Mechanism of beam cleanup by stimulated Brillouin scattering in multimode fibers
Qilin Gao, Zhiwei Lu*, Chengyu Zhu, and Jianhui Zhang
National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin 150001, China
E-mail: zw_lu@sohu.com
Received March 7, 2015; accep ted Mar ch 24, 2015; published online April 13, 2015
The conditions leading to the beam cleanup effect induced by stimulated Brillouin scattering (SBS) have been debated by many researchers. In
this paper, we firstly reveal that the key point of the beam cleanup induced by SBS is at the select ive excitation of the fundamental mode of the
Stokes beam, which is amplified by the counterpropagating high-order-mode pump beam along multimode fibers while preserving the fundamental
mode and subsequently realizing the beam cleanup. This mechanism is theoretically modeled and experimentally demonstrated in this work.
©
2015 The Japan Society of Applied Physics
T
he process of stimulated Brillouin scattering (SBS) in
optical fibers has been investigated by many research-
ers over the last three decades. On the one hand,
efforts to mitigate the SBS effec t have been reported.
1–3)
On
the other hand, the SBS effect has been implemented in many
fields such as sensors,
4)
optical delays,
5)
and fiber lasers.
6)
Moreover, an interesting effect caused by SBS called “beam
cleanup”, which can transform a poor-quality pump beam
into a high-quality one, is very attractive but the mechanism
is unclear until now.
In the case of the SBS processes in bulk media, the
reflected Stokes beam is always phase-conjugated to the
pump beam, which exhibits a replication of the pump profile.
In step-index (SI) multimode (MM) fibers, phase conjugation
with high fidelity has also been achieved either by pulsed
7)
or continuous wave
8)
laser sources. Nevertheless, in 1993,
Bruesselbach observed a beam cleanup phenomenon that
contradicted the properties of phase conjugation.
9)
After that,
some researchers have employed this effect to combine laser
beams,
10)
but they have not explained it theoretically. In
2006, Lombard et al. firstly modeled the beam cleanup,
showing that the behaviors of either beam cleanup or phase
conjugation were linked to the fiber geometry.
11)
They sug-
gested that in MM fibers, it would lead to phase conjugation
in SI fibers and to beam cleanup in long (>30 m) graded-
index (GI) fibers. Experimentally, they have also realized
good-quality beam cleanup in GI fibers.
11,12)
However, there
are still many other experimental results that contradicted the
model. For example, phase conjugations have also been ob-
served in GI fibers with lengths of several hundred meters
13)
and several kilo meters.
14,15)
As was debated, Massey et al.
suggested that these two behaviors are interrelated.
8,16)
They
indicated that when the length of the MM fiber increases,
phase conjugation would be inhibited, while the beam
cleanup effect would be enhanced. Mocofanescu and Shaw
even considered that their cleanup results are related to a
spatial filter of a poo r-quality incident beam, not to a SBS
process.
14)
The modeling
17,18)
and experiment
19)
of intermo-
dal Brillouin scattering also predict a way of explai ning the
problem; however, the mechanism of beam cleanup in MM
fibers is still unclear so far.
In this paper, we propose a new mechanism to explain
the beam cleanup process based on the SBS in MM fibers.
We indicate that under the SBS amplification process, the
output Stokes beam would preserve the same modes of the
seed beam, which are externally excited. The so-called beam
cleanup effect is actually an amplification of the low-order
mode, especially the fundamental mode of the seed beam
that is based on the selective excitation in MM fibers. The
mechanism is modeled theoretically and proved experimen-
tally in this work.
To model the SBS process in MM fibers, we begin with the
expression of the electric field as
E
p,s
ðr; ; z; tÞ
¼
1
2
fA
p,s
ðzÞ f
p,s
ðr; Þ exp½iðk
p,s
z !
p,s
tÞ þ c :c:g; ð1Þ
where the scalar field approximation has been used, A
p,s
ðzÞ
represent the amplitudes of the pump and Stokes fields,
respectively, f
p,s
ðr; Þ are the transverse field profiles in
cylindrical coordinates, k
p,s
are the wave vectors, and !
p,s
are
the angular frequencies. Then, we use the acoustic equation
in elast ic solid as
20)
r
2
t
u
z
þ
þ
@
2
u
z
@z
2
þ
þ
@
2
u
z
@x@z
þ
@
2
u
z
@y@z
@
2
u
z
@t
2
¼
"
0
e
2
rðE
p
E
s
Þ; ð2Þ
where u
z
is the longitudinal acoustic displacement, λ and
μ are the lame constants, "
0
is the permittivity in vacuum,
γ
e
is the electrostrictive constant, and ρ is the material
density.
In order to facilitate the calculation, we make an approxi-
mation to neglect the effect of the transverse profiles in
Eq. (2). Under such an approximation, the acoustic wave can
be expressed as
u
z
ðz; tÞ¼
"
0
e
2h
0
A
p
A
s
exp½iðqz tÞ þ c:c:; ð3Þ
where in SBS, the frequency of the driven acoustic wave
is given by ¼ !
p
!
s
, the acoustic wave vector can be
expressed as q ¼ k
p
þ k
s
2k
p
, and the coefficient in the
denominator is defined as h
0
¼ð=qÞ
2
ð þ 2 Þ=.In
addition, the Helmholtz equation is expressed as
r
2
Eðr; ; z; tÞ
n
2
ðrÞ
c
2
@
2
@t
2
Eðr; ; z; tÞ
¼
0
@
2
@t
2
P
NL
ðr; ; z; tÞ; ð4Þ
where n(r) is the refractive index, which is equal to a constant
in SI fibers but a variable of radius in GI fibers, μ
0
is
the magnetic permeability in vacuum, and the nonlinear
polarization density can be expressed as P
NL
ðr; ; z; tÞ¼
"
0
e
½r uðr; ; z; tÞEðr; ; z; tÞ.
Applied Physics Express 8, 052501 (2015)
http://dx.doi.org/10.7567/APEX.8.052501
052501-1
©
2015 The Japan Society of Applied Physics