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Gaussian Beam Propagation in Generic Anisotropic Wavenumber Profiles
Igor Tinkelman and Timor Melamed
Abstract
The propagation characteristics of the scalar Gaussian Beam in a homogeneous anisotropic medium is considered.
The medium is described by a generic wavenumber profile wherein the field is formulated by a Gaussian plane wave
distribution and the propagation is obtained by saddle point asymptotics to extract the Gaussian Beam phenomenology
in the anisotropic environment. The resulting field is parameterized in terms of the spatial evolution of the Gaussian
beam curvature, beam width, etc., which are mapped to local geometrical properties of the generic wavenumber
profile.
I. INTRODUCTION AND STATEMENT OF THE PROBLEM
Anisotropic materials are of interest for optical waveguides, microwave devices, plasma science, and dif-
ferent propagation environments. Comprehensive studies have been performed for the problem of Gaussian
Beam (GB) 2D and 3D propagation for specific wavenumber profiles [1], [2], [3], [4], [5], [6], [7], [8], [9],
[10], [11], [12], [13], [14], [15], [16], [17], [18]. The generic profiles of the three-dimensional problem asso-
ciated with a GB with complex wavenumber spectral constituents, are of fundamental significant, since GB
propagation is a practical problem as well as theoretically significant, since GBs form the basis propagators
for the “phase-space beam summation method”, which is a general analytical framework for local analy-
sis and modelling of radiation from extended source distributions [19]. Using GBs as basis waveobjects for
modelling different anisotropic propagation and scattering problems were introduced in [20], [21]. The prop-
agation of GB over a generic anisotropy has been studied in [22] using the Complex Source Point method.
By applying the saddle point asymptotic technique, approximated over the zaxis, the authors arrived at a
close form analytic solution for the GB field. In our view, the Complex Source Point method cannot account
for astigmatic effects which are present in our analysis of the generic wavenumber profiles, and therefore the
results in [22] may be applied only to the case of uniaxially anisotropic medium. Alternatively, by applying
a planewave spectral representation for the propagation problem, an alternative rigorous solution for the GB
field is presented here, which is suitable for any generic wavenumber profile (see discussion following (8)) .
The current study is concerned with the effects of anisotropy on the propagation characteristics of the scalar
Gaussian beam field in a homogeneous medium described by the generic wavenumber profile kz(ξ)where
ξis wavenumber normalized coordinate. The field is formulated in the frequency-domain via a plane wave
spectral integral and is evaluated asymptotically by saddle point technique. Given a field, u(r), where a
exp(−iωt)time dependence is assumed and suppressed, and r= (x1, x2, z)are conventional Cartesian
coordinates, the wavenumber spectral (plane wave) transform pairs on the z= 0 initial surface are given by
euo(ξ) = Z∞
−∞
d2x uo(x)e−ikξ·x(1 a)
uo(x) = (k/2π)2Zd2ξeuo(ξ)eik ξ·x(1 b)
where ξ= (ξ1, ξ2)is the normalized spatial wavenumber vector, x= (x1, x2),kis the homogenous media
wavenumber k=ω/c , with cbeing the phase velocity of the homogeneous media, and eidentifies a
wavenumber spectral function. The normalization with respect to the wavenumber krendering ξfrequency-
independent. Therefore, the GB field is formulated via the Gaussian distribution
u0(x) = exp[−1
2kx2/β](2)
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel.
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Fig. 1. Fig. 1. The local beam coordinate frame for the Gaussian beam propagating in the anisotropic medium. The beam axis is directed along
the unit vector b
κ. The local transverse coordinates, xb, are given by the transformation in (13). The transverse local coordinates are located over
the (x1, x2)plane which is in general non-orthogonal to the beam axis. The rotation transforation of (x1, x2)to (xb1, xb2)by αis carried out
so that the resulting field in (16) exhibits Gaussian decay in the local coordinates.
were β=βr+iβiwith βr>0is a parameter, and x2=x·x=x2
1+x2
2. By inserting (2) into (1), we obtain
the plane wave spectral distribution corresponding to (2)
eu0(ξ) = (2πβ/k) exp[−1
2kβξ2].(3)
The plane wave distribution in (3) can be propagated into the z > 0half space using the generic anisotropic
propagator exp[ikζ(ξ)z], where, as in (1), we normalize the longitudinal wavenumber ζby k. Using this
propagator, the field propagating into the z > 0half space is given by
u(r) = βk
2πZd2ξexp[ikΦ(ξ,r)],Φ(ξ,r) = (ξ·x+ζ(ξ)z+i
2βξ2).(4)
The field in (4) cannot be evaluated in close form. In the next section we shall evaluate it asymptotically to
obtain an analytic expression for the beam field in the high frequency regime.
II. ASYMPTOTIC EVALUATION AND PARAMETERIZATION
The field representation in (4) may be evaluated asymptotically by the saddle point technique. The stationary
point ξssatisfies
∇ξΦ = x+∇ξζ(ξ)|ξsz+iβξs= 0.(5)
Equation (5) has a real solution only if ξs= 0 and for observation points
x+z∇ξζ0= 0,(6)
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where, here and henceforth, subscript 0 denotes sampling at ξ= 0, i.e., ∇ξζ0≡ ∇ξζ|ξ=0. The condition in
(6) defines the beam axis being as a tilted line in the configuration space, with a anisotropy-dependent tilt.
For isotropic materials, where ζ=p1−ξ2, giving ∇ξζ0= 0, the beam axis coincides with the z-axis. For
the generic ζ(ξ)wavenumber profile, the beam axis is directed along the unit vector
b
κ= (cos ϑ1,cos ϑ2,cos ϑ3)(7)
where ϑ1,2,3, are the beam axis angles with respects to the (x1, x2, z)axes, respectively. In view of (6), they
are given by
cos ϑ1,2=−cos ϑ3∂ξ1,2ζ0,cos ϑ3=1
p(∂ξ1ζ0)2+ (∂ξ2ζ0)2+ 1.(8)
Note, that the beam axis direction, as defined in (6), is different from the definition given in [22] where the
propagation of Gaussian beam in a generic wavenumber profile was investigated using the Complex Source
Point method. The beam axis as defined in [22] is directed along the complex source bparameter which,
for our problem, coincides with the zaxis. Clearly, from (6), the zaxis may serve as the beam axis only for
symmetrical ζ, where ∇ξζ0= 0, as in isotropic or uniaxially anisotropic medium. The condition in (6) may
,therefore, serves as a more generalized definition for the beam axis which accounts for astigmatic effects
due to the medium anisotropy (see figure1).
For off-axis observation points, equation (5) could not be solved explicitly. Furthermore, the off-axis sta-
tionary point is complex and the solution requires analytic continuation of the wavenumber profile ζ=ζ(ξ)
for complex ξ. In order to obtain a closed form analytic solution for the beam field, we notice that the beam
field decays away from the beam axis. Therefore, we may apply a Taylor expansion of the phase Φabout
the on-axis stationary point ξs= 0.
Φ≈Φ0+Φ1·ξ+1
2ξΦ2ξ(9)
with
Φ0= Φ|ξ=0=ζ0z, Φ1=∇Φ|ξ=0=∇ξζ0z+x,(10)
and
Φ2=iβ +∂2
ξ1ζ0z ∂2
ξ1ξ2ζ0z
∂2
ξ1ξ2ζ0z iβ +∂2
ξ2ζ0z,(11)
where subscript 0implies sampling at ξ= 0. Using (9), one finds that the saddle point for both on- and
off-axis observation points is ξs=−Φ2
−1Φ1and the field in (4) may be evaluated asymptotically by
u(r) = β
√−det Φ2
exp[ikS(r)], S(r) = Φ0−1
2Φ1Φ2
−1Φ1.(12)
The beam field in (12) may be presented in terms of local beam coordinates over which the field exhibits a
Gaussian decay away from the beam axis. The local beam coordinates, rb= (xb1, xb2, zb), are defined by the
non-orthogonal transformation
rb=Tr,T=
cos αsin α(−cos ϑ2sin α−cos ϑ1cos α)/cos ϑ3
−sin αcos α(−cos ϑ2cos α+ cos ϑ1sin α)/cos ϑ3
0 0 1/cos ϑ3
(13)
where cos ϑ1,2,3are defined in (8), and the angle αis given by
tg2α=−2∂2
ξ1ξ2ζ0/(∂2
ξ2ζ0−∂2
ξ1ζ0).(14)
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The transformation in (13) is consist of a rotation transformation in the (x1, x2)plane by α, in which the
phase S(r)in (12) exhibit Gaussian decay, followed by tilting the z-axis to the beam-axis direction b
κin (7)
(see figure 1). The inverse transform is given by
T−1=
cos α−sin αcos ϑ1
sin αcos αcos ϑ2
0 0 cos ϑ3
.(15)
Using the beam coordinate system (13) in (12), the field may be presented in the Gaussian form
u(r) = β
√−Γ1Γ2
exp ik ζ0zbcos ϑ3+1
2x2
b1
Γ1
+x2
b2
Γ2 (16)
where
Γ1,2=−
∂2
ξ1ζ0z+∂2
ξ2ζ0z+ 2iβ ∓q(∂2
ξ1ζ0z−∂2
ξ2ζ0z)2+ 4(∂2
ξ1ξ2ζ0z)2
2.(17)
Using (14) in (17), we obtain
Γ1,2=zba1,2−iβ, a1,2=cos ϑ3
cos(2α)∂2
ξ1ζ0−cos2α
sin2α+∂2
ξ2ζ0sin2α
−cos2α (18)
Equation (16) has the form of a Gaussian beam, propagating along the beam axis zb. The beam field exhibits
a Gaussian decay in the transverse local coordinate xbwhich are, in general, tilted with respect to the beam
axis direction b
κ. In order to parameterize the beam field, we rewrite the element of Γ1,2in the form
1
Γ1,2
=1
R1,2
+i
kD2
1,2
(19)
where
D1,2=qF1,2/kq1 + a2
1,2(zb−Z1,2)2/F 2
1,2(20)
R1,2=a1,2(zb−Z1,2) + F2
1,2/[a1,2(zb−Z1,2)].(21)
with
Z1,2=−βi/a1,2, F1,2=βr.(22)
By substituting (19) into (16) one readily identifies D1,2as the beam width in the (z, xb1,2)plane, while R1,2
is the phase front radius of curvature. The resulting GB is therefore astigmatic; its waist in the (z, xb1,2)
plane, is located at zb=Z1,2, while F1,2is the corresponding collimation length. This astigmatism is caused
by the beam tilt which reduces the effective initial beam width in the xb1,2directions.
The compact presentation in equations (16)–(22), parameterizes the GB field in terms of local properties
of the generic wavenumber profile about the stationary point ξ= 0. This general parameterization can be
compared to the isotropic profile where ζ(ξ) = √1−ξ·ξ. In which case ζ0= 1,∂ξ1,2ζ0= 0, and the beam
axis in (7) coincides with the z-axis. Furthermore, using ∂2
ξ1ξ2ζ0= 0 in (14), we obtain α= 0, and therefore,
from (18), Γ1,2=z−iβ. By inserting Γ1,2into (16) we obtain the well-known isotropic asymptotic GB
field [19]
u(r) = −iβ
z−iβ exp[ik(z+1
2x2/(z−iβ))].(23)
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III. CONCLUSION
In this paper, we have been concerned with the parameterization of the effects of spectral anisotropy on the
propagation characteristics of a paraxially approximated Gaussian beam in a medium with generic wavenum-
ber profile. Various beam parameters have been systematically found to quantify the effect of anisotropy on
various observables associated with the GB field. Introducing the anisotropy-dependent non-orthogonal lo-
cal beam coordinate system enables the beam parameterization to be quantified in terms of local properties
of the anisotropic surface ζ(ξ), at the stationary on-axis point ξ= 0.
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