Re ection and non-specular eå ects of 2D Gaussian beams ininterfaces between isotropic and uniaxial anisotropic media

LILIANA I. PEREZ

Laboratorio de OÂ ptica, Departamento de Fõ Â sica, Facultad de CienciasExactas y Naturales, Universidad de Buenos Aires, (1428) Buenos

Aires, Argentina

(Received 18 October 1999; revision received 21 February 2000)

Abstract. In the present paper bidimensional Gaussian linearly polarizedbeams, which are incident on an interface between an isotropic medium and anon-absorbing non-magnetic uniaxial crystal are considered. The relationbetween the re¯ ection coeæ cients and the non-specular eå ects are studied.The expressions for the ® rst order eå ects in the case of a beam linearly polarizedin a plane perpendicular to the one containing the crystal optical axis and thenormal to the interface are analytically determined. Two highly symmetricgeometries are analysed and the results are compared to those obtained withisotropic interfaces.

1. IntroductionThe method of describing a beam with Gaussian variation of intensity in terms

of a continuous spectrum of plane wavefronts has enabled the analytical calcula-

tion, up to second order, of non-specular eå ects when the beam is incident on an

interface between two isotropic media. Horowitz and Tamir used this method for

the ® rst time in 1971 [1] and studied the lateral displacement of a Gaussian beamfor incidence angles near the total re¯ ection one. Many works followed this and in

these three other non-specular eå ects, of ® rst and second order for symmetric

beams, were determined: widening of the beam, focal shift and angular shift. Their

nature was investigated and their magnitudes at isotropic interfaces for bi- and

tridimensional beams were evaluated [2± 7].

If we consider ® nite but not excessively thin beams (so that the angularspectrum is not too wide), polarized in one of the eigenmodes and incident

on an interface between two dielectric isotropic media, there are three situations

for the constituent re¯ ected waves: all, none or some of them undergo total

re¯ ection. In the ® rst case the amplitude of each re¯ ected wave is equal to that

of the corresponding incident one (the re¯ ection coeæ cient has unitary modulus)and only the phase changes. This gives rise to a lateral displacement (® rst order

eå ect) and to a focal shift (second order eå ect). In the second case, none of

the constituent waves undergoes total re¯ ection, and the modi® cation of the

characteristics of the re¯ ected beam with respect to the incident one is due to

the diå erent amplitudes of each component wave relative to the incident one(variation of the modulus of the re¯ ection coeæ cient with the incidence direction

but not of phase). This causes an angular shift of the centre of the beam (® rst

Journal of Modern Optics ISSN 0950± 0340 print/ISSN 1362± 3044 online # 2000 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

JOURNAL OF MODERN OPTICS, 2000, VOL. 47, NO. 10, 1645 ± 1658

order eå ect) and a variation of its width (second order eå ect). In the case inwhich some of the waves undergo total re¯ ection, the four eå ects are present

(as for re¯ ection in absorbing media). Similar situations are encountered under

conditions near those of Brewster [8].

As is known, if a bidimensional beam with Gaussian distribution of amplitude

and halfwidth w is incident on an isotropic interface, the amplitude of the re¯ ected® eld E¤ is given by

E¤…¹r ;²r† ˆ 1

2º

…1

¡1r…k²i

†exp ¡w2k¤2

²i

4… † exp ‰i …k¤¹r¹r ‡ k¤

²r²r†Š dk²r ; …1†

where * indicates that the variable corresponds to the re¯ ected beam, ¹r is themean direction of specular re¯ ection, ²i and ²r the directions perpendicular to the

mean incidence and re¯ ection ones and r…k²i† is the re¯ ection coeæ cient either for

the perpendicular or parallel mode (® gure 1).

When isotropic interfaces and incident beams with arbitrary linear polarization

are taken into account, the re¯ ected beam can be considered as the sum of twobeams with orthogonal polarizations (those corresponding to the eigenmodes of the

isotropic media). If the incidence mean angle is less than the total re¯ ection one,

then for each re¯ ected beam there is only angular shift of its centre and modi® ca-

tion of its width (diå erent for each polarization). If the mean incidence angle is

suæ ciently greater than the total re¯ ection one, then both beams undergo lateraldisplacement and focal shift (diå erent for each polarization){. Conversely, when

1646 L. I. Perez

z

x

y

z3

d

J

g i

x iz i

x r

g r

z r

p

e

e o , e e

Figure 1. Surface coordinate systems. The angle ¯ determines the incidence plane andthe angle # the direction of the crystal optical axis z3; " is the dielectric constant ofthe isotropic medium, "o the ordinary principal dielectric constant and "e theextraordinary one. The coordinates ¹i and ¹r correspond to the mean direction of theincident beam and of the beam specularly re¯ ected respectively; ²i and ²r are in theincidence plane and ±i and ±r are perpendicular to it.

{ We consider a beam with an angular spectrum such that all the waves that signi® cantlycontribute to the intensity are incident with angles less than or greater than the totalre ection one.

the interface corresponds to an isotropic medium and a crystal the behavioursexpected for the two re¯ ected beams are diå erent from those corresponding to

isotropic interfaces, even for simple geometries, since not only the re¯ ection

coeæ cients diå er signi® cantly but also there are two total re¯ ection angles: the

ordinary one ¬0T and the extraordinary one ¬

00T.

The relation between the ® rst and second order non-specular eå ects for a

Gaussian beam and the re¯ ection coeæ cients is well known for isotropic surfaces[5] but not for an isotropic medium± anisotropic medium interface. In section 2 of

the present paper we study the relation between these eå ects and the re¯ ection

coeæ cients for this kind of beam when they are incident on the interface between

an isotropic medium and a uniaxial crystal. We consider a bidimensional beam

with arbitrary linear polarization and orientation of the optical axis with respect tothe mean direction of incidence. In section 3 we analyse the characteristics of the

re¯ ected ® elds and we calculate explicitly the magnitudes of the ® rst order eå ects

for the case in which the optical axis is in the plane perpendicular to the plane

formed by the mean incidence direction and the normal to the interface. Finally, in

section 4, we determine the complex lateral displacement for highly symmetricgeometries.

2. Re¯ ected ® elds and non-specular eå ectsWhen a wave is incident on an interface between isotropic and anisotropic

media, in the general case the re¯ ected ® eld E¤ is a superposition of two ® eldsorthogonally polarized, even if the incident beam is polarized in one of the

eigenmodes of the isotropic medium. Then

E¤p

E¤s

… †ˆRpp Rsp

Rps Rss… † Ep

Es… †; …2†

where, for each incident wave, Ep and Es are the ® eld components parallel and

perpendicular to the incidence plane respectively and the re¯ ection coeæ cients are

given explicitly in equations (69)± (77) of [9]. Moreover if the incident wave islinearly polarized, we can de® ne the polarization angle ® by

Es

Ep

ˆ tan®: …3†

We consider a bidimensional Gaussian beam, of halfwidth w, which is incident

on the isotropic dielectric± uniaxial dielectric interface and we call the plane where

the distribution is Gaussian the ` incidence plane’ . If the beam is not too thin, as

has been shown in [10], a scalar treatment of the parallel ® eld can be performedand consequently the re¯ ected ® eld can be written in terms of the co-ordinates

which coincide with those corresponding to the mean direction of specular

re¯ ection, so

E¤…¹r ;²r† ˆ 1

2º

…1

¡1…E¤

s e±r‡ E¤

pe²r†

£ exp ¡w2k¤2

²i

4… †exp ‰i…k¤¹r ¹r ‡ k¤

²r ²r†Šdk¤²r ; …4†

Re¯ ection and non-specular eå ects of 2D Gaussian beams 1647

where E¤s and E¤

p are given in (2), ²i, ¹r and ²r are the coordinates used inequation (1) and ±r is the direction perpendicular to the incidence plane (because

of boundary conditions k²iˆ ¡k¤

²r). In ® gure 1 we represent the pro® le of the

incident beam (which has a mean direction given by ¬ and ¯). The plane yz is the

interface, º is the incidence plane and z3 is the direction of the optical axis

(characterized by #) which is contained in the plane xz.

As is known, an incident linearly polarized wave which is incident on anisotropic medium± uniaxial crystal interface, usually originates two linearly polar-

ized waves in the anisotropic medium: an ordinary one (with phase velocity uo and

electric ® eld with no component along the direction of the optical axis) and an

extraordinary one (with phase velocity u 00 and electric ® eld in the plane containing

the optical axis and the normal to the extraordinary wavefront) and a re¯ ectedwave with linear polarization and polarization angle diå erent to the incident one

[11± 13]. We use primes to indicate quantities referring to the ordinary wave and

double prime for those of the extraordinary wave.

The re¯ ection coeæ cients are usually complicated functions of the incidence

direction and of the characteristics of the media (dielectric constants and directionof the optical axis) [9, 13]. However, if the beam is not too thin then we ® nd that

not only equation (4) is valid but also we can make approximations for the

re¯ ection coeæ cients and solve it analytically. To determine explicitly the ® rst

and second order eå ects we apply the method used by Tamir in [6], expanding in

Taylor series (up to second order) the re¯ ection coeæ cients and the components of

the wavenumber vectors around those corresponding to the mean angle of theincident beam (k¤

²rˆ 0). Then the re¯ ected ® eld is

E¤…¹r ;²r† ˆ Es

p1=2

R0p

w2 ‡ i2…¹r ¡ Fp†

k

1=2exp

¡…²r ¡ Lp†2

w2 ‡ i2…¹r ¡ Fp†

k

264

375e²r

‡ Es

p1=2

R0s

w2 ‡ i2…¹r ¡ Fs†

k

1=2exp

¡…²r ¡ Ls†2

w2 ‡ i2…¹r ¡ Fs†

k

264

375e±r

…5†

with

R0p ˆ …Rpp ‡ tan® Rsp† k¤²r

ˆ0 ; R0s ˆ …tan® Rss ‡ Rps† k¤²r

ˆ0 ; …6†

Lp ˆ i@‰ln …Rpp ‡ tan®Rsp†Š

@k¤²r k¤

²rˆ0

; Ls ˆ i@‰ln …tan®Rss ‡ Rps†Š

@k¤²r k¤

²rˆ0

…7†

and

Fp ˆ ¡i2k@2‰ln …Rpp ‡ tan®Rsp†Š

@k¤2²r k¤

²rˆ0

; Fs ˆ ¡i2k@2‰ln …tan®Rss ‡ Rps†Š

@k¤2²r k¤

²rˆ0

:

…8†As has been shown by Tamir in 1986 [6], the imaginary parts of the lateral

displacement and of the focal shift have a physical signi® cance. The imaginary part

of the focal shift F leads to a modi® cation of the beam width given by

1648 L. I. Perez

w2m º w2 ‡ 2 Im …F†

k : …9†

On the other hand, the imaginary part of the lateral displacement appears as an

angular deviation of the re¯ ected beam with respect to the direction corresponding

to the specular re¯ ection

D ¬ º 2 Im …L†kw2

m: …10†

As expected, the characteristics of the non-specular eå ects strongly depend on

the real, imaginary or complex character of the re¯ ection coeæ cients (as in the case

of isotropic interfaces). However, contrary to what occurs in isotropic interfaces,

equations (7) and (8) cannot be solved easily because of the complexity of the

expressions for the re¯ ection coeæ cients. Nevertheless for certain orientations ofthe optical axis z3 with respect to the incidence direction, the re¯ ection coeæ cients

are simpler. We ® rst analyse a case with certain symmetry (optical axis in the plane

perpendicular to the incidence plane but forming any angle with the interface and

with the mean incidence direction) and then two geometries with high symmetry

(optical axis perpendicular to the mean incidence direction and optical axis

perpendicular to the interface). The ® rst case corresponds to ¯ ˆ 908 and anyvalue of #; the second to ¯ ˆ 908 and # ˆ 08, and the third to # ˆ 908 and, because

of symmetry reasons, to any value of ¯ (see ® gure 1).

3. Optical axis in the plane perpendicular to the incidence planeFrom the expressions of the re¯ ected ® elds obtained in [13], the re¯ ection

coeæ cients can be rewritten as a function of the characteristics of the media and of

the components of the incident ordinary transmitted and extraordinary trans-

mitted wavenumber vectors.With the restriction to the case ¯ ˆ 908 (kz ˆ 0) it results in:

Rpp ˆk2

y…kx ‡ k 0x†…"k 00

x ¡ "okx†…x·z3†2 ‡ ·!2"o…kx ‡ k 00

x †…"k 0x ¡ "okx†…z·z3†2

k2y…kx ‡ k 0

x†…"k 00x ‡ "okx†…x·z3†2 ‡ ·!

2"o…kx ‡ k 00

x †…"k 0x ‡ "okx†…z·z3†2 ; …11†

Rss ˆk2

y…kx ¡ k 0x†…"k 00

x ‡ "okx†…x·z3†2 ‡ ·!2"o…kx ¡ k 00

x †…"k 0x ‡ "okx†…z·z3†2

k2y…kx ‡ k 0

x†…"k 00x ‡ "okx†…x·z3†2 ‡ ·!

2"o…kx ‡ k 00

x †…"k 0x ‡ "okx†…z·z3†2 ; …12†

Rsp ˆ ¡Rps

ˆ2"o…·!

2"†1=2kykx…k 00

x ¡ k 0x†…z· z3†…x· z3†

k2y…kx ‡ k 0

x†…"k 00x ‡ "okx†…x·z3†2 ‡ ·!

2"o…kx ‡ k 00

x †…"k 0x ‡ "okx†…z·z3†2 ; …13†

with

k2x ‡ k2

yˆ ·!

2"; …14†

k 0x

2 ‡ k2y

ˆ ·!2"o ; …15†

k 00x

2‰"o…z· z3†2 ‡ "e…x· z3†2Š‡ "ok2y

ˆ ·!2"o"e ; …16†

Re¯ ection and non-specular eå ects of 2D Gaussian beams 1649

where …z·z3† ˆ cos # and …x· z3† ˆ ¡ sin #; ", "o, "e are the dielectric constants ofthe isotropic, ordinary principal and extraordinary principal media respectively

and kx, k 0x and k 00

x are the components perpendicular to the interface of the incident,

ordinary transmitted and extraordinary transmitted wavenumber vectors (® gure 1).

As is evident from equations (11)± (13), the expressions for the re¯ ection

coeæ cients for this geometry are simpler than for the general case althoughthe crossed coeæ cients Rsp and Rps are not zero as in cases of high symmetry.

In ® gure 2 we plot the moduli of the coeæ cients when the optical axis forms

an angle of 308 with the interface. Contrary to the general case [14], when

¯ ˆ 908 the moduli of Rsp and Rps are also symmetrical with respect to the

incidence angle.Using equations (2), (3), (11)± (16), we plot the real and imaginary parts of the

parallel and perpendicular ® elds corresponding to a re¯ ected wave for three

diå erent polarizations of the incident wave (® gure 3). The asymmetry with respect

to the incidence angle is due to the asymmetry in the phase of Rsp and Rps. In the

plot we observe that the only singular points are for the ordinary and extraordinary

total re¯ ection angles independent of the polarization angle of the incident wave.Consequently, the development done to calculate the non-specular ® rst order

eå ects is valid for a beam with any mean incidence angle if this is not too close to

§¬0T or to §¬

00T or angles where one of the ® eld components is zero.

Since for the Gaussian beam we consider an arbitrary incidence mean direction

(in the plane xy), a rotation of coordinates is convenient to evaluate equations(6)± (8) explicitly using the method proposed by Tamir for isotropic media. If k¤

²ris

the ²r component of the re¯ ected wavenumber vector for each component of the

beam and ¬ is the incidence angle of each constituent wave, then

ky ˆ ‰·!2" ¡ k¤

²r

2Š1=2 sin ¬ ¡ k¤²r

cos ¬ …17†

and the components perpendicular to the interface of the incident and refracted

wavenumber vectors corresponding to the mean incidence direction are

1650 L. I. Perez

Figure 2. Absolute values of the re¯ ection coeæ cients for ¯ ˆ 908 and # ˆ 308.("

1=2 ˆ 1:755, "1=2o ˆ 1:6584, "

1=2e ˆ 1:4865† . . . jRppj, - - - jRss j, Ð jRpsj ˆ jRspj.

Re¯ ection and non-specular eå ects of 2D Gaussian beams 1651

Figure 3. Real (Ð ) and imaginary (- - -) parts of the components which are parallel andperpendicular to the re¯ ected ® eld for three diå erent polarizations of the incidentbeam and for the same interface of ® gure 2. (a) Real and imaginary parts of E¤

p for

® ˆ 08; ¬10 ˆ 37:138. (b) Real and imaginary parts of E¤

s for ® ˆ 08; ¬20 ˆ 08. (c) Real

and imaginary parts of E¤p for ® ˆ 458; ¬

30 ˆ ¡27:178. (d) Real and imaginary parts

of E¤s for ® ˆ 458. (e) Real and imaginary parts of E¤

p for ® ˆ 908; ¬40 ˆ 08. ( f ) Real

and imaginary parts of E¤s for ® ˆ 908.

kx†k¤²r

ˆ0ˆ …·!

2"†1=2 cos ¬; …18†

k 0x†k¤

²rˆ0

ˆ …·!2†1=2‰"o ¡ " sin2

¬Š1=2; …19†

k 00x †k¤

²rˆ0

ˆ ·!2… 1=2 "o…"e ¡ " sin2

¬†"o…z· z3†2 ‡ "e…x·z3†2

" #1=2

: …20†

To calculate the complex lateral displacements explicitly from equation (7),the derivatives of the re¯ ection coeæ cients must be evaluated. After a long

calculation we obtain

@Rpp

@ky

ˆ ¡2"2o"

sin ¬cos ¬

Dpp

D2 ; …21†

@Rss

@ky

ˆ 2sin ¬cos ¬

Dss

D2 ; …22†

@Rsp

@ky

ˆ ¡ @Rps

@ky

ˆ 2""o…x· z3†…z· z3†Dsp

D2 ; …23†where

Dpp ˆ 2" cos2¬…kx ‡ k 0

x†…kx ‡ k 00x †…k 0

x ¡ k 00x †…x· z3†2…z· z3†2

‡ A " sin2¬…kx ‡ k 0

x†2…" ¡ "e†…x· z3†2

k 00x ‰"o…z· z3†2 ‡ "e…x· z3†2Š

‡ …" ¡ "o†…kx ‡ k 00x †2…z·z3†2

k 0x

( )

…24†

Dss ˆ 2""o cos2¬k 00

x …"okx ‡ "k 00x †…"okx ‡ "k 0

x†…x·z3†2…z·z3†2

‡ A

(" sin2

¬…"okx ‡ "k 00x †2…" ¡ "o†…x·z3†2

k 0x

‡ "2o…" ¡ "e†…"okx ‡ "k 0

x†2…z·z3†2

k 00x ‰"o…z· z3†2 ‡ "e…x· z3†2Š

)

…25†

Dps ˆ cos ¬…k 00x ¡ k 0

x†‰"o…kx ‡ k 00x †…"okx ‡ "k 0

x†…z· z3†2

¡ " sin2¬…kx ‡ k 0

x†…"okx ¡ "k 00x †…x· z3†2Š

‡ A""o sin2¬

…" ¡ "o†…"ok2x ‡ "k 00

x2†

"okxk 0x

(

‡ …"e ¡ "†…"ok2x ‡ "k 0

x2†

kxk 00x ‰"o…z· z3†2 ‡ "e…x· z3†2Š

‡ …"e ¡ "o†…"o ‡ "†k2x

k 0xk 00

x ‰"o…z· z3†2 ‡ "e…x· z3†2Š

)

; …26†

D ˆ " sin2¬…kx ‡ k 0

x†…"k 00x ‡ "okx†…x· z3†2 ‡ "o…kx ‡ k 00

x †…"k 0x ‡ "okx†…z·z3†2

; …27†where kx, k 0

x and k 00x are given in equations (18)± (20) and

A ˆ ·!2‰" sin2

¬…x· z3†2 ‡ "o…z· z3†2Š; …28†From equations (7), (11)± (13) and (21)± (28) and taking into account that

1652 L. I. Perez

@ky

@k¤²r k¤

²rˆ0

ˆ ¡ cos ¬;

we obtain the expressions for the lateral displacements of the two re¯ ected beams

corresponding to an incident beam linearly polarized:

Lp ˆ i2"o"D

"o sin ¬Dpp ¡ tan® cos ¬…x· z3†…z· z3†Dsp

Npp ‡ 2"o" tan® …x·z3†…z· z3†Nsp; …29†

Ls ˆ ¡i2

D

tan® sin ¬Dss ¡ "o" cos ¬…x· z3†…z· z3†Dsp

tan®Nss ¡ 2"o"…x·z3†…z· z3†Nsp; …30†

where Npp, Nss and Nsp are given by the numerators of (11)± (13) except for a factor

…·!2†2

{. Notwithstanding the lengths of formulae (29) and (30) they are easy to

handle and to understand. As expected, the lateral displacements for both beamsare zero (that is there is only an angular shift) only in the case in which the mean

incidence angle is less than both ordinary and extraordinary total re¯ ection angles,

¬0T and ¬

00T where, for the symmetry here considered:

sin ¬0T

ˆ "o

"

1=2

; sin ¬00T

ˆ "e

"

1=2

: …31†

Conversely, for angles greater than one of the total re¯ ection angles (greater

than ¬0T when the crystal is positive and than ¬

00T when it is negative), the

expressions (29) and (30) are complex (except for high symmetry geometries

which we consider in section 4). That is, both re¯ ected beams, polarized parallel

and perpendicular to the incidence plane, undergo not only lateral displacement

but also angular shift. This is because, even if all the constituting waves undergo

total re¯ ection, the existence of the re¯ ection coeæ cients Rsp and Rps causes any

pair of constituent waves which have the same aperture with respect to the meandirection of geometrical re¯ ection to have diå erent intensities. In ® gure 4 we

schematize and compare the ® rst and second order eå ects for a dielectric isotropic±

isotropic interface (for angles of incidence lesser and greater than the total

re¯ ection angle), and for a dielectric isotropic± anisotropic interface with the

optical axis contained in a plane perpendicular to the plane of the ® gure (forangles less than both the ordinary and extraordinary re¯ ection angles and for

angles greater than the minimum of them).

In ® gure 5 we plot the real lateral displacements (Goos± HaÈ nchen eå ect) of the

two re¯ ected beams (in wavelengths units) when the optical axis forms an angle of

308 with the interface and for three diå erent linear polarizations of the incidentbeam (parallel and perpendicular to the incidence plane and forming an angle of

458 with it). The discontinuities of the curves only correspond to the ordinary and

extraordinary total re¯ ection angles. In all the cases, the lateral displacement of the

beam appears for mean incidence angles greater than the smaller of the total

re¯ ection angles (extraordinary for negative crystals and ordinary for positive

Re¯ ection and non-specular eå ects of 2D Gaussian beams 1653

{ If the beam is p-polarized …® ˆ 908† it can be deduced that

Lp ˆ ¡i cos ¬1

D

Dsp

Nspand Ls ˆ ¡2i sin ¬

1

D

Dss

Nss

(from (4), (5) and (7) or from the limits of (29) and (30)).

crystals). Conversely, there is an angular shift of the beam for any mean incidence

angle not necessarily lesser than the smaller of the total re¯ ection angles. In ® gure 6we plot the angular shifts up to ® rst order (that is considering w ˆ wm in equation

(10)) of each re¯ ected beam (expressed in minutes). Figure 6 (a) corresponds to

angular shifts undergone by the re¯ ected beam polarized parallel to the incidence

plane, we consider the same three polarizations as in ® gures 3 (a), (c) and (e) and an

incident beam of width equal to 200 wavelengths. As can be seen, besides the

discontinuities corresponding to incidence angles equal to those of ordinary andextraordinary total re¯ ection, there are also discontinuities for the mean incidence

angles ¬j0. Contrary to what happens in isotropic interfaces, ¬

j0 depends not only on

the characteristics of the media but also on the polarization of the incident beam.

The non-validity of equations (29) and (30) for these angles was to be expected

since the series expansion up to second order applied in equation (5) is not valid for

mean incidence angles near ¬j0 (for these angles the real part of the ® eld is zero and

the ® rst order eå ects are intimately related to the logarithmic derivative of there¯ ected ® eld). As in the case of mean incidence near Brewster’ s angle for isotropic

media, this corresponds to a beam which is divided in two distinct peaks and

therefore we cannot speak of angular shift of the centre of the beam. In ® gure 6 (b)

1654 L. I. Perez

Figure 4. Comparison between ® rst and second order eå ects for a dielectric isotropic±isotropic interface (with dielectric constants " and "

0) and for a dielectric isotropic±anisotropic interface (with dielectric constants ", "o and "e and optical axis in a planeperpendicular to the xz plane). (In none of the situations have we drawn the changein w due to propagation.) Notice the existence of an angular shift only for isotropic±anisotropic interfaces when the angle of incidence is greater than one or both anglesof total re¯ ection.

we plot the shifts for the re¯ ected beam with polarization perpendicular to the

incidence plane for an incident beam of width equal to 100 wavelengths and the

other three polarizations.

The expressions for the complex displacements for bidimensional beams which

are incident on an interface between isotropic and anisotropic media, besides their

dependence on the orientation of the optical axis with respect to the interface, show

another fundamental diå erence in the case of isotropic interfaces (® gure 7). In the

case of isotropic media, if the bidimensional incident beam is linearly polarized,

forming a certain angle ® with the incidence plane, the magnitudes of the non-

specular eå ects for both re¯ ected beams (parallel and perpendicular modes) do not

Re¯ ection and non-specular eå ects of 2D Gaussian beams 1655

Figure 5. Lateral shift of the re ected beam (a) with p polarization and (b) withs polarization (in vacuum wavelengths units) for three incident polarizations:Ð ® ˆ 08, - - - ® ˆ 458, ¢ ¢ ¢ ® ˆ 908 and the interface of ® gure 2.

Figure 6. Angular shift (in minutes) of the re¯ ected beam at the same interface as in® gure 2: (a) with p polarization for: Ð ® ˆ 08, - - - ® ˆ 458, ¢ ¢ ¢ ® ˆ 908 andw ˆ 100¶; (b) with s polarization for: Ð ® ˆ 08, - - - ® ˆ 158, ¢ ¢ ¢ ® ˆ 908 andw ˆ 50¶.

depend on ® . Conversely, if the second medium is anisotropic the displacements

strongly depend on the polarization angle. Moreover, for each mean incidence

angle, using equations (29) and (30), we can obtain the polarization of the incident

beam required to make the ® rst order eå ect equal to zero for one of the re¯ ected

beams. In ® gure 8 we plot the polarization angles of the incident beam which make

the angular shifts of the two re¯ ected beams equal to zero as a function of the mean

incidence angle for # ˆ 308. Moreover for an arbitrary # we can obtain no angular

shift for both re¯ ected beams. Not only the angular shift but also the lateral

displacement can be made zero: from ® gure 7 we obtain that, if the polarization of

the incident beam and the orientation of the optical axis with respect to the

interface are given, it can be zero for a certain incidence angle.

1656 L. I. Perez

Figure 7. Lateral shift of the re¯ ected beam with p polarization for ® ˆ 308 and diå erentorientations of the optical axis.

Figure 8. Polarization angle as a function of the incidence angle which makes theangular shift of the re¯ ected beam with p polarization (Ð ) and s polarization (- - -)(# ˆ 308) zero.

4. High symmetry geometries4.1. Optical axis perpendicular to the incidence plane

In this case Rps and Rsp are zero for every incidence angle and, therefore, the

parallel and perpendicular modes are separated, that is, a beam polarized in one

of the eigenmodes of the isotropic media gives rise to a re¯ ected beam polarized

in the same eigenmode. The expressions (21) and (23) for the lateral displacements

are simple:

Lp ˆ 2i"o sin ¬

…·!2†1=2‰"o ¡ " sin2

¬Š1=2‰"o cos2¬ ¡ " sin2

¬Š ; …32†

Ls ˆ ¡2i sin ¬

…·!2†1=2…"e ¡ " sin2

¬†1=2 : …33†

Then if the incident beam is polarized in a certain direction, which does

not correspond to an eigenmode, the lateral displacement of each of the re¯ ected

beams does not depend on the polarization angle. Equation (32) corresponds to

the displacement for the parallel mode of a beam which is incident on anisotropic interface with dielectric constants " and "o, and equation (33) to

the displacement for the perpendicular mode for an isotropic interface with

dielectric constants " and "e. This result is to be expected since for this orientation

of the optical axis with respect to the incidence plane, the ordinary refracted

waves have p polarization and phase velocity uo (corresponding to a dielectric

constant "o) and the extraordinary ones s polarization and phase velocity ue

(corresponding to "e).

4.2. Optical axis perpendicular to the interfaceWhen the optical axis is perpendicular to the interface, because of symmetry

reasons, all the incidence planes are equivalent. Since the ® elds corresponding to

the extraordinary waves only have p component and phase velocity u 00, which

depends on uo and ue, the lateral displacement for the re¯ ected p-polarized beam is

expected to depend on the dielectric constant of the isotropic medium and on the

ordinary and extraordinary dielectric constants of the crystal. Replacing the value

…z· z3† ˆ 0 in equation (29) the expression for the displacement is

Lp ˆ 2i…"o"e†1=2…"e ¡ "† sin ¬

…·!2†1=2‰"e ¡ " sin2

¬Š1=2‰""e ¡ "e"o cos2¬ ¡ "

2 sin2¬Š : …34†

On the other hand, since the ® eld corresponding to an ordinary wave has no

component in the direction of the optical axis, it is always polarized perpendicu-

larly to the incidence plane and has velocity uo. Then the lateral displacement of

the s-polarized beam only depends on the ordinary dielectric constant

Ls ˆ ¡ 2i sin ¬

…·!2†1=2…"o ¡ " sin2

¬†1=2 : …35†

From equation (34) it is easy to see that the expression is not valid when themean incidence angle corresponds to the extraordinary total re¯ ection angle and to

Brewster’ s angle ¬B (which in this con® guration, as in the previous one, corre-

sponds to the incidence angle for which Rpp is zero).

Re¯ ection and non-specular eå ects of 2D Gaussian beams 1657

5. ConclusionsUsing the description of a bidimensional Gaussian beam in terms of a

continuous spectrum of plane waves and considering beams which are not too

thin, we relate the non-specular ® rst and second order eå ects to the re¯ ection

coeæ cients corresponding to interfaces formed by an isotropic medium and an

uniaxial one which has any orientation of the optical axis with respect to the mean

incidence direction. We apply the method to the case in which the optical axis is ina plane perpendicular to the one de® ned by the incidence mean direction and the

normal to the interface, and we obtain explicit formulas for the ® rst order eå ects

for re¯ ected beams with p polarization and with s polarization. The formulas are

valid for mean incidence angles not too close to total re¯ ection ones and for angles

such that the re¯ ected ® elds are zero (which except in cases of high symmetries donot correspond to the Brewster angle). There are substantial diå erences with the

complex displacements corresponding to isotropic surfaces. These diå erences are

due to the existence of the re¯ ection coeæ cients Rps and Rsp: for incidence angles

greater than the ordinary and/or extraordinary total re¯ ection ones the two

re¯ ected beams do not only undergo lateral displacement but also angular shift.On the other hand, contrary to what takes place in isotropic media, the lateral

displacement and angular shift of each of the re¯ ected beams depend on the

polarization of the incident beam. Because of this, for a given polarization and

mean incidence direction, there are re¯ ected beams with no angular shift.

AcknowledgmentsThis work has been done with the support of CONICET and UBA.

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[10] PEREZ, L. I., and CIOCCI, F., 1998, J. mod. Optics, 45, 2487.[11] SIMON, M. C., 1983, Appl. Optics, 22, 354.[12] SIMON, M. C., and ECHARRI, R. M., 1986, Appl. Optics, 25, 1935.[13] SIMON, M. C., and PEREZ, L. I., 1991, J. mod. Optics, 38, 503.[14] PEREZ, L. I., 1993, An. Afa (Asoc. Fõ Âs. Argentina), 5, 224.

1658 Re¯ ection and non-specular eå ects of 2D Gaussian beams