Nonlinear Fiber Optics - 4 ed. Agrawal

12.4. Temporal and Spectral Evolution 477
order is governed by Eq. (5.2.3), or by N =(γP 0 L D ) 1/2 , where L D = T 2
0 /|β 2| is the dispersion
length. As discussed in Section 5.5, such solitons are perturbed considerably
by higher-order effects, such as third-order dispersion and intrapulse Raman scattering,
that lead to their fission into much narrower fundamental solitons. It turns out that the
phenomenon of soliton fission plays a critical role in the formation of a supercontinuum
in highly nonlinear fibers [28].
As discussed in Section 12.1.1, the fission of a higher-order soliton produces multiple
fundamental solitons whose widths and peak powers follow Eq. (12.1.2). Almost
all of these solitons are shorter than the original input pulse, the shortest one being
narrower by a factor of 2N − 1. For femtosecond input pulses, the individual solitons
have a relatively wide spectrum (∼10 THz) and are thus affected by intrapulse Raman
scattering that shifts the soliton spectrum toward longer and longer wavelengths
with further propagation inside the fiber. As a result, many new spectral components
are added on the long-wavelength side of the original pulse spectrum. This process is
different from cascaded SRS that creates multiple Stokes bands for picosecond pulses.
The important remaining question is what process creates spectral components on
the short-wavelength side of the pulse spectrum. This is where the dispersive properties
of the fiber play a critical role. As discussed in Section 12.1.2, ultrashort solitons,
created through the fission process and perturbed by third- and higher-order dispersion,
emit the NSR in the form of dispersive waves whose wavelengths fall on the shortwavelength
side in the normal dispersion region of the fiber. Remarkably, as seen in
Figure 8.22, such NSR was observed in a 1987 experiment [3], although it was not
identified as such at that time.
12.4 Temporal and Spectral Evolution
In this section we consider in more detail the temporal and spectral evolution of femtosecond
pulses and focus on the physical phenomena that create the supercontinuum
at the fiber output. A numerical approach is necessary for this purpose. We discuss
first the model based on the generalized NLS equation and then focus on the process
of soliton fission and the subsequent generation of NSR.
12.4.1 Numerical Modeling of Supercontinuum
The supercontinuum generation process can be studied by solving the generalized NLS
equation of Section 2.3.2. As it is important to include the dispersive effects and intrapulse
Raman scattering as accurately as possible, one should employ Eq. (2.3.36) for
modeling supercontinuum generation, after generalizing it further by adding higherorder
dispersion terms. The resulting equation can be written as
∂A
∂z + 1 2
= i
(
∂
α + iα 1
∂t
(
γ + iγ 1
∂
∂t
)
A +
M
∑
m=2
)(
A(z,t)
i m−1 β m
m!
∫ ∞
0
∂ m A
∂t m
R(t ′ )|A(z,t −t ′ )| 2 dt ′ ), (12.4.1)