January 10, 2010 / Vol. 8, No. 1 / CHINESE OPTICS LETTERS 89
Analytical self-similar solutions of the Ginzburg-Landau
equation with three-order dispersion effect
Jie Feng (¾¾¾ ###)
1∗
, Wencheng Xu (MMM©©©¤¤¤)
2
, Weici Liu (444aaa)
2
,
and Songhao Liu (444wwwÍÍÍ)
2
1
Scho ol of Physics and Telecommunication Engineering, South China Normal University,
Guangzhou 510006, China
2
Lab oratory of Photonic Information Technology, School of Information and Optoelectronic Science and Engineering,
South China Normal University, Guangzhou 510006, China
∗
E-mail: fengjie@shnu.edu.cn
Received January 9, 2009
Based on the technique of the symmetry reduction, we find the asymptotic self-similarity analytical reso-
lutions from the constant coefficient Ginzburg-Landau equation considering both influences of the third-
order dispersion and gain dispersion on the evolution of pulses. We have obtained the self-similar pulse
amplitude function, phase function, strict linear chirp function, and the effective temporal pulse width.
Numerical simulations show qualitative agreement with these theoretical results.
OCIS co des: 190.4370, 060.2310.
doi: 10.3788/COL20100801.0089.
The latest progress of linear-chirp self-similar pulses of
parabolic asymptotic evolution has been obtained in an-
alytic theories, numerical simulations, and experimen-
tal results at rare-earth ions doped fibers amplifiers and
lasers for a decade. These results show that the self-
similar pulses have three significant properties
[1−3]
and
may be widely used in fibers communications, nonlin-
ear optics, ultra-fast optics, transient optics, and laser
process
[4−8]
. Because of self-similar features of linear
chirp and robust resisting pulse broken, the evolution
pulse energy is gradually increasing with pulse transmis-
sion. This can result in the nonlinearity and high-order
dispersion effects. Especially when the second group ve-
locity dispersion (GVD) is small in dispersion manage-
ment optic fiber amplifiers, the third-order dispersion
(TOD, β
3
) will play an importance role. At present,
the experimental studies of self-similar parabolic asymp-
totic pulse evolution have transformed to measure the
self-similar amplitude shape, strict linear chirp feature,
and effective temporal width. Theoretical analyses have
focused on nonlinear Schr¨odinger equation (NLSE)
[8−14]
only considering the infinite frequency bandwidth, and
on constant and varying coefficients Ginzburg-Landau
equation (G-LE) considering the realistic influence of
doped elements effect, i.e., effect of gain media finite fre-
quency bandwidth
[15,16]
. There are only numerical sim-
ulations results for high-order dispersion in NLSE and
G-LE
[17−22]
but no report on study of the analytical res-
olutions of self-similar pulses with high-order dispersion
in G-LE to date.
In this letter, we will further theoretically investigate
self-similar pulse evolution features with the third-order
dispersion of G-LE under affection of high-order disper-
sion and gain media finite frequency bandwidth in doped
fibers. Especially, we will discuss the dynamic mecha-
nism of self-similar pulse of analytical results, and then
compare the analytical results with the simulations in
high-order dispersion of G-LE.
Suppose that the incident pulse has larger energy (but
smaller than the gain saturation energy) in the rare-earth
ions doped gained fibers, the evolution of the pulse can
be described by general G-LE
[1,3,22]
expressed as
∂Ψ
∂z
= iγ |Ψ|
2
Ψ − i
β
2
2
∂
2
Ψ
∂T
2
+
β
3
6
∂
3
Ψ
∂T
3
+
g(T )
2
Ψ +
g(T )
2Ω
2
∂
2
Ψ
∂T
2
, (1)
where Ψ = Ψ (z, T ) is a slowly varying amplitude of
the pulse envelope in a co-moving frame, T = t−β
1
z is
retarded time, z is the propagation distance. β
2
(>0),
β
3
, γ, and g(T ) are GVD, TOD, nonlinearity parameter,
and gain coefficient, respectively.
g( T )
2Ω
2
∂
2
Ψ
∂T
2
is called gain
dispersion factor which originates from the frequency de-
pendence of the gain, where Ω is the bandwidth of the
doped gained fibers.
We suppose Eq. (1) has the self-similar probe solution
of
[1−3]
½
Ψ(z, T ) = A(z, T ) exp iΦ(z, T )
Φ(z, T ) = B(z) + CT
2
, (2)
where A(z, T ), Φ(z, T ), B(z), and C are amplitude func-
tion, function phase, phase offset function and chirp pa-
rameter, respectively. Substituting Eq. (2) into Eq. (1)
and comparing the real with the image part, we can get
∂A
∂z
= β
2
∂A
∂T
∂Φ
∂T
+
β
2
A
2
∂
2
Φ
∂T
2
+
β
3
6
∂
3
A
∂T
3
+
β
3
6
∂A
∂T
(
∂Φ
∂T
)
2
−
β
3
2
A
∂
2
Φ
∂T
2
∂Φ
∂T
+
g( T )A
2
+
g( T )
2Ω
2
∂
2
A
∂T
2
−
g( T )A
2Ω
2
(
∂Φ
∂T
)
2
,
A
∂Φ
∂z
= −
β
2
2
∂
2
A
∂T
2
+
β
2
A
2
(
∂Φ
∂T
)
2
+
β
3
2
∂
2
A
∂T
2
∂Φ
∂T
+
β
3
2
∂A
∂T
∂
2
Φ
∂T
2
−
β
3
A
6
(
∂Φ
∂T
)
3
+ γA
3
+
g (T )
Ω
2
∂A
∂T
∂Φ
∂T
+
g( T )A
2Ω
2
∂
2
Φ
∂T
2
. (3)
In order to separate variation further, we write the am-
1671-7694/2010/010089-04
c
° 2010 Chinese Optics Letters