaplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379(1959).

15. J. W. Goodman, Introduction to Fourier Optics, 1st ed.(McGraw-Hill, San Francisco, 1968).

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Electromagnetic Diffraction in Optical Systems. I. an Integral Representation of the Image Field