2. Simulation of Dynamics of a Chirped Pulse
in a Graded-Index Waveguide
[5] Simulation of the process of propagation of a short
weakly nonlinear pulse in a graded-index waveguide with a
weak longitudinal irregularity was performed similarly to that
done by
Bisyarin and Molotkov [2002].
In dimensionless variables
r (radial coordinate),
s (stretched longitudinal coordinate) and
t (time),
the model equation acquires the form
 | (2) |
The field amplitude
f is assumed to be the magnitude of the
order of
d; in this case the pulse duration is the magnitude of
the order of
1/d. The high-frequency carrier and envelope of
the pulse evolve with different phases. For this reason, the
envelope phase is given by a separate relation
in which the
Q(s) function should be defined in the process of
the problem solution. It follows from (1) that the term
describing the quadratic phase modulation is of the order of
d3. The solution of (2) is finally sought in the form
 | (3) |
Note that in a waveguide with a longitudinal irregularity, the
modulation coefficient
m(s) 
1 depends on the longitudinal
coordinate. The complex amplitude
F is expanded in a power
series of the small parameter
 | (4) |
If the refractive index is independent of angle
j, the
expansion terms of the zero and first orders do not depend on
this coordinate as well. Nevertheless, if a spatial bending of
the waveguide channel axis is taken into account, the
dependence on the azimuthal angle will appear in expansion
(4) beginning from the term of the order of
d2 even in the
case of an azimuthally symmetric distribution of the
refractive index in the waveguide cross section. The complex
amplitude of the wave process is concentrated in the vicinity
of the channel axis, and therefore all
Fj satisfy the boundary
condition
Fj 
0 at
r 
.
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