2 QUALITATIVE AND
QUANTITATIVE
CONSIDERATIONS
As we mentioned above, during their propagation in
a FOWG, light pulses are subject to a perturbed en-
vironment. Perturbations may have two main origins:
They may be induced by the waveguide, i.e., be con-
substantial to the system. Other perturbations can be
external to the system, i.e., related to the action of
an agent external to the system. For sake of clarity
of presentation, we shall discuss separately these two
situations.
2.1 Perturbation Induced by the
Periodic Structure of the Waveguide
As is well known, the conventional optical soliton
is the result of a delicate balance between two ef-
fects that compensate exactly, namely, the self-phase
modulation and the anomalous dispersion of the fiber
(Hasegawa and Tappert, 1973; L. F. Mollenauer and
Gordon, 1980). In a FOWG, as the pulse propagates
through the system, it undergoes losses which gradu-
ally reduce the self-phase modulation. Consequently,
the balance between self-phase modulation and dis-
persion can no longer be maintained in the real sys-
tem. In this context, several alternative strategies have
been developed in order to stabilize pulse propaga-
tion in FOWGs, leading to the emergence of partic-
ularly robust pulses such as the guiding-center soli-
ton (GCS) or the dispersion-managed soliton, which
are able to propagate in a highly stable manner over
several thousands of kilometers. All of those strate-
gies have as common general feature, the fact of be-
ing based on periodically structured waveguides, i.e.,
which are made up of the repetition of the same ba-
sic structure called amplification span. Within each
span, the pulse executes a relatively fast internal dy-
namics, before going back (at the end of the span)
to a profile identical or close to the one it was hav-
ing at the beginning of the span. Most of the periodi-
cally structured waveguides, admit stationary pulses.
Here, the terminology of stationary pulse (SP) refers
to a pulse that propagates while executing, in a pe-
riodic manner, a deformation of profile whose peri-
odicity corresponds exactly to that of the waveguide
(say, Z
A
). If one disregards the internal dynamics
within each amplification span (i.e., the fast dynamic
induced by the combined actions of the exponential
attenuation of the pulse peak power, the dispersion,
and the amplification process), and if one considers
only the pulse profile at the end of each span (slow
dynamic), then the SP will display a behavior identi-
cal to that of a conventional soliton (i.e., a propagation
without change of profile). However, at this juncture,
we wish to point out a crucial point. Indeed, there is
a major qualitative difference between the ideal sys-
tem, where the (conventional) soliton propagates with
a perfectly smooth profile (of Sech shape), and the pe-
riodically structured waveguides, where the profile of
the SP is never smooth. In fact, the internal dynamic
of the pulse, which is closely related to the structure
of the amplification span, significantly alters the pro-
file of the SP, which becomes rough. To illustrate the
perturbation induced by the waveguide on the pro-
file of SPs, we will use a waveguide corresponding
to a GCS (guiding-center soliton) (Hasegawa and Ko-
dama, 1990). In this waveguide the amplification span
consists of only one section of fiber with anomalous
dispersion, followed by an amplifier. The choice of
this waveguide is dictated only by a concern of sim-
plicity, and does not restrict in any way the generality
of the tools that we will develop afterward. Note that
the technique of GCS is based on the compensation
of dispersion by the self-phase modulation, but not in
an instantaneous way. Indeed, as the self-phase mod-
ulation decreases gradually as the pulse propagates
along the amplification span, the pulse is initially en-
dowed with a peak power P
0
larger than that of the
conventional soliton in the same waveguide (say, P
m
),
so that in the beginning of the span, the nonlinearity is
stronger than the dispersion, and that, afterwards, the
situation gets reversed within the span. So, the power
P
0
is chosen so that the balance between the two ef-
fects is thus globally reached at the end of each ampli-
fication span. In this waveguide, the pulse dynamics
may be described by the generalized generalized non-
linear equation (NLSE) which follows
A
z
= −i
β
2
(z)
2
A
tt
+ iγ
|
A
|
2
A −
α
2
A
+
√
G −1
×
N
∑
n=1
δ(z −nZ
A
)A, (1)
where A refers to the electric field of the pulse, β
2
,
γ and α designate the dispersion, non-linearity, and
linear-attenuation coefficients, respectively. The pa-
rameter G = exp(αZ
A
) refers to the gain of each
amplifier. Here, it is worth noting that in the lit-
erature, there exists no exact analytical expression
for the profiles of SPs in FOWGs. Consequently,
the profiles of SPs in real waveguides are accessi-
ble only numerically, by means of specialized tech-
niques. By following the procedure of Ref.(J. H.
B. Nijhof and knox, 1997), we have obtained the re-
sults depicted in figures 1, for the following typical
parameters: α = 0.24dB/km, D = 1ps/nm/km, β
2
=
SpontaneousEmissionofRadiationbySolitonsinFiber-opticWaveguides
75