Home Search Collections Journals About Contact us My IOPscience Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial crystal interfaces This content has been downloaded from IOPscience. Please scroll down to see the full text. 2002 J. Opt. A: Pure Appl. Opt. 4 640 (http://iopscience.iop.org/1464-4258/4/6/308) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 157.92.4.6 This content was downloaded on 06/02/2015 at 21:13 Please note that terms and conditions apply. INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS J. Opt. A: Pure Appl. Opt. 4 (2002) 640–649 PII: S1464-4258(02)35548-X Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial crystal interfaces L I Perez Laboratorio de Óptica, Departamento de Fı́sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina E-mail: [email protected] Received 11 April 2002, in final form 9 September 2002 Published 31 October 2002 Online at stacks.iop.org/JOptA/4/640 Abstract The modifications undergone by a bidimensional Gaussian beam incident on an isotropic–anisotropic uniaxial medium interface when it is reflected and transmitted are analytically determined by developing one of the components of the wavenumber vectors up to second order. The description in terms of the spectrum of plane waves is used as a starting point. The case of uniaxial crystals with optical axis in any orientation relative to the interface but along the plane containing the mean incidence direction and the normal to the interface is considered. Moreover, the influence of the polarization modes and of the characteristics of the media in the appearance of the different non-geometrical effects is analysed. Keywords: Gaussian beam, anisotropic materials, reflection, refraction 1. Introduction The understanding and determination of non-geometric effects appearing in the reflection and transmission of limited beams incident on every kind of interface are important because of the applications in linear and non-linear optics, both under partial and total reflection conditions [1]. Starting from the description of a bidimensional Gaussian beam in terms of a continuous spectrum of plane waves, taking into account the precursory paper of Horowitz and Tamir [2] and considering not excessively thin beams (i.e. some wavelengths wide), non-specular effects of the first and second order of beams (symmetric or not) incident on isotropic interfaces with or without losses have been established both qualitatively and quantitatively by various methods [3–12]. On the other hand, the case of propagation in anisotropic media and of the reflection and transmission of limited beams on isotropic–birefringent crystal interfaces has not been studied so extensively [13–15]. As is known, a plane wave, polarized or not, incident on an interface isotropic–anisotropic uniaxial medium usually originates a polarized or a partially polarized reflected wave (of polarization different to that of the incident wave). Furthermore, contrary to what happens at isotropic interfaces, when the waves propagate through the uniaxial medium, there are two linearly polarized waves, an ordinary one (with electric displacement vector without a component along the optical axis direction) and an extraordinary one (with a component along the optical axis direction). If the optical axis is along the incidence plane and the incident wave is polarized along one of the principal modes of the isotropic medium then the reflected wave is also polarized in this way and, additionally, only one refracted wave (ordinary if the polarization of the incident beam is perpendicular to the incidence plane or extraordinary if it is parallel to it) can exist. In spite of the similarity of this behaviour to that appearing at isotropic interfaces, since in anisotropic media the direction of energy propagation (i.e. of the ray) and of the wavenumber vector of the extraordinary wave do not coincide, there are remarkable differences between the phenomena appearing at both kinds of interfaces. In the present paper we determine and analyse analytically and explicitly the first- and second-order nonspecular effects in reflection and the non-geometric effects in the transmission of a bidimensional Gaussian beam which is incident with mean angle α on an isotropic medium–uniaxial crystal interface, 1464-4258/02/060640+10$30.00 © 2002 IOP Publishing Ltd Printed in the UK 640 Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial crystal interfaces the optical axis being along the incidence plane but forming an arbitrary angle with the interface. To do so we use the description of the beam in terms of the superposition of plane waves with an angular spectrum not too wide (that is equivalent to, say, a not very thin beam), in such a way that we can do approximations up to second order in the fields and in the wavenumber vectors (paraxial approximation). This only limits the validity of some results to mean incidence angles not too near to those of ordinary or extraordinary total reflection or Brewster’s angle. In section 2 we calculate the magnitudes of the first- and second-order effects for the case of reflection and in section 3 those for transmission. In section 4 we analyse the influence of the various parameters in the characteristics of the non-geometric effects and the differences of the resulting effects with those for isotropic interfaces. We denote with primes those magnitudes referred to the ordinary wave and with double primes those of the extraordinary one. The direction of the optical axis is termed z̆ 3 while (z̆ · z̆ 3 ) and (x̆ · z̆ 3 ) are the projections of the optical axis of the crystal on the interface and on the normal to it, respectively (figure 1). The components of the incident wavenumber vectors k x , the ordinary transmitted one k x and the extraordinary transmitted one k x , perpendicular to the interface and the component parallel to the interface, k z , are related by [16] k x = (µω2 ε − k z2 )1/2 (1) k x = (µω2 εo − k z2 )1/2 kx = kz (εo − εe )(z̆ · z̆ 3 )(x̆ · z̆ 3 ) + √ εo εe µω2 (εo (z̆ · z̆ 3 )2 + εe (x̆ · z̆ 3 )2 ) − kz2 (εo (z̆ · z̆ 3 )2 + εe (x̆ · z̆ 3 )2 ) (2) Figure 1. Coordinate systems and characteristics of the media. The z axis is contained in the interface. The angle ϑ (positive in the figure) determines the direction of the optical axis of the crystal z 3 (that is, in the incidence plane); ε is the dielectric constant of the isotropic medium, εo the ordinary principal dielectric constant and εe the extraordinary one. k is the incident wavenumber vector, k∗ is the reflected one, k is the ordinary wavenumber vector, R̆ is the direction of the associated ray, k is the extraordinary wavenumber vector and R̆ is the associated ray. The coordinates x i and xr correspond to the mean directions of the incident and the specularly reflected beams, respectively (determined by the angle α). The coordinates x t and x t correspond to the ordinary and extraordinary transmitted beams (determined by the angles β and β ). The coordinates z i , z r , z t and z t are in the incidence plane. (3) 2. Effects on reflection where µ is the permeability of vacuum, ε is the permeability of the isotropic medium, εo and εe are the principal dielectric constants of the crystal and ω is the frequency of light. As is well known, the direction of energy propagation for an ordinary wave, R̆ , coincides with the direction of the corresponding normal to the wavefront, k , while for the extraordinary wave, the relation between the direction of energy propagation, R̆ , and of the extraordinary wavenumber vector, k , is given by [17] R̆ = 1 2 (ε k + (εo2 − εe2 )(k · z̆ 3 )z̆ 3 ) fn e (4) where f n is a normalization factor. Thus, if α is the incidence angle and β and β R are, respectively, the angles formed by the normal to the extraordinary wavefront [18] and by the corresponding ray with the normal to the interface, then tan β = sin α × ε(εo − εe )(z̆ · z̆ 3 )(x̆ · z̆ 3 ) sin α + √ εεo εe εo (z̆ · z̆ 3 )2 + εe (x̆ · z̆ 3 )2 − ε sin2 α εo εe − ε[εo (x̆ · z̆ 3 )2 + εe (z̆ · z̆ 3 )2 ] sin2 α (5) and tan β R = [εo (x̆ · z̆ 3 )2 + εe (z̆ · z̆ 3 )2 ] tan β − (εo − εe )(z̆ · z̆ 3 )(x̆ · z̆ 3 ) . [εo (z̆ · z̆ 3 )2 + εe (x̆ · z̆ 3 )2 ] − (εo − εe )(z̆ · z̆ 3 )(x̆ · z̆ 3 ) tan β (6) From equations (5) and (6) we can see that, depending on the incidence angle and the characteristics of the interface (dielectric constant and optical axis direction), β R can be greater than, less than or equal to β and, consequently, the incident ray and the extraordinary refracted one can be on the same or on a different semi-plane relative to the normal to the interface. If the incident beam of width w is not excessively thin (w 50 µm for common optical materials), a scalar treatment of the field parallel to the incident plane can be done [12] and, consequently, the reflected field, written in terms of the coordinates related to the mean direction of specular reflection, is ∞ 1 2 ∗2 ∗ ∗ E ∗ (xr , z r ) = (E s∗ ĕ y +E ∗p ĕzr )e−w kzr /4 ei(kxr xr +kzr zr ) dk z∗r 2π −∞ (7) where the symbol ∗ indicates that the quantity refers to the reflected beam, xr is the mean direction of specular reflection, z i and zr are the directions perpendicular to the mean incidence and reflection directions, respectively, E s∗ and E ∗p are the components of the electric field of each reflected wave perpendicular and parallel to the incidence plane and ∗ R pp Rsp Ep Ep = . (8) E s∗ R ps Rss Es Even in cases of geometries with symmetry where modes are separated (i.e. Rsp = R ps = 0), the reflection coefficients R pp and Rss [19] are complicated functions of the incidence direction and of the characteristics of the media (dielectric constant of the isotropic medium, ε; ordinary principal, εo , and extraordinary, εe , constants of the crystal and direction of the optical axis z 3 ). Nevertheless, to solve equation (7) analytically, the reflection coefficients can be approximated up to second order around the mean incidence direction of the Gaussian beam (k z∗r = 0). Applying the method proposed by Tamir [7] to interfaces formed by an isotropic medium and a 641 L I Perez uniaxial one with the optical axis on the incidence plane [11], we obtain R0 p = R pp |kz∗r =0 ∂(ln R pp ) Lp = i ∂k z∗r kz∗ =0 r ∂ 2 (ln R pp ) Fp = −i2k ∗ ∂k z∗2r kz =0 R0s = Rss |kz∗r =0 ∂(ln Rss ) Ls = i ∂k z∗r kz∗ =0 (9) (10) ∂ 2 (ln Rss ) Fs = −i2k ∂k ∗2 zr kz∗r =0 . (11) Replacing equations (9)–(11) in (7) and integrating, results in R0 p 2(x −F ) w2 + i √r 2 p µω ε Ep + tan γ √ π R0s √r −Fs ) w2 + i 2(x 2 µω ε 2 −(zr − L p ) exp ĕ 2(x −F ) p r p w2 + i √ 2 µω ε 2 − L ) −(z r s ĕs exp √r −Fs ) w2 + i 2(x 2 µω ε (12) where k indicates the modulus of the wavenumber vector in the isotropic medium and γ is the polarization direction of the incident beam with respect to the incidence plane, which is tan γ = Es . Ep (13) Equation (12) is formally identical to the one corresponding to isotropic interfaces and the characteristics of the nonspecular effects at the interfaces considered here also depend strongly on the real, imaginary or complex character of the reflection coefficients. In the case considered here the reflection coefficients R ps and Rsp are zero and the reflection coefficient Rss (R pp ) is associated with the ordinary (extraordinary) transmitted wave. Rewriting the expression for the reflection coefficient for the case in which the optical axis is along the incident plane in terms of the components of the wavenumber vector [19] we get Rss = R pp k x − k x k x + k x ε[εo (z̆ · z̆ 3 )2 + εe (x̆ · z̆ 3 )2 ]kx + ε(εe − εo )(z̆ · z̆ 3 )(x̆ · z̆ 3 )kz − εe εo kx = . ε[εo (z̆ · z̆ 3 )2 + εe (x̆ · z̆ 3 )2 ]kx + ε(εe − εo )(z̆ · z̆ 3 )kz + εe εo kx (14) 642 F p = −i 2 sin α (µω2 )1/2 (ε o − ε sin2 α)1/2 2εo cos α (µω2 )1/2 (εo − ε sin2 α)3/2 √ εo εe (ε ∗ − ε) 2 sin α 2 2 1/2 ∗ 1/2 (µω ) (ε − ε sin α) (ε(ε ∗ − ε sin2 α) − εe εo cos2 α) ∗ √ sin2 α)(ε2 −εe εo ) sin2 α εo εe (ε ∗ − ε)(ε ∗ + 2 (ε −ε ) 2 cos α ε(ε∗ −ε sin 2 α)−εe εo cos2 α (µω2 )1/2 (ε ∗ − ε sin2 α)3/2 (ε(ε ∗ − ε sin 2 α) − εe εo cos2 α) ε∗ ≡ εo (z̆ · z̆ 3 )2 + εe (x̆ · z̆ 3 )2 . (18) (19) (20) For the perpendicular mode, the lateral shift, L s , and the focal shift, Fs , are independent of the optical axis direction and of the principal extraordinary constant, this being in accordance with the fact that the ordinary wave has s-polarization and its phase velocity only depends on εo . In contrast, for the parallel mode (which corresponds to the extraordinary waves), even in cases of high symmetry (optical axis perpendicular to the interface or optical axis parallel to the interface and along the incidence plane), L p and Fp depend on both principal dielectric constants and so does the phase velocity of each of the extraordinary waves composing the beam [17]1 . As is well known, the real parts of the first- and secondorder effects correspond to the lateral shift of the beam maximum (Goos–Hänchen effect) and to the focal shift, while their imaginary parts are related to the angular deviation α of the direction of maximum intensity of the reflected beam (with respect to the corresponding direction of specular reflection) and to the width of the reflected beam wm [7]: α ≈ 2Im(L) kwm2 wm2 = w2 + 2Im(F) . k (21) (22) As is to be expected, for the case in which the electric field is polarized in the direction perpendicular to the incidence one, the results coincide with those obtained for isotropic interfaces, there can be only one singularity in L s and Fs and it corresponds to the angle of ordinary total reflection: sin αT = (15) (16) (17) where we have defined The coefficients given by equations (14) and (15) are symmetric with respect to the incidence angle and R pp depends on the orientation of the optical axis with respect to the interface. When the dielectric constant of the isotropic medium ε is between εo and εe , depending on ϑ, there can or cannot be extraordinary total reflection. This gives rise to important differences between isotropic–isotropic and isotropic–uniaxial interfaces. Replacing equations (14) and (15) in (9)–(11) we obtain the explicit expressions for the lateral shift, L, and the focal shift, F, in the incidence plane for both polarization modes: L s = −i L p = −i r r Ep E ∗ (xr , z r ) = √ π Fs = −i εo . ε (23) In figure 2 we plot the lateral shift and the angular displacement for the p-mode in a glass–calcite interface with different orientations of the optical axis. If the dielectric constant of the isotropic medium, ε, is greater than εo and εe (figure 2(a)) there are two singularities and they correspond to the extraordinary total reflection angle (see e.g. [19]): ε∗ sin αT = (24) ε 1 When the optical axis is in the plane of incidence and is parallel to the interface, the expression√ for the lateral displacement for the p-mode is εo εe (ε−εo ) 2i sin α L p = (µω 2 )1/2 (ε −ε sin 2 α)1/2 (εε −ε ε cos2 α−ε2 sin 2 α) , which depends on both o o e o principal dielectric constants. An analogous result is obtained for the complex focal shift. Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial crystal interfaces (a) (b) Figure 2. Angular shift (in minutes) (in vacuum wavelength units) of the reflected beams (w = 50 λ √vacuum ) with √ and lateral displacement √ p-polarization on a glass–calcite ( εo = 1.6584, εe = 1.4865) interface with different orientations of the optical axis. (a) ε = 1.7550, √ (b) ε = 1.5800. and to Brewster’s angle [16] sin α B = εo εe − εε∗ . εo εe − ε 2 (25) On the other hand, if ε has a value between εo and εe (figure 2(b)) there can be none (an impossible situation in isotropic interfaces), one or two singularities which correspond to the total reflection and Brewster’s angles. In spite of these, the expressions obtained still hold for mean incidence angles very close to those corresponding to the singularities, even in the case in which the beam is ten or more wavelengths wide. If we take into account only those waves that contribute more than 2% to the amplitude of the beam and we consider that the isotropic medium is air, for beams of more than 50 µm, the approximation is valid for mean incidence angles that differ from that of Brewster by half a degree (for mean incidence angles very close to Brewster’s the reflected beam stops having a Gaussian distribution since it is divided into two peaks and the approximation does not hold) [12]. As can be derived from equations (16)–(20), the effects cannot be complex; they are only real or imaginary. Thus, if the mean incidence angle is less than both angles of total reflection αT and αT , the result is solely imaginary and, consequently, both reflected beams only undergo angular deviation with respect to the geometric mean reflection direction and changes in width (figure 3). For angles between αT and αT , the reflected beam is usually a superposition of two beams: one with only lateral shift and focal displacement and another with only angular deviation and change in width. If, for example, the crystal is negative then αT < αT and the p-polarized reflected beam undergoes lateral shift and focal displacement while the s-polarized one undergoes angular deviation and change in width. In contrast, for mean incidence angles greater than both total reflection angles, the reflected beam is the sum of two beams, which only have different lateral shifts and focal displacements. The modification of the beam width as a result is less than 1% for all mean incidence angles such that the approximations are valid, except for mean incidence angles close to that of Brewster, for which it can be up to 10%. The characteristics of the reflected beams obtained for the kind of interface considered here differ not only from those obtained in the case of isotropic interfaces [6, 20] but also from those obtained in the case in which the interface is an isotropic medium–uniaxial crystal with the optical axis along the plane perpendicular to the incidence one [11]. In the case of isotropic interfaces, the effects corresponding to the perpendicular mode coincide with those obtained in equations (16) and (17) when the second medium has a dielectric constant εo and this is in accordance with the fact that, when the optical axis is along the incidence plane, the perpendicular mode corresponds to ordinary rays. In contrast, for the parallel mode the effects obtained in equations (18) and (19) correspond to no isotropic interface, i.e. we cannot find an equivalent dielectric constant. 3. Effects on transmission The existence of double refraction with different wavenumber vectors, though with separation of modes, enables us to write 643 L I Perez the electric field corresponding to the ordinary beam: 2 2 E p tan γ ∞ Es (xt , z t ) = Tso e−w kzi /4 ei(kxt x t +kzt z t ) dk z t ĕ y 2π −∞ (26) and the one composed by extraordinary waves is 2 2 Ep ∞ E p (xt , z t ) = T pe e−w kzi /4 ei(kxt x t +kzt z t ) dk zt ĕz t 2π −∞ (27) where xt denotes the propagation direction of the ordinary wave corresponding to the mean incidence wave, xt is the direction of the extraordinary one (figure 1), k z i is the component of each wave that composes the incident beam in the direction perpendicular to the mean incidence direction, Tso is the transmission coefficient corresponding to the ordinary wave (polarized perpendicular to the incidence plane) and T pe is the one corresponding to the extraordinary one (polarized parallel to the incidence plane) [17]: Tso = 2 T pe = 2 k x kx + kx (28) ε∗ k x + (εe − εo )(z̆ · z̆ 3 )(x̆ · z̆ 3 )k z . ε∗ k x + (εe − εo )(z̆ · z̆ 3 )(x̆ · z̆ 3 )k z + εεe ε∗o k x (29) As can be derived from equations (1)–(3) and (29), both coefficients are symmetrical with respect to the incidence direction. The ordinary case is equivalent to the case of a perpendicular mode at isotropic interfaces with constants ε and εo and analysis of the effects appearing enables us to compare them to those corresponding to the extraordinary case. Developing up to second order in k z t and using a method similar to that for reflection, we obtain the expression for the bidimensional transmitted ordinary beam in terms of the characteristics of the incident beam: √ 1 2E p tan γ ε cos α E s (xt , z t ) = √ √ 2 ε cos α + εo − ε sin α π √ 2 ei µω εo x t × √ t −Fo ) Mo w2 + i 2(x µω2 εo × exp − with (z t − L o )2 Mo w2 2(x t −Fo ) + i√ (30) µω2 εo 1 ∂ 2 k z2i 2 ∂k z2t kz =0 t ∂ ln Tso Lo = i ∂k Mo = zt (31) (32) Figure 3. First-order effects for a reflected bidimensional Gaussian beam that impinges on a dielectric isotropic–uniaxial interface (with dielectric constants ε, εo and εe and optical axis in the plane xz). (In none of the situations have we drawn the change in w due to propagation.) The incident beam is linearly polarized (not an eigenmode). L o = −i kzt =0 ∂ 2 ln Tso Fo = −i2 µω2 εo ∂k z2t kz t =0 . (33) Fo = i (εo − ε) sin α √ √ (µω2 εo )1/2 ε cos2 α ε cos α + εo − ε sin2 α (35) √ 2(εo − ε)[εo (1 + sin2 α) − 2ε sin4 α + ε cos α εo − ε sin2 α(1 + 2 sin 2 α)] . √ 2 (µω2 εo )1/2 ε cos4 α ε cos α + εo − ε sin2 α (36) After a little algebra, we get (εo − ε) sin2 α Mo = 1 + εo cos2 α 644 (34) The origin of the parameters L o and Fo is the asymmetry of the transmission coefficient Tso with respect to the mean incidence direction (except for normal incidence). Since there is no phase difference between the different components Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial crystal interfaces of the beam, they are purely imaginary and, consequently, they represent an angular shift from the maximum and a modification of the ordinary beam width equivalent to those appearing in partial reflection. From equation (35) we obtain that the angle between the direction of maximum intensity of the ordinary beam and the normal to the interface can be smaller or greater than the angle formed by the refracted geometric ordinary ray corresponding to the mean angle of the incident beam. This is in accordance with the fact that the transmission coefficient Tso decreases or increases when the incidence angle is ε < εo or ε > εo , respectively. Moreover the transmitted beam undergoes a widening or a narrowing (depending on the relation between the dielectric constant of the isotropic medium and the ordinary principal one) in Mo of order zero (due to the difference between the mean angle of the incident beam and of the transmitted ordinary beam given by Snell’s law) and one of second order given by −Fo /2 µω2 εo . The latter results in the widening of the beam if ε < εo since, in this case, the transmitted beam has an angular spread larger than that of the incident beam. For normal incidence, the angular shift and the geometric modification of the beam width are zero but the second-order effects still appear. Nevertheless for any other mean incidence angle this effect is, even for thin beams, at least three orders of magnitude less than the geometric effect. For the transmitted extraordinary beam we obtain 0 E p (x t , z t ) = T pe Ep √ eiK 1 xt π Me w 2 + i with 2(xt −Fe ) Ke (z − Ve x t − Se )2 exp− t 2(x −F ) Me w 2 + i tK e e 1 ∂ 2 k z2i Me = 2 ∂k z2t kz =0 (37) (38) t ∂k xt Ve = − ∂k z t kz =0 (39) t Ke = − ∂ 2 k xt ∂k 2 zt −1 (40) kzt =0 ∂ ln T pe Se = i ∂k zt Fe = i ∂ 2 ln T pe ∂kz2t ∂ 2 kxt ∂kz2t (41) kzt =0 kzt =0 . (42) kzt =0 0 Tep (43) ε∗ − ε sin2 α √ εo εe cos α + ε ε∗ − ε sin2 α (44) sin2 α ε (ε∗ − ε sin2 α) εo εe cos2 α sin2 β (45) √ = 2 ε√ Me = sin α µω2 ε sin β Ve = ε(εe − εo ) sin2 α [(1 − 2(x̆ · z̆ 3 )2 ) sin β cos β + (z̆ · z̆ 3 )(x̆ · z̆ 3 )(1 − 2 sin2 β )] εo εe sin 2 β (47) Se = i sin α 2 µω2 sin β cos α ε ∗ − ε sin2 α Ke = √ ∗ (ε − ε) εo εe cos α + √ ∗ ε ε − ε sin 2 α (48) µω2 εo εe sin3 β . (49) ε3/2 sin3 α As can be derived from equation (37), the complex lateral shift undergone by the extraordinary beam is given by L e = Se + Ve xt . The first term is purely imaginary, it corresponds to the asymmetry of the transmission coefficient T pe with respect to the mean direction of propagation and it results in zero in the case of normal incidence since T pe (α) = T pe (−α). The second term is real and, as can be derived from equations (5), (6) and (47), it is Ve = tan(β R − β ). That is, it corresponds to the difference between the direction of the normal to the wavefront and the direction of energy propagation and it appears even for normal incidence. In spite of the fact that, for the reflected beam, the real lateral shift corresponds to the Goos–Hänchen effect, for the transmitted beam it determines an angular shift of the maximum. This is so because, as can be derived from equation (37), the location of maximum intensity of the extraordinary beam corresponds to Se =0 (50) z t − xt tan(β R − β ) − 2i K e wm2 i.e. the angle formed by the direction of maximum intensity of the extraordinary beam with xt verifies Se (51) tan β = tan(β R − β ) − 2i 2 K e wme where K e is real, Se is purely imaginary and wme is the (real) width of the extraordinary beam transmitted in the interface given by Fe 2 = Me w2 − 2i . (52) wme Ke Since the difference between β and β R is less than 7◦ for natural media, the shift of the maximum of the beam can be approximated by β ∼ = (β R − β ) − 2i After a lot of algebra we get K1 = √ Fe = −i (ε∗ − ε) εo εe sin β (ε∗ − ε sin2 α)3/2 − 2ε1/2 √ √ × sin4 α εo εe cos α + ε ε∗ − ε sin2 α µω2 √ × sin α cos3 αε(ε∗ − ε sin2 α) εo εe cos α 2 −1 √ (46) + ε ε∗ − ε sin2 α Se . 2 K e wme (53) The second term of the right-hand side is much less than the first, except for β R − β ≈ 0, i.e. when the normal to the wavefront is perpendicular to the direction of the optical axis (equation (4)). Nevertheless, also in this case the contribution to the angular shift is negligible (of thousandths of a degree). In figure 4 we show the difference between the angle in degrees formed by the extraordinary beam and that corresponding to the direction of the mean wave, for an interface formed by glass (with a dielectric constant between the principal constants of 645 L I Perez extraordinary beam is given by 2 wme = w2 + 2 √ √ (ε∗ − ε) εo εe √ 2 . √ εε∗ εo εe + εε∗ (54) For normal incidence, the correction of second-order results is negligible for beams where the approximation holds since the right-hand side of equation (54) is less than 20 λ2vacuum for natural media. From the results (37)–(53), for normal incidence and highly symmetric geometries (optical axis parallel or perpendicular to the interface), we obtain the result of [14]. Moreover, if the transmission coefficient is equal to one [13, 23, 24], the results agree with those corresponding to bidimensional Gaussian beams, which propagate in uniaxial media when the optical axis is along the plane where the intensity distribution is Gaussian. 4. Geometric beams Figure 4. β (in degrees) for a refracted extraordinary beam as a function of the mean angle of incidence for different values of ϑ. The interface is formed by glass (ε1/2 = 1.5500) and calcite. the crystal) and calcite for different orientations of the optical axis with respect to the interface in terms of the incidence angle. From this figure we see that two incident beams, which impinge symmetrically with respect to the normal to the interface, are refracted (in contrast to what happens when the incident beam is s-polarized or when the interface is isotropic) in non-symmetric directions in spite of the symmetry of the transmission coefficients (except in cases of high symmetry in which the optical axis is parallel or perpendicular to the interface). As in the ordinary case, the extraordinary beam width is modified. The beam gets wider or narrower depending on, not only the dielectric constant of the media and the width of the incident beam, but also on the incidence angle and the orientation of the optical axis with respect to the interface (figure 5). The second term on the second side of equation (52) can be negligible or not, compared to the first term (geometric effect) depending on the geometry of the interface, the mean incidence angle and the beam width. For example, when the isotropic medium is glass with ε1/2 = 1.755, the anisotropic medium is calcite with ϑ = 30◦ and the beam has approximately 65 µm, the second term contribution is 45% when α = 67◦ and is totally negligible for normal incidence. Because of this, even in the case in which the dielectric constant of the isotropic medium is less than εo and εe , the beam gets narrower for some incidence mean angles and wider for others. As can be derived from equations (45), (48) and (52), when the beam is incident normally to the interface, the width of the 646 The properties of anisotropic media yield the possibility of choosing adequately the characteristics of the interface to obtain a reflected beam polarized parallel to the incidence plane and a refracted extraordinary beam with no non-geometric effects. That is, if the principal dielectric constant and the direction of the optical axis corresponding to the way in which the crystal has been cut are known, we can choose as the isotropic media one with dielectric constant ε = ε∗ (equation (20)). The reflection coefficient for the parallel mode is √ ε ∗ − εo εe (55) R pp = ∗ √ ε + εo εe and the transmission coefficient is T pe = 2 ε∗ + ε∗ √ εo εe (56) i.e. they are independent of the incidence angle. Thus, if a beam linearly polarized parallel to the incidence plane is incident on the interface, the amplitude of each of the constituent waves is diminished by the same amount when it is reflected and transmitted while the phase difference is the same for all the waves in each beam (the reflection and Brewster’s angles are π/2 for any orientation of the optical axis and are independent of the birefringence of the crystal [21]). Then there are no nongeometric effects neither of the first nor the second order. If, in contrast, a beam linearly polarized in an arbitrary direction is incident on the interface which has these characteristics, the reflected p-polarized beam is reflected with a mean angle equal to the incident one while the s-polarized beam is laterally shifted (if α > αT ) or has an angular displacement and modification in its width (if α < αT ) depending on the mean incidence angle and on whether the birefringence of the crystal is negative or positive. Since an interface of these characteristics mostly transmits light, we will analyse the characteristics of the extraordinary refracted beam. In figure 6 we plot the percentage difference between the width of the extraordinary beam and that of the incident one for different interfaces of an isotropic medium– uniaxial crystal with the optical axis forming an angle of 30◦ Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial crystal interfaces Figure 5. Percentage difference between the widths of the extraordinary refracted and incident (w = 50 λvacuum ) beams as a function of the mean angle of incidence for different isotropic–calcite interfaces. The full curve corresponds to isotropic–isotropic interfaces with dielectric constants ε − εe1/2 . The dotted and broken curves correspond to isotropic–calcite interfaces with ϑ = 30◦ and 60◦ , respectively. (a) (b) Figure 6. Percentage difference between the widths of the extraordinary refracted and the incident beams for different isotropic–uniaxial interfaces (w = 50 λvacuum ). The broken curve corresponds to ε = ε∗ . (a) isotropic–vaterite interfaces (εo1/2 = 1.5500, εe1/2 = 1.6500) and ϑ = 30◦ , (b) isotropic–calcite interfaces (εo1/2 = 1.6584, εe1/2 = 1.4865) and ϑ = 30◦ . with the interface. Figure 6(a) corresponds to a negative crystal (calcite) and figure 6(b) to a positive crystal (vaterite). We see that, when the isotropic medium has a dielectric constant equal to ε∗ , the extraordinary transmitted beam width changes by less than 9% for every mean incidence angle and this change corresponds to the geometric modification of the width. This relative modification depends on the orientation of the optical axis but it is almost independent of the beam width (within the restrictions of our model) and less than 12% for every mean angle of incidence and orientation of the optical axis (figure 7). In figure 8 we plot the difference between the mean incidence angle and the mean angle of the extraordinary beam for different isotropic media–calcite interfaces with ϑ = 30◦ . We see that, when ε is equal to ε∗ , the deviation is less than 6◦ for every mean incidence angle (that corresponds to the relation between the ray and the normal to the wavefront). This 647 L I Perez Figure 7. Percentage difference between the widths of the extraordinary refracted and the incident beams for different isotropic–uniaxial interfaces (w = 50 λvacuum ). The index of the isotropic medium is given by ε∗ . maximum deviation is almost independent of the orientation of the optical axis. As can be seen from figures 6 and 8, when the value of ε is very close to ε∗ (i.e. it differs by approximately <15%) the change in the width of the beam and the angle of deviation correspond fundamentally to the geometrical effects for mean angles of incidence less than 60◦ . Moreover it is possible to choose a mean angle of incidence for each interface such that the geometrical effects are also negligible. 5. Conclusions The properties of anisotropic media lead to the appearance of substantial differences between the complex displacement obtained at isotropic–isotropic interfaces and those obtained at isotropic–anisotropic interfaces and this is so even when the beams are limited in one direction and along an incidence plane that is a symmetry plane of the crystal. Starting from the description of a Gaussian beam in terms of a continuous spectrum of plane waves and in the case of bidimensional beams not excessively thin (>50 µm approximately), we have determined the characteristics of the reflected and transmitted beam for interfaces formed by an isotropic medium and a uniaxial one in the case in which the mean incidence direction, the crystal optical axis and the normal to the interface are in the same plane. We have applied the method to linearly polarized beams, obtaining explicit formulae for the first- and second-order effects for the reflected beam (one with p-polarization and another with s-polarization) and for both transmitted beams (ordinary and extraordinary). We have found that, in contrast 648 Figure 8. Difference between the mean angle of the refracted extraordinary and of the incident beams (in degrees) as a function of the mean angle of incidence for isotropic–calcite interfaces with different values of ε and ϑ = 30◦ . The broken curve corresponds to ε∗ . to what happens at isotropic interfaces, there are mean incident angles (of values between the ordinary total reflection one and the extraordinary total reflection one) for which one of the reflected beams (p- or s-polarized) undergoes a lateral shift and a focal displacement relative to those of the specularly reflected beam and the other (s- or p-polarized) undergoes an angular displacement and a modification of width. Which one undergoes the latter or former effects depends on the positive or negative character of the birefringence. On the other hand, the ordinary transmitted beam undergoes the same changes as those of a beam polarized perpendicular to the incidence plane refracted at an isotropic interface of dielectric constants ε and εo , i.e. the angular displacement and the widening or narrowing of the beam due to nonspecular effects are negligible for beams which are not excessively thin. The extraordinary transmitted beam has behaviour notably different to that of the transmitted one with p-polarization at an isotropic interface and, similar to what happens for the reflected beam with this polarization, it is not possible to simulate the isotropic–uniaxial interface by an equivalent isotropic one. We have found that both the real longitudinal shift and the imaginary one correspond to an angular displacement of the direction of maximum intensity of the beam. The real displacement, appearing because of the relation ray-normal to the wavefront, leads to the fact that two p-polarized beams, which are incident symmetrically with respect to the normal to the interface, give rise in the uniaxial medium to two non-symmetrical p-polarized beams. For this kind of interface the first-order effect is negligible but Modifications of geometric parameters of Gaussian beams reflected and transmitted on isotropic–uniaxial crystal interfaces the second-order effect can be negligible or not compared to the geometric effect, depending on the width of the original beam, the principal constants of the media, the orientation of the optical axis and the mean incidence direction. Moreover, in contrast to what happens at isotropic interfaces and depending on the mean angle of incidence, the p-polarized refracted beam can be wider or narrower for values of the dielectric constant of the isotropic medium greater or smaller than the principal dielectric constants. Furthermore we have found that, for beams with p-polarization, we can find a value of the dielectric constant of the isotropic medium for which the non-geometric effects do not appear for all the mean incidence angles. Moreover these effects are negligible for many mean angles of incidence when the value of the dielectric constant is very close to it. This suggests that the use of an isotropic medium verifying the abovementioned characteristics could be of advantage for the coupling of devices made of uniaxial crystals. Acknowledgments This work has been done with the support of CONICET and UBA. References [1] Lotsch H K V 1970 Optik 32 116 Lotsch H K V 1970 Optik 32 189 Lotsch H K V 1971 Optik 32 299 Lotsch H K V 1971 Optik 32 553 [2] Horowitz B and Tamir T 1971 J. Opt. Soc. Am. 61 586–94 [3] Ra J W, Bertoni H L and Felsen L B 1973 SIAM J. Appl. Math. 24 396–413 [4] Mc Guirk M and Carniglia C K 1977 J. Opt. Soc. Am. 67 103–7 [5] Carniglia C K and Brownstein K R 1977 J. Opt. Soc. 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