534 CHINESE OPTICS LETTERS / Vol. 5, No. 9 / September 10, 2007
Effects of group-velocity mismatch and cubic-quintic
nonlinearity on cross-phase mo dulation instability in
optical fibers
Xianqiong Zhong (
) and Anping Xiang (
ËËË
)
Department of Optoelectronics and Technology, Chengdu University of Information Tec hnology, Chengdu 610225
Received January 17, 2007
The synthetic effects of group-velocity mismatch and cubic-quintic nonlinearity on cross-phase modulation
induced modulation instability in loss single-mode optical fibers have been numerically investigated. The
results show that the quintic nonlinearity plays a role similar to the case of neglecting the group-velocity
mismatch in modifying the modulation instability, namely, the positive and negative quintic nonlinearities
can still enhance and weaken the modulation instability, respectively. The group-velocity mismatch can
considerably change the gain spectrum of modulation instability in terms of its shape, peak value, and
p osition. In the normal dispersion regime, with the increase of the group-velocity mismatch parameter,
the gain spectrum widens and then narrows, shifts to higher frequencies, and the peak value gets higher
b efore approaching a saturable value. In the abnormal dispersion regime, two separated spectra may occur
when the group-velocity mismatch is taken into account. With the increase of the group-velocity mismatch
parameter, the peak value of the gain sp ectrum gets higher and shorter before tending to a saturable value
for the first and second spectral regimes, respectively.
OCIS codes: 190.4370, 190.3100, 190.5940, 190.3270.
It is well known that the interplay between the nonlin-
earity (self-phase modulation or cross-phase modulation)
and dispersion effects inside optical fibers can result in a
breakup of the continuous wave (CW) or quasi-CW op-
tical wave into a train of ultra-short pulses. Since 1968,
this so-called modulation instability has been extensively
studied in various cases
[1−20]
,foritcanbeusedinmany
fields, such as optical switching
[4]
, formation of chains
of soliton like optical pulses
[5,6]
, generation of super-
continuum spectrum
[7]
, measurement of nonlinear and
chromatic dispersion parameters of optical fibers
[8]
,etc..
On the other hand, however, previous researches have
also shown that, whether self-phase or cross-phase mod-
ulation induced modulation instability may considerably
degrade the performances of the optical fiber communi-
cation systems
[9,10]
.
Self-phase modulation
[11−13]
and cross-phase
modulation
[14−16]
induced modulation instability in op-
tical fibers have both been extensively studied in the
case of cubic nonlinearity of the refractive index. How-
ever, as previous work proposed
[21]
, as high incident
optical intensities or materials with very high nonlinear
coefficients such as semiconductor doped glass optical
fibers are considered, it is necessary to take high-order
nonlinearities into consideration. The lowest high-order
nonlinearity in optical fibers is quintic one. Thus, Tanev
et al. investigated the solitary wave propagation and
bistability behavior in optical fibers with cubic-quintic
nonlinearity
[21]
. In addition, self-phase
[18,19]
and cross-
phase modulation
[3,20]
induced modulation instability
have also been investigated in the case of cubic-quintic
nonlinearity. However, to our best knowledge, there
are no reports on the synthetic effects of group-velocity
mismatch and cubic-quintic nonlinearity on cross-phase
modulation instability in optical fibers. Agrawal in-
dicated that group-velocity mismatch can considerably
influence cross-phase modulation instability
[17]
.
When two optical pulses co-propagate in optical fibers,
the linearised coupled nonlinear Schr¨odinger equation
can be taken as
[20]
∂a
j
∂z
+
1
V
gj
∂a
j
∂t
+
i
2
β
2j
∂
2
a
j
∂t
2
= iγ
1j
P
j
exp (−α
j
z)
a
j
+ a
∗
j
+i2γ
1j
P
1
P
2
exp (−α
3−j
z)
a
3−j
+ a
∗
3−j
+i2γ
2j
P
j
exp (−α
j
z)[P
j
exp (−α
j
z )
+3P
3−j
exp(−α
3−j
z)](a
j
+ a
∗
j
)
+i6γ
2j
P
1
P
2
exp(−α
3−j
z)[P
j
exp(−α
j
z)
+P
3−j
exp(−α
3−j
z)](a
3−j
+ a
∗
3−j
), (1)
where a
j
(j =1, 2) is the perturbation amplitude, V
gj
the group velocity, β
2j
the second-order group velocity
dispersion coefficient, t the time coordinate, z the prop-
agating distance, α
j
the loss coefficient of the optical
fiber, P
j
the incident optical power, and γ
1j
and γ
2j
the
third- and fifth-order nonlinear coefficients, respectively.
Adopting the well-known linear stability analysis
[15,20]
,
the dispersion relation can be obtained as
k −
Ω
V
g1
2
− f
1
k −
Ω
V
g2
2
− f
2
= C
XPM
, (2)
where k and Ω are the wave number and angle frequency
of the modulation wave, respectively. It is obvious that
this expression is completely the same as the case of cubic
1671-7694/2007/090534-04
c
2007 Chinese Optics Letters