Temporal structure of stimulated-Brillouin-scattering reflectivity considering transversal-mode
development
Shahraam Afshaarvahid,
1,
* Axel Heuer,
2,²
Ralf Menzel,
2
and Jesper Munch
1
1
Department of Physics and Mathematical Physics, Adelaide University, SA 5005, Australia
2
Lehrstuhl fu ¨r Photonik, Universita ¨t Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany
Received 9 January 2001; published 11 September 2001
The time-resolved reflectivity of optical phase conjugation by stimulated Brillouin scattering SBS is
investigated both theoretically and experimentally. A three-dimensional and transient model of SBS is devel-
oped to compare the experimental and theoretical results. Noise initiation of the SBS process is included in the
model to simulate the shot-to-shot variation in the reflectivity and the Stokes temporal profile.
DOI: 10.1103/PhysRevA.64.043803 PACS numbers: 42.65.Es, 42.65.Hw
I. INTRODUCTION
Optical phase conjugation by stimulated Brillouin scatter-
ing SBS has been extensively studied, both theoretically
and experimentally, and used in laser systems to improve
beam quality since the early 1970s. Extensive experimental
studies of SBS have revealed many aspects of this phenom-
enon and SBS now is used in commercial laser systems. The
limitations of the SBS are, by now, well known. However, in
spite of all of the extensive theoretical studies 1–16,a
single unified numerical model that can include all aspects of
SBS phenomena especially transient and transverse effects
has not, until recently 17, been developed. Thus, no single
model has been able to produce results in good agreement
with experiments, even for simple cases such as the temporal
profile of the Stokes output or the reflectivity curve. To the
best of our knowledge a detailed comparison with a good
agreement between the SBS experiment and theory has not
been published before.
In this paper we present a detailed comparison of results
from our transient three-dimensional model 17 and experi-
ments performed in identical-parameter regime. SBS experi-
ments were carried out using two materials with extreme
phonon lifetimes, Freon 113 with =0.84 ns and SF
6
with
=17.4 ns. The total reflectivity vs input energy and the
temporal profile of the Stokes output were examined experi-
mentally and numerically. An excellent agreement between
the experimental and numerical results has been achieved.
II. THEORY
The equations describing the SBS process are derived
from Maxwell’s equations for the electric fields and the
Navier-Stokes equation for the acoustic field inside the ma-
terial. Using the slowly varying envelope approximation, the
three-dimensional transient-SBS equations describing the
acoustic, Stokes and pump fields propagating along the z
direction, can be written as 18,19,12
t
+
Q =-ig
1
E
l
E
s
* , 1a
i
2 k
s
“
t
2
+
n
c
t
+
z
E
s
=-ig
2
Q * E
l
, 1b
-i
2 k
l
“
t
2
+
n
c
t
-
z
E
l
=-ig
2
QE
s
. 1c
Here “
t
2
refers to the derivatives in the transverse direc-
tions x and y, g
1
and g
2
are coupling constants given by
g
1
=
2
n
q
c
2
,
g
2
=
4 cn
0
,
n is the refractive index of the medium and k
s
k
l
are the
Stokes and pump wave numbers, respectively. In the trans-
verse directions, we use a decomposition method
20,4,5,11,14 to expand the electric fields in terms of ortho-
normal bases modes A
m
and B
m
,
E
l
r
, z , t =
m
a
m
z , t A
m
r
, z , 2
E
s
r
, z , t =
m
b
m
z , t B
m
r
, z , 3
where r
is the position vector in transverse directions and
the particular set of A
m
and B
m
used in our model will be
given below. By substituting Eqs. 2 and 3 into Eqs. 1a –
1c and assuming that A
m
and B
m
satisfy the homogeneous
Maxwell equations, Eqs. 1b and 1c can be rewritten as
17
n
c
t
+
z
b
n
=g
1
g
2
i , j , k
C
ij
* a
k
g
knij
, 4
n
c
t
-
z
a
n
=-g
1
g
2
i , j , k
C
ij
b
k
g
knij
* , 5
*Electronic address: shahraam@physics.adelaide.edu.au
²
Electronic address: heuer@rz.uni-potsdam.de
PHYSICAL REVIEW A, VOLUME 64, 043803
1050-2947/2001/644/0438035/$20.00 ©2001 The American Physical Society 64 043803-1