Incoherent self-similarities of the coupled amplified nonlinear Schrödinger equations
Guoqing Chang, Herbert G. Winful, Almantas Galvanauskas, and Theodore B. Norris
FOCUS Center and Center for Ultrafast Optical Science, University of Michigan, Ann Arbor, Michigan 48109-2099, USA
Received 13 September 2005; revised manuscript received 5 December 2005; published 25 January 2006
Self-similar propagation in a system of coupled amplified nonlinear Schrödinger equations is studied. We
find that each individual amplified nonlinear Schrödinger equation can sustain a component similariton with a
quadratic phase, which is the asymptotic self-similar solution of the corresponding equation. Under a width-
matching condition, the incoherent summation of all the component similaritons leads to another similariton
with parabolic profile. Numerical simulations show that this incoherent parabolic similariton maintains all the
characteristics of its coherent counterpart.
DOI: 10.1103/PhysRevE.73.016616 PACS numbers: 42.81.Dp, 42.25.Fx, 42.65.Jx
Recently the discovery of incoherent solitons 1,2 has
revived the investigation of coupled nonlinear Schrödinger-
like equations 3–12. Among them, coupled nonlinear
Schrödinger equations CNLSEs have been extensively
studied since the self-trapping of light beams in a slow Kerr-
type nonlinear medium is well characterized by those equa-
tions 6–12. As a relatively established subject, CNLSEs
describe many interesting and important physical phenom-
ena, which include multicomponent Bose-Einstein conden-
sates BECs13, temporal incoherent solitons 14,15, and
nonlinear interactions of optical waves with different polar-
izations or with different wavelength 16.
The incoherent soliton solution of CNLSEs can be re-
garded as the extension of the coherent soliton governed by a
single nonlinear Schrödinger equation NLSE. It has been
well known for decades that the anomalous dispersion and
Kerr nonlinearity of an optical fiber accommodate temporal
solitons. Recently, another temporal self-similar propagation
solution, the parabolic similariton, has been observed in a
fiber amplifier with normal dispersion and has been inten-
sively studied both theoretically and experimentally 17–23.
As the asymptotic self-similar solution of the amplified
NLSE, the similariton is formed and maintained due to the
interplay of normal dispersion, Kerr nonlinearity, and gain,
which transforms an arbitrary input pulse into an amplified,
linearly chirped pulse with a parabolic temporal profile. The
temporal parabolic similariton has found important applica-
tions in high-power fiber amplifiers and lasers 21–23.
Based on the strong analogies between the diffraction of
paraxial optical beams and the dispersive propagation of
quasi-monochromatic pulses in dielectric media, the spatial
parabolic similariton, i.e., parabolic beam, has been proposed
and demonstrated theoretically for the amplified NLSE 24;
interestingly and in contrast to solitons, self-similar propa-
gation is possible in 2+1 dimensions. Besides nonlinear
optics, such a parabolic similariton has also been predicted in
the growth of BECs 25. Since the concept of solitons has
now been extended to include incoherent solitons, it is of
interest to ask whether an incoherent parabolic similariton
IPS can exist in amplified CNLSEs. In this paper, we show
theoretically that such an excitation indeed exists.
The N-component amplified CNLSEs incorporating a lin-
ear gain term g have the form
j
+ ia
2
j
2
= i
m=1
N
|
m
|
2
j
+
g
2
j
. 1
In nonlinear optics, Eq. 1 describes the propagation of
pulses through a fiber amplifier or of a cw incoherent
paraxial beam in a dielectric planar waveguide amplifier with
a Kerr nonlinearity, and
j
is the electric field envelope. The
first term appearing on the right side of the Eq. 1 denotes
the incoherent coupling among N components due to the
intrinsic nonlinearity. The second-order derivative term ac-
counts for group velocity dispersion GVD for the temporal
pulse or geometrical diffraction for the spatial beam. In the
context of the growth of multicomponent BECs,
j
repre-
sents the wave-function description of the condensate so that
|
j
|
2
is the particle number density. The second term on the
left side of Eq. 1 corresponds to the kinetic energy contri-
bution. We start by considering a more general two-
component temporal amplified CNLSE of the form
s
z
+ i
2s
2
2
s
T
2
= i
s
|
s
|
2
+ |
p
|
2
s
+
g
s
2
s
, 2
p
z
+ i
2p
2
2
p
T
2
= i
p
|
p
|
2
+ |
s
|
2
p
+
g
p
2
p
, 3
where
s
and
p
are the slowly varying envelopes associ-
ated with two mutually incoherent pulses in the following,
we call them s-pulse and p-pulse to make a distinction. Here
incoherent means that there is no interference between such
two pulses, and therefore, the total power of the two pulses is
a direct sum of the powers of both pulses, i.e., P
tot
= |
s
|
2
+ |
p
|
2
. In contrast, this number would be P
tot
= |
s
+
p
|
2
for
two coherent pulses.
2 j
,
j
, and g
j
denote group velocity
dispersion coefficient, the nonlinear parameter, and the con-
stant gain, respectively, for the s-pulse j = s and p-pulse
j = p. Multiplying Eq. 2 by
s
*
, subtracting from the com-
plex conjugate of the same equation, and integrating both
sides of this expression yields
-
|
s
z, T|
2
dT = expg
s
z
-
|
s
0, T|
2
dT .
We can conclude that the total power in each pulse increases
exponentially with propagation distance. There is no power
PHYSICAL REVIEW E 73, 016616 2006
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