Modifications of geometric parameters of Gaussian beams reflected

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Modifications of geometric parameters of Gaussian beams reflected and transmitted on
isotropic–uniaxial crystal interfaces
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2002 J. Opt. A: Pure Appl. Opt. 4 640
(http://iopscience.iop.org/1464-4258/4/6/308)
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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
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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.
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