Fourier description of the propagation and focusing of an extraordinary beam in a planar uniaxial medium

Fourier description of the propagationand focusing of an extraordinary beamin a planar uniaxial medium

Miklos Barabas and Gabor Szarvas

The scalar-angular-spectrum approach is used to examine a focused extraordinary wave propagatingalong an arbitrary direction in a homogeneous uniaxial planar medium, and the Fraunhofer diffractionformula is generalized for this case. The size of the focal spot is found to be inversely proportional to aneffective index, depending on the refractive indices and the propagation direction. The validity of theparaxial model is checked by nonparaxial 1but scalar2 numerical calculations. They show that theparaxial formulas predict the spot size correctly, but if the beam propagates neither parallel with norperpendicular to the optic axis, they do not reproduce the symmetries of the amplitude distribution in thefocal line.Key words: Propagation, focusing, diffraction, crystal optics, planar waveguides, aberrations, lens

design.

1. Introduction

The design of planar lenses for anisotropic wave-guides1–3 makes the evaluation of wave aberrationsnecessary. In conventional optical design4 the easi-est way to understand and compute the effect of smallaberrations is to use the Fraunhofer diffraction for-mula, according to which the focal-plane amplitudedistribution is proportional to the scaled Fouriertransform of the complex amplitude in the exit pupil.4–6In this paper we generalize this Fourier-transformrelationship for the case of a focused extraordinarywave beam propagating along an arbitrary directionof a planar uniaxial medium.7General and exact diffraction formulas for electro-

magnetic waves in three-dimensional, homogeneous,uniaxial crystals have been discussed extensively inthe literature.8–12 A rigorous generalization of theHuygens principle for both ordinary and extraordi-nary waves was given by Bergstein and Zachos8 whoextended the derivation of the Rayleigh–Sommerfelddiffraction formula and the angular spectrum repre-sentation. Kujawski and Petykiewicz9 used the

The authors are with the Department of Atomic Physics, Insti-tute of Physics, Technical University of Budapest, H-1111, Buda-foki ut 8, Budapest, Hungary.Received 24 September 1993; revised manuscript received 9

June 1994.0003-6935@95@010011-11$06.00@0.

r 1995 Optical Society of America.

Green’s function approach to generalize the Helm-holtz–Kirchhoff integral pertaining to the Cartesian-field components, and thus they obtained anotherexact formulation of the Huygens principle for wavesin uniaxial media. Stamnes and Sherman10 devel-oped the Laplace–Fourier representation of fieldsradiated by an arbitrary current density in a 1possiblyeven conducting2 three-dimensional uniaxial me-dium, and from there they also derived the correspond-ing exact angular-spectrum representation. The fullelectromagnetic theory of reflection and refraction ofgeneral 1not only plane2 waves on plane boundariesseparating two arbitrarily oriented uniaxial mediawas described by the same authors.11Fleck and Feit12 derived from Maxwell’s equations

paraxial-wave equations for extraordinary waves inuniaxial crystals. Their equations show that theangle between the propagation direction13 and thecentral wave vector14 is the same as the angle be-tween the ray vector and the wave vector in the case ofplane waves and that the field of the wave as afunction of the coordinate perpendicular to the central-wave vector is a scaled and translated version of thefield that would be obtained in an isotropic mediumwith an appropriate effective index of refraction.Because the rigorous vectorial theories8–12 are quite

involved, it is of some interest to develop a simplifiedheuristic scalar treatment of diffraction in two-dimensional 12D2 uniaxial media.15 Therefore in Sec-tion 2 we rederive the results in Ref. 12 cited above

1 January 1995 @ Vol. 34, No. 1 @ APPLIED OPTICS 11

from the three assumptions that 112 the extraordinaryfield is scalar, 122 it is the superposition of plane wavesthat carry energy in the forward16 direction, and 132the end points of the 2D wave vectors of the plane-wave components lie on the ellipse of the extraordi-nary wave vectors.Assumption 112means that the field is characterized

by a single scalar function and that the intensity iscomputed as the square modulus of this quantity.Consequently this assumption restricts the validity ofthe model to fields with a narrow angular spectrumbecause extraordinary waves are TM with respect tothe optic axis8,10 and the scalar description ignoresthe angles enclosed by the field vectors of the indi-vidual plane waves. Assumption 122 excludes fromthe superposition those plane waves whose Poyntingvectors have a backward16 component, but it stillincludes some backward waves, i.e., plane waves withwave vectors with a component in the backwarddirection. 1The energy transport associated with thebackward vector waves occurring in the exact treat-ment is discussed in Ref. 10, and a detailed analysis ofthe role of scalar backward waves in isotropic mediawas presented recently by Vassallo17 in connectionwith beam-propagation-method calculations.2 As-sumption 132 is physically clear: It states only thatthe direction dependence of the phase velocity of theplane-wave components is the one required for extraor-dinary waves. This last assumption is used to ex-press the transfer function6 of a homogeneous, uni-axial 2Dmedium in terms of the relationship betweenthe two rectangular components of the wave vectors ofplane waves.We take into account the paraxial nature of the

wave field under consideration by keeping only thelinear and quadratic terms in the power-series expan-sion of the phase of the transfer function. Theparaxial propagation law is cast in the form of aconvolution integral expressing the required Fourier-transform relationship between the amplitude distri-butions along two parallel lines.In Section 3 we apply the propagation law to a

focused paraxial extraordinary beam. We assumethat, far from the focal region, the beam is adequatelyrepresented by a homocentric pencil of extraordinaryrays and that the elliptical wave frontmay be approxi-mated by a parabolic one. In the paraxial approxima-tion we obtain the following three results:

112 In the vicinity of the geometrical focal point, aline exists along which we may calculate the complexamplitude distribution by first removing from theincident amplitude the quadratic phase factor thatcorresponds to focusing and by subsequently takingthe Fourier transform of the expression remainingfrom the incident amplitude. This means that theamplitude in this line, which is perpendicular to theguide axis10,16,18 and which is referred to below as theFourier line, is obtained from a simple generalizationof Fraunhofer diffraction. The distance between theFourier line and the geometrical focal point depends

12 APPLIED OPTICS @ Vol. 34, No. 1 @ 1 January 1995

on the ratio of the ordinary and extraordinary refrac-tive indices as well as on the orientation of themedium, and it is zero if the beam propagates in thedirection of the ordinary or the extraordinary axis.7In the case of a planar waveguide on LiNbO3 thisdistance is less than 0.2% of the geometrical focaldistance for all orientations. This fact assures usthat it is a reasonable approximation to compute thepoint-spread functions of planar waveguide lenses3 bythe common one-step Fourier relation.

122 The beam displacement that might be expectedfrom the angle between the ray vector and the wavevector is almost 1but not exactly2 canceled by theelliptical shape of the wave front of the focused beam.Like the shift of the Fourier line, the uncompensatedpart of the angular beam displacement is also afunction of the ratio of the refractive indices and theorientation of the medium, and it too vanishes for abeam propagating parallel with or perpendicular tothe optic axis. For a planar waveguide on LiNbO3the maximum of this residual angular shift is lessthan 0.0013°, which means that the diffraction focusof an off-axis beam leaving the exit pupil of a planarlens coincides to a high accuracy with the geometricalfocus. Consequently the merit functions used dur-ing the optimization of anisotropic planar lenses3 maybe constructed in the same way as those in theconventional optical design.

132 The width of the intensity distribution in theFourier line depends on the refractive indices and theorientation of the medium. This width is inverselyproportional to an effective index assigned to theuniaxial medium. The width of the focal spot of abeam propagating along the extraordinary axis7 isne2@no2 times the width of the focal spot of a beam1with the same incident far-field amplitude and phase2that propagates in an isotropic medium with refrac-tive index ne.

In Section 4 we illustrate and check the aboveresults by numerical calculations. We carried outthese calculations by implementing the initial formu-las of Section 1 directly through fast Fourier trans-forms 1FFT’s2 and by using both the paraxial and thenonparaxial transfer function. As noted in connec-tion with assumption 112, the nonparaxial scalar calcu-lation cannot be taken too seriously. Nevertheless, ifit is applied numerically to a beam with a narrowangular spectrum, it can serve as a check on theanalytically derived paraxial results. The direct nu-merical calculations based on the paraxial transferfunction support the analytically obtained formulasfor the shift of the Fourier line, but the nonparaxialresults show that the Fourier line coincides with thegeometrical focal line and that the residual lateralbeam shift is also exactly zero.Finally, in Section 5, we compare our analysis with

the results of Fleck andFeit.12 The principal theoreti-cal difference between the two analyses is in thetransition from general waves to paraxial beamwaves.In Ref. 12 Fleck and Feit made this transition by

neglecting certain derivatives in the exact wave equa-tion, whereas in the approach in this paper paraxial-ity means that the ellipse of wave vectors is replacedby a parabola and elliptic wave fronts are approxi-mated by parabolic ones. Although these differentapproximations lead to different formulas, the numeri-cal differences are negligible for practical materials.

2. Angular Spectrum of an Extraordinary WavePropagating in a Planar Uniaxial Medium

In this section we consider a monochromatic scalarwave of vacuum wave number k0 and amplitudeu1x, z2 propagating along the z axis in a planaruniaxial medium whose extraordinary axis enclosesan angle c with the z axis and whose ordinary andextraordinary refractive indices we denote by no andne, respectively. Our goal is to find the relationshipbetween the incident amplitude u1x, 02 and the imageamplitude u1x, z2 1see Fig. 1 for notation and geom-etry2. We assume that the time dependence of theamplitude is given by the 1suppressed2 factor exp1ick0t2and that the wave is the superposition of propagatingand evanescent plane waves:

u1x, z2 5 e A1kx2exp32i1kzz 1 kxx24dkx, 112

where kz and kx denote the z and x components of thewave vectors of the plane waves used to form thesuperposition. Therefore kz and kx satisfy the equa-tion of the ellipse of wave vectors,

1kz cos c 1 kx sin c22

@ne2 1 12kz sin c 1 kx cos c22@no2 5 k02, 122

and kz is a function of kx.This equation has in general two solutions kz for

given c and kx. When solving the equation for kz, wekeep only the solution with the greater real part,because this is the one that describes a plane wavewhose Poynting vector is directed into the half-space

Fig. 1. Notation and geometry. The end points of the wavevectors k of the plane waves in the superposition integral Eq. 112 lieon the elliptical arc shown by the full line. Since the Poyntingvector S of a plane wave is perpendicular to the ellipse of wavevectors, these plane waves carry no energy in the negative zdirection even if kz , 0.

z $ 0. Although this restriction permits some back-ward waves10,17 with kz , 0, they correspond torelatively large values of kx. If the angular spectrumof u1x8, 02 is narrow, these backward waves and theevanescent waves with even greater kx values play nosignificant part in the paraxial approximation.Equation 112 implies that

A1kx2 5 12p221 e u1x8, 02exp1ikxx82dx8, 132

and consequently

u1x, z2 5 e 312p221 e u1x8, 02exp1ikxx82dx84Pz1kx2

3 exp12ikxx2dkx, 142

where

Pz1kx2 5 exp52i3zkz1kx246 152

is the transfer function of the medium. Equations112–152 are convenient if we want to compute u1x, z2from u1x, 02 numerically because they may be easilyimplemented by FFT’s. Nevertheless, if we want toderive analytically a simpler, one-step Fourier rela-tion between u1x, 02 and u1x, z2, it is easier to rewriteEq. 142 into the following equivalent form by invokingthe convolution theorem:

u1x, z2 5 12p221 e u1x8, 02 Pz1x 2 x82dx8, 162

where

Pz1w2 5 e Pz1kx2exp12ikxw2dkx 172

is the Fourier transform of the transfer functionPz1kx2 defined by Eq. 152.In the remaining part of this section 1and in Sec-

tion 32 we restrict our analysis to paraxial waves.This means that we assume that the angular spec-trumA1kx2 is sufficiently narrow to permit the phase ofPz1kx2 to be approximated by a quadratic functionof kx.Solving Eq. 122 for kz and expanding the result to

the second power in kx@k0 yield

kz 5 k03F 1 Gkx@k0 1 H1kx@k0224, 18a2

where

F 5 1cos2 c@ne2 1 sin2 c@no2221@2, 18b2

G 5 1ne2 2 no22sin c cos c@1no2 cos2 c 1 ne2 sin2 c2,

18c2

H 5 211@22none1no2 cos2 c 1 ne2 sin2 c223@2. 18d2


From Eqs. 152, 172, and 182 we obtain for the Fouriertransform of the transfer function

Pz1w2 5 K11z2exp3ik01zG 1 w22@14zH24, 192

where

K11z2 5 12pk0@Hz21@2exp32i1k0zF 2 p@424. 1102

Substituting Eq. 192 into Eq. 162 and introducing thenotation

K21x, z2 5 K11z2exp3ik01x2 1 2zGx 1 z2G22@14zH24,

1112

we find that

u1x, z2 5 1K2@2p2 e u1x8, 02exp1ik0x82@4zH2

3 exp32ik01x 1 Gz2x8@12zH24dx8. 1122

The contents of this equation are the following.Propagation from the line z 5 0 to a line at a distancez affects the x-dependent complex amplitude in threeways:

112 The incident amplitude u1x8, 02 is first multi-plied by a quadratic phase factor.

122 Next, the resulting amplitude distribution isFourier transformed with respect to the shifted andscaled x coordinate. The shift of the x coordinate1i.e., the additive term Gz2 moves the amplitudedistribution away from the origin by

Dx 5 2Gz ; z tan d, 113a2

where on account of Eq. 18c2

tan d 5 1no2 2 ne22tan c@1no2 1 ne2 tan2 c2. 113b2

It is remarkable that the angular beam displacementd is equal to the angle between the wave vector k andthe ray vector s of a plane wave if k is parallel to the zaxis. 1This result was derived by Fleck and Feit12from the paraxial-wave equation.2

132 Finally, the amplitude is multiplied by thefactor 1K2@2p2 that accounts for the spreading of thebeam energy 3see z21@2 in K11z2, Eq. 11024 and thatcauses a tilt and a curvature in the image phase frontas expressed by the exponential in K21x, z2 3Eq. 11124.Besides these effects, the phase varies linearly with z3because of k0zF in Eq. 1102 and z2G2 in Eq. 11124, asone might expect from a propagating wave. The factthat K2 appears outside the integral has a noteworthyconsequence: If the incident beam is scanned bytilting its wave front—irrespective of the exact formof u1x, 02—the phase of the image does not movetogether with the image intensity. In other words, alinear and parabolic phase variation over the imageline is due only to the propagation and is independentof the profile of the incident beam.


3. Propagation of a Geometrically FocusedExtraordinary Beam

In this section we consider the consequences of theabove results relating to a converging, geometricallyfocused incident extraordinary beam. We say that abeam is geometrically focused if, far from the focalregion, its equiphase lines coincide with those of ahomocentric pencil of extraordinary rays convergingtoward focal point F. 1We use the term geometricallyfocused to distinguish this kind of beam from theperfectly focused waves defined for isotropic media byStamnes.19,20 Geometrical focusing means only thatthe phase of the beam is ideal in the above sense,whereas perfect focusing requires also that the modu-lus of the incident amplitude have a prescribed func-tional form.2 Along the line z 5 0 the complexamplitude of such a geometrically focused wave isgiven by

u1x8, 02 5 a1x82exp3if1x824, 114a2

where

f1x82 5 k0nray1 f, x82l1 f, x82, 114b2

where f denotes the distance of the geometrical focusF 1lying on the z axis2 from the line z 5 0,

l1 f, x82 5 1x21 f 221@2 114c2

is the length of the ray connecting point P1z 5 0, x82with focus F,

nray1 f, x82 5 31no sin b22 1 1ne cos b2241@2, 114d2

is the refractive index of the ray1 PF,

b 5 g 1 c, 114e2

tan g 5 x8@f 114f 2

Fig. 2. Incident wave front with an extraordinary ray and theangles used in Eqs. 114d2–114f 2.

1see Fig. 22. The function a1x82 is introduced to takeaccount of the finite width and the possible inhomoge-neity and aberrations of the incident wave, and wecall it the pupil function. We assume that thebeamwidth 1i.e., the spatial extension along the xdirection2 is small enough to justify the paraxialapproximation in which the phase f1x82 is replaced bya quadratic function of x8@f. Therefore we writef1x82 in the form

f1x82 5 k0 f 3A 1 B 1x8@f 2 1 C1x8@f 224, 115a2

where the factors A, B, and C are found from thepower-series expansion of f1x82with respect to x8@f:

A 5 nray1g 5 02 5 31no sin c22 1 1ne cos c2241@2, 115

B 5 1no2 2 ne22sin c cos c@A, 115c2

C 5 A@2 1 31no22 ne22@12A243cos 2c 2 B sin 2c@12A24.

115d2

To determine the combined effect of propagationand focusing, we rewrite Eq. 1122 with the aid of Eqs.1152 so that it contains the transform of the product ofthe pupil function a1x82 and a single phase factorexp3ik0 F 1x824:

u1x, z2 5 1K2@2p2 e a1x82exp3ik0 F 1x824

3 exp32ik0xx8@12zH24dx8, 116a2

where

F 1x82 5 F0 1 F1x8 1 F2 x82, 116b2

where

F0 5 fA, 116c2

F1 5 12BH 2 G2@12H2, 116d2

F2 5 1@14zH2 1 C@f. 116e2

From this modified form 3Eq. 116a24 of the equation itis clear that if the quadratic term containing F2 in thephase F 1x82 is zero, the image amplitude u1x, z2 isproportional to the displaced Fourier transform of thepupil function a1x82 and that the lateral displacementis determined by F1.

3.A. Shift of the Fourier Line

To find the shift z 2 f of the Fourier line, that is, thedistance between the geometrical focus and the zcoordinate of the line along which the one-step Fou-rier relation connecting the pupil function and theimage amplitude is true, we must solve the equation

1@14zH2 1 C@f 5 0 1172

b2

for the unknown z. Taking into account the defini-tions of C andH 3Eqs. 115d2 and 18d24 and using

t 5 tan c, 118a2

r 5 no@ne, 118b2

dz 5 1z 2 f 2@f, 118c2

we obtain

dz 5 31r2 1 t221r2t2 1 1243@2@5r11 1 t2231r2t2 1 122

1 1r2 2 1211 2 t221r2t2 1 12 2 1r2 2 12246 2 1.

1192

Thus the relative shift dz of the Fourier line dependsonly on the ratio r of the refractive indices and theorientation c of the medium. If the principal ray ofthe incident pencil encloses only a small angle c withthe extraordinary axis, the relative shift of the Fou-rier line depends quadratically on c, which meansthat, to the first order in c, the Fourier line coincideswith the geometrical focal line. 3The quadratic depen-dence on c follows from the fact that the function dz1c2is smooth and invariant under the sign change of c.4Equation 1192 is plotted in Fig. 3 for the case of aplanar waveguide on LiNbO3 with effective refractiveindices no 5 2.293 and ne 5 2.213 1r 5 1.036, the thickcontinuous curve2 and for the case of calcite withrefractive indices no 5 1.658 and ne 5 1.486 1r 5 1.116,the thick dotted curve2. These curves show that theFourier line is shifted from the geometrical focus inthe positive z direction by less than 0.2% of the focaldistance f in the LiNbO3 waveguide and by less than2% in calcite. 1The two thin curves labeled Fleckwere computed from the paraxial approximation ofRef. 12; see Section 5.2

3.B. Lateral Beam DisplacementNext we examine the lateral beam shift associatedwith F1. By analogy with the interpretation of Eqs.

Fig. 3. Relative shift of the Fourier line dz 5 1z 2 f 2@f versus theorientation c of the medium. Thick curves, Eq. 1192; thin curves,computed fromEq. 1352; solid curves, the waveguide of LiNbO3 withrefractive indices no 5 2.293, ne 5 2.213; dotted curves, calcite withrefractive indices no 5 1.658, ne 5 1.486.


1122 and 1132, we may conclude from Eqs. 116a2 and116d2 that the image amplitude u1x, z2 is displacedfrom the origin by 12BH 2 G2z because of the presenceof F1. The substitution of B, H, and G from Eqs.115c2, 18d2, and 18c2 gives the following for the tangentof the angular beam shift d8 in the case of a focusedwave:

tan d8 5 2BH 2 G

5 3t1r2 2 12@1r2 1 t224511 1 t22

@31r2 1 t2211@r2 1 t2241@2 2 16, 1202

where t and r are defined in Eq. 1182. The power-series representation of tan d8 as a function of creveals that the lowest-order nonvanishing term inthe series is the third one. This means that thelateral beam displacement incurred by propagation ina uniaxial medium according to Eqs. 1122 and 1132 iscanceled by geometrical focusing to the second orderin c. Thus Eq. 1202 gives only a residual beam shift,which in practical cases remains small even for finitevalues of c. This is demonstrated in Fig. 4 where weplotted the angular beam shift d8 as a function of c forthe previously considered LiNbO3 waveguide 1thethick continuous curve2 and for calcite 1the thickdotted curve2. The angular shift is smaller then0.0013° for the LiNbO3 waveguide and smaller than0.038° for calcite. 1The two thin curves labeled Fleckwere computed from the paraxial approximation ofRef. 12; see Section 5.2

3.C. Effective Index and the Width of the IntensityDistribution along the Fourier Line

Themeaning of Eq. 1162 becomes clear if we compare itwith the corresponding relation pertaining to anisotropic medium. Equation 116a2 contains a Fouriertransform with respect to the variable xk0@ 122zH2.For an isotropic medium with refractive index n thisvariable would be [email protected] we may introduce an approximate effec-

tive refractive index neff for the beam by defining it as

Fig. 4. Residual angular shift of a focused extraordinary beamversus the orientation c of the medium. Thick curves, Eq. 1202;thin curves, computed from Eq. 1352; continuous curves, waveguideon LiNbO3 with refractive indices no 5 2.293, ne 5 2.213; dottedcurves, calcite with refractive indices no 5 1.658, ne 5 1.486.


the quantity that takes the place of n. 1We wouldarrive at the same effective index by defining neff asthe quantity that plays the role of the isotropic indexn in the transfer function.2 Taking into account Eq.18d2 and using z < f give

neff 5 1@122H2 5 31no cos c22 1 1ne sin c2243@2@1none2.

1212

The beamwidth measured in the Fourier line isinversely proportional to this effective index. If thepropagation direction 1i.e., the z axis2 is parallel to theextraordinary axis, the width of the intensity distribu-tion in the Fourier line is 1ne@no22 times the widththat would arise for a focused beam with the samefocal point and far-field incident intensity distribu-tion in a homogeneous medium of refractive indexn 5 ne. This is just the result derived in Ref. 15.

4. Numerical Experiments

In this section we present numerical results to illus-trate the findings in Sections 2 and 3 and to check thevalidity of the paraxial approximation. We com-puted the image amplitude directly from Eq. 142,which we implemented through the FFT algorithm.The two FFT’s were computed over a grid of 8192points, and the spatial integration interval was twicethe width of the pupil function.

4.A. Symmetrical Incident Phase Front: Demonstration ofthe Lateral Beam Shift and the Effect of Defocus

In the first set of examples we consider a beam ofuniform intensity emerging from a finite apertureand focused onto a point of the z axis, with theassumption that it propagates along the extraordi-nary axis 3vacuumwavelength, l0 5 633 nm; apertureradius, a 5 0.1; focal length, f 5 1 mm; f number,f@12a2 5 5; approximate Fresnel number,21,22a2@1le f 2 5 7.14. The only approximation 1besides theunavoidable truncation and sampling2 in these calcu-lations is that we use a scalar model and that weneglect evanescent waves. Instead of the paraxialequation 3Eq. 18a24 we use the exact solutions to Eq. 122for kz1kx2 in the transfer function 3Eq. 1524.To demonstrate the lateral beam displacement

described by Eq. 1132, we vary the orientation of themediumwithout modifying the incident phase and wecompute the intensity distribution along both theoriginal geometrical focal line 3Fig. 51a24 and theFourier line1s2 3Fig. 51b24 for various orientations of awaveguide on LiNbO3 1no 5 2.293, ne 5 2.2132.The intensity in Fig. 5 is normalized so that its peak

value would be unity if the medium were isotropicwith a refractive index of ne.We construct the phase of the beam without taking

into account the angle c between the z axis and theextraordinary axis, so that for nonzero values of c thebeam will not be geometrically focused. The ampli-tude u1x8, 02 of this beam is zero if 0x80 . a, and,

Fig. 5. 1a2 Intensity distributions 3computed by FFT’s from Eqs. 122, 142, and 1524 along the z 5 1 line for a homogeneous beam that would begeometrically focused onto the point z 5 1.0, x 5 0 if the z axis were the e axis. We obtained the different curves by varying the direction ofthe e axis without modifying the incident phase. Aperture radius, 0.1; vacuum wavelength, 633 nm; no 5 2.293, ne 5 2.213. C is theangle between the z and e axes. The intensity is normalized so that its peak value would be unity for an isotropic mediumwith a refractiveindex of ne. 1b2 Intensity distributions 3computed by the FFT’s from Eqs. 122, 142, and 1524 along the Fourier lines for a homogeneous beamthat would be geometrically focused onto the point z5 1.0, x5 0 if the z axis were the e axis. The incident phases are the same as in 1a2, butthe image lines are at the position where the Fourier spectrum of the pupil function is expected. Aperture radius, 0.1; vacuumwavelength, 633 nm; no 5 2.293, ne 5 2.213. C is the angle between the z and e axes. The normalization is the same as in 1a2. The zcoordinates of the five Fourier lines corresponding to increasing values of C are as follows: z 5 1 mm, z 5 0.9967 mm, z 5 0.9871 mm, z 5

0.9725 mm, z 5 0.9547 mm.

according to Eqs. 114a2–114d2, for 0x80 # a it is givenby

u1x8, 02 5 exp5ik031nox822 1 1nef 2241@26. 1222

The phase that plays the role of F 1x82 in Eq. 116a2 isnow 1to the second order in x8@f 2

F 1x82 5 const 1 3no2@12nef 2 1 1@14Hz24x82, 1232

so that 3similarly to Eq. 11724 the condition of theapplicability of the one-step Fourier relation is

no2@12nef 2 1 1@14Hz2 5 0. 1242

Hence the z coordinate of the Fourier line may befound as the solution of Eq. 1242, and the associatedrelative defocus df is given by

df ; 1z 2 f 2@f 5 1no@ne22

2 31no cos c22 1 1ne sin c2243@2@1none22. 1252

The dependence of df on c is shown in Fig. 6 for theexample medium.Our motivation for considering this special beam,

which would be rather hard to realize physically fornonzero c 2 s, is threefold:

112 The calculation of the position of its Fourierline 3i.e., the derivation of Eq. 12524 provided anopportunity to check the algebra leading to the earlierformulas.

122 The lateral beam displacement predicted byEq. 1132 is expected to appear for all incident complexamplitudes that are symmetrical around the z axis, sothis special beam does equally well for the purpose ofdisplaying the lateral displacement.

132 By constructing the phase with the help of Eq.1222, we proceed in the same way as if we weredesigning a focusing lens for a beam that is offset fromthe extraordinary axis by requiring that its wavefront have the same shape as a geometrically focusedbeam that is centered on the extraordinary axis.Thus the current example also shows that such adesign error would result in large first- and second-order aberrations 1i.e., distortion and defocus2 of theoutput beam.

4.B. Geometrically Focused Incident Wave: Comparison ofParaxial and Nonparaxial Results

As a second set of numerical examples we present theamplitude and phase distributions along the geometri-cal focal line and along the Fourier line of a beam thatis geometrically focused onto a point of the z axis in

Fig. 6. Relative defocus df 5 1z 2 f 2@f versus the angle c betweenthe z and e axes. This defocus arises because the phase of thebeam is not adjusted as would be required for geometrical focusingif the e axis is not parallel with the z axis: no 5 2.293, ne 5 2.213.


calcite. We consider calcite instead of the practicallymore important LiNbO3 because its higher refractive-index ratio makes it more suitable for the study of thedifferences between the nonparaxial and the paraxialdescriptions. We use two values for the angle be-tween the extraordinary axis and the z axis: c 5 45°and c 5 0°. The remaining parameters of the calcu-lations in this section are the following: vacuumwavelength, l0 5 633 nm; aperture radius, a 5 0.1;focal length, f 5 1 mm; f number, f@12a2 5 5; approxi-mate Fresnel number,21,22 a2@1le f 2 5 10.6.To check the validity of the paraxial approxima-

tion1s2, we compute the complex amplitude in threedifferent ways.First, we use the nonparaxial expressions for both

the incident phase and the phase of the transferfunction. The resulting curves are shown by thethick continuous curves in Figs. 71a2–81b2. Figure 7shows the amplitude and phase along the geometricalfocal line; Fig. 8 shows the same quantities along theFourier line. In the geometrical focal line the modu-lus of the amplitude is symmetrical with respect tothe z axis, whichmeans that the lateral beamdisplace-ment predicted by the paraxial formula 3Eq. 12024 ismissing. That the focal-line amplitude drops to zeroand that the focal-line phase displays jumps of 180° atthe zeros of the amplitude suggest that the amplitudeis proportional to a sinc function, i.e., to the Fouriertransform of the homogeneously illuminated slit.This means that, in spite of the paraxial expectationsin Subsection 3.A, the line, along which the complexamplitude is the Fourier transform of the pupil func-tion, is not shifted away from the geometrical focalline. These findings are corroborated by the continu-ous curves in Fig. 8 that display the nonparaxialresults in the Fourier line. The amplitude is obvi-ously not the Fourier transform of the rectangularpupil function. Therefore we must conclude thatthe paraxial model gives slightly misleading resultsfor focused beams whose average wave vector en


closes a large angle with the extraordinary axis of astrongly birefringent medium.The amplitude obtained by using the paraxial form

of both the incident phase 3Eqs. 115a2–115d24 andkz1kx2 3Eqs. 18a2–18d24 is shown by the thick dottedcurves in Figs. 7 and 8. They show that, according tothe paraxial formulas, the amplitude in the Fourierline is indeed proportional to the shifted and scaledFourier transform of the pupil function and that theposition of the Fourier line, as well as the lateralbeam shift, is correctly given by Eqs. 1192 and 1202.Note that these curves indicate nothing about thetrue physical situation beyond the paraxial modelbut, being computed directly from the FFT’s withoutEqs. 1122 or 1162–1212 taken into account, they supportthe correctness of the derivations that lead from Eqs.142 to 1122 and from Eqs. 1152 to 1212.We computed the third pair of curves, the thin

dotted curves in Fig. 7, for the geometrical focal lineby using the paraxial form of the incident phase 3Eq.11524 and the expression for kz1kx2 3Eq. 13524 derived fromthe paraxial-wave equation of Fleck and Feit12 inSection 5. These curves 1and also those labeled Fleckin Figs. 3 and 42 show that the Fourier line is closer tothe geometrical focal line and the residual lateralbeam shift is of the opposite sign and smaller inmagnitude than the one obtained from the equationsin Section 3. Thus the results of the approximationof Ref. 12 are closer to the nonparaxial results thanthose of the paraxial model introduced in Sections 2and 3. However, this approximation also fails toreproduce the symmetry observed in the exact numeri-cal results.The final curves to be discussed are those that

relate to a geometrically focused beam that is sym-metrical around the extraordinary axis. In this casethe focal-line amplitude 3the thin continuous curve inFig. 71a24 and phase 1Fig. 92 distributions computed bythe three different methods cannot be distinguishedfrom one another.

Fig. 7. 1a2 Normalized amplitude disttribution along the geometrical focal line for a homogeneous incident beam that is geometricallyfocused onto the point z 5 1.0, x 5 0. Aperture radius, 0.1; vacuum wavelength, 633 nm; no 5 1.658; ne 5 1.486. PSI is the anglebetween the z and e axes. The amplitude is normalized so that its peak value would be unity for an isotropic medium with a refractiveindex of ne. 1b2 Phase distribution along the geometrical focal line for a homogeneous incident beam that is geometrically focused onto thepoint z5 1.0, x5 0. Aperture radius, 0.1; vacuumwavelength, 633 nm; no 5 1.658; ne 5 1.486. PSI is the angle between the z and e axes.

Fig. 8. 1a2 Normalized amplitude distribution along the Fourier line 1z 5 1.018 mm2 for a homogeneous incident beam that isgeometrically focused onto the point z 5 1.0, x 5 0. Aperture radius, 0.1; vacuumwavelength, 633 nm; no 5 1.658; ne 5 1.486. PSI is theangle between the z and e axes. The amplitude is normalized so that its peak value in the geometrical focal line is unity for an isotropicmediumwith a refractive index of ne. 1b2 Phase distribution along the Fourier line 1z 5 1.018mm2 for a homogeneous incident beam that isgeometrically focused onto points z 5 1.0, x 5 0. Aperture radius, 0.1; vacuum wavelength, 633 nm; no 5 1.658; ne 5 1.486. PSI is theangle between the z and e axes.

5. Comparison with the Analysis of Fleck and Feit

Fleck and Feit12 have derived a paraxial-wave equa-tion for the propagation of extraordinary wave beamsin uniaxial media. They found the same lateralbeam displacement as the one given in Eq. 113b2, buttheir result for the coefficient of the quadratic term inkz1kx2 is different from ours. To clarify the cause andto calculate the magnitude of this difference, werewrite the relevant equations of Ref. 8 using thenotation in this paper. We conclude that the differ-ences are due to the use of two not exactly equivalentdefinitions of paraxiality and that the maximum ofthe direction-dependent difference between the effec-tive indices 1and the corresponding beamwidths men-tioned in Subsection 3.C2 provided by the two analysesis ,0.1% for the LiNbO3 waveguide and ,1% forcalcite.The exact equation for the x component of the

electric-field vector of an extraordinary wave is Eq.

Fig. 9. Phase distribution along the focal line for a homogeneousincident beam that is geometrically focused onto the point z 5 1.0,x 5 0. Aperture radius, 0.1; vacuum wavelength, 633 nm;no 5 1.658; ne 5 1.486. The e axis coincides with the z axis, i.e.,C 5 0. Both paraxial methods yield the same result as thenonparaxial calculation.

1322 of Ref. 12. Omitting the coordinate perpendicu-lar to the principal plane and rearranging this equa-tion, we find that in our notation 3see Eqs. 1132 and118b24 it is equivalent to

≠2u@≠z2 1 A11≠2u@≠z≠x 1 A02≠

2u@≠x21 A00u 5 0,

126a2

where

A11 5 2 tan d, 126b2

A02 5 1r2 sin2 c 1 cos2 c2@1r2 cos2 c 1 sin2 c2, 126c2

A00 5 1k0ner22@1r2 cos2 c 1 sin2 c2. 126d2

Equation 126a2 has plane-wave solutions of the form

exp32i1kzz 1 kxx24, 1272

where kz and kx are connected by the equation of theellipse of wave vectors 3Eq. 1224. Consequently Eqs.132–172 are applicable without any changes, and thewave equation 3Eq. 12624 contains the same informa-tion on wave propagation as the transfer functionPz1kx2 defined by Eq. 152 or the function kz1kx2.In Section 2 we defined paraxiality by requiring

that the quadratic expression 3Eq. 18a24 be a validapproximation of kz1kx2. In Ref. 12, on the otherhand, the paraxial-wave equation is derived from Eq.126a2 by the introduction of a trial function of the form

u1x, z2 5 u01x, z2exp52i3zkz102 1 xkx46, 1282

where

kz102 5 A001@2, 1292

and by the assumption that u01x, z2 is the amplitude ofa paraxial wave, which is expressed by the require-


ments that

0≠2u0@≠z20 9 kz1020≠u0@≠z 0, 130a2

0≠2u0@≠z≠x 0 9 kz1020≠u0@≠z 0. 130b2

Equations 1282 and 1292 and inequalities 130a2 and130b2 lead to the paraxial-wave equation

2inek01cos2 c 1 sin2 c@r221@21≠u0@≠z 1 tan d≠u0@≠x2

5 1sin2 c 1 cos2 c@r22≠2u0@≠x2. 1312

To find those plane-wave solutions of Eqs. 1262 3andthe corresponding relation kz1kx24 that are in accor-dance with Eq. 1312, we write

kz 5 kz102 1 kz112, 1322

u01x, z2 5 exp52i3zkz112 1 xkx46, 1332

and we obtain

kz112 5 211@22A11kx 2 5A02@32kz10246kx2. 1342

Thus the paraxial-wave equation 3Eq. 13124 yieldsfor kz

kz1Fleck–Feit2 5 kz102 2 1A11@22kx 2 5A02@32kz10246kx2. 1352

For the sake of the current comparison we write ourEq. 18a2 in terms of kz102, A11, and A02:

kz < kz102 2 1A11@22kx 2 514A02 2 A1122@38kz10246kx2.

1362

The first two terms in Eq. 1352 and approximation 1362are obviously equal, but the quadratic one is different.Thus the definition of paraxiality used in the currentpaper and the definition in Ref 12 3i.e., our Eqs.1282–13024 are not equivalent.To assess the difference quantitatively between the

two kinds of paraxiality, we compute the relativedifference between the two resulting effective indicesneff 3given by Eq. 12124 and neff1Fleck–Feit2 given by

neff1Fleck–Feit2 5 kz102@1k0A022. 1372

From Eqs. 1212, 18d2, 1362, and 1372, we obtain

dneff ; 3neff 2 neff1Fleck–Feit24@neff 1Fleck–Feit2

5 1A02 2 A112221 2 A02

21. 1382

Substitution of the example values used above for therefractive-index ratio 3r 5 1.036 1LiNbO3 waveguide2and r 5 1.116 1calcite24 shows that dneff 5 0 if thebeam is propagating in the direction of the ordinaryor the extraordinary axis and that the maximum ofdneff is 1.22 3 1023 for the LiNbO3 waveguide and11.5 3 1023 for calcite. These values indicate that,by choosing one definition of paraxiality instead ofthe other, we obtain values for the beamwidth thatdiffer from one another by ,1% or less. The numeri-


cal calculations in Section 4 show that the shift of theFourier line 1which corresponds to a focal shift in thecase of large Fresnel numbers2 and the residualangular displacement depend also on the definition ofparaxiality. However, according to the nonparaxialresults, these two shifts are merely computationalartifacts, so the question concerning which of the twoparaxial approximations is better from the point ofview of these shifts is rather academic. Summariz-ing the above comparison, we may say that as long aswe are concerned with practical questions concerningthe width and the position of a paraxial beam, it isirrelevant which definition of paraxiality we accept.

6. Summary and Conclusions

We generalized the Fourier optical description ofbeam propagation for the case of extraordinary wavesin planar uniaxial media, and we applied it to thepropagation of geometrically focused wave beams.Starting from the angular spectrum representation ofa scalar wave field, we found a one-step paraxialFourier relationship that may be regarded as a gener-alization of the Fraunhofer diffraction formula of theoptics of isotropic media. According to this one-stepFourier relationship, the amplitude of a focused beamalong a certain line 1the Fourier line2 is proportional tothe shifted and scaled Fourier transform of the far-field incident amplitude.Our numerical calculations based on the scalar

nonparaxial starting formulas show that if the anglebetween the extraordinary axis and the mean wavevector of the focused incident beam is not small and ifthe medium is strongly birefringent, the paraxialtheory gives qualitatively wrong results for the focal-line amplitude. Nevertheless the beam width maybe computed from the paraxial formulas with anaccuracy of ,1% even for calcite, and in the case ofbeams that are only slightly tilted with respect to theextraordinary axis the paraxial and the exact calcula-tions yield practically the same results. For theselatter cases the effect of anisotropy may be taken intoaccount by use of a weakly direction-dependent effec-tive refractive index that is a function of only the ratioof the two refractive indices and the propagationdirection.Although we used a definition of paraxiality that is

not exactly equivalent to the one accepted by previousworkers, we have shown that, as far as numericalresults for practical materials are concerned, thisdifference is negligible.The results in this paper are important because of

their implications concerning the design of lenses inanisotropic planar waveguides. The fact that thedeviations from isotropic optics are small assuresthat, if the correct effective index is used, the methodsof conventional aberration theory are applicable tothe evaluation of anisotropic waveguide lenses.

References and Notes1. P. Jiang, P. J. R. Laybourn, and G. C. Righini, ‘‘Homogeneous

planar lens on an anisotropic waveguide,’’ J. Mod. Opt. 39,121–132 119922. 3The angle b in Ref. 1 is measured from the

ordinary axis. The b in our Eqs. 114d2 and 114e2 and Fig. 2 ismeasured from the extraordinary axis. Therefore the indiceso and e are interchanged.4

2. W. Jiang, and V. M. Ristic, ‘‘Study of anisotropy effect inplanar lenses for integrated optics,’’ J. Mod. Opt. 35, 849–862119882.

3. G. Szarvas, M. Barabas, P. Richter, and L. Jakab, ‘‘Design ofmultielement acircular waveguide lens systems in anisotropicmedia,’’ Opt. Eng. 32, 2510–2516 119932.

4. W. T. Welford, Aberrations in Optical Systems 1Hilger, London,19862.

5. M. Born and E. Wolf, Principles of Optics 1Pergamon, Oxford,UK, 19862, Chap. 9, pp. 459–490.

6. J. W. Goodman, Introduction to Fourier Optics 1McGraw-Hill,NewYork, 19882, Chap. 4, pp. 57–62; Chap. 5, pp. 77–96.

7. The optic axis 3called also the ordinary 1or o2 axis4 is parallel tothe plane of the 2Dmedium. For brevity we call the directionperpendicular to the optic axis the extraordinary 1or e2 axis.

8. L. Bergstein and T. Zachos, ‘‘A Huygens principle for uniaxi-ally anisotropic media,’’ J. Opt. Soc. Am. 56, 931–937 119662.

9. A. Kujawski and J. Petykiewicz, ‘‘A Huygens principle forextraordinary waves in uniaxially anisotropic media,’’ Opt.Commun. 3, 23–25 119712.

10. J. J. Stamnes and G. C. Sherman, ‘‘Radiation of electromag-netic fields in uniaxially anisotropic media,’’ J. Opt. Soc. Am.66, 780–788 119762.

11. J. J. Stamnes and G. C. Sherman, ‘‘Reflection and refraction ofan arbitrary wave at a plane interface separating two uniaxialcrystals,’’ J. Opt. Soc. Am. 67, 683–695 119772.

12. J. A. Fleck, Jr., and M. D. Feit, ‘‘Beam propagation in uniaxialanisotropic media,’’ J. Opt. Soc. Am. 73, 920–926 119832.

13. The propagation direction is the direction along which thecentral peak of the intensity profile of a beam travels. In thecase of a paraxial beam it coincides with the central 1oraverage2 direction of the Poynting vector and also with thegroup velocity.

14. The central wave vector plays the same role in the paraxialdescription as the guide axis defined in Ref. 10.

15. M. Barabas and G. Szarvas, ‘‘Diffraction of a focused extraordi-nary beam in a planar uniaxial medium,’’ in Proceedings of theEuropean Conference on Integrated Optics 1European Confer-ence on Integrated Optics, Neufchatel, France, 19932, pp.13-18–13-19. 3This reference contains three disturbing typ-ing errors: The sign of the phase of the incident beam in Eq.112 should be 1, and 1no@ne22 should be replaced by 1ne@no22 inSections 1 and 5.4

16. The forward and backward directions are referred to the guideaxis, which will be the z axis 1see also Fig. 1, Ref. 10, and notes14 and 182.

17. C. Vassallo, ‘‘Reformulation for the beam-propagationmethod,’’J. Opt. Soc. Am.A 10, 2208–2216 119932.

18. The guide axis as defined in Ref. 10 is the direction aroundwhich the angular spectrum representation is constructed.In the notation in this paper it is the z axis as shown in Fig. 1.

19. J. J. Stamnes, ‘‘Focusing of a perfect wave and theAiry patternformula,’’ Opt. Commun. 37, 311–314 119812.

20. J. J. Stamnes, ‘‘The Luneburg apodization problem in thenonparaxial domain,’’ Opt. Commun. 38, 325–329 119812.

21. J. J. Stamnes, ‘‘Focusing of two-dimensional waves,’’ J. Opt.Soc. Am. 71, 15–31 119812.

22. Y. Li, ‘‘Dependence of the focal shift on Fresnel number and fnumber,’’ J. Opt. Soc. Am. 72, 770–774 119822.


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Fourier description of the propagation and focusing of an extraordinary beam in a planar uniaxial medium