408 CHAPTER 9. SOLITON SYSTEMS
Figure 9.2: Evolution of a Gaussian pulse withN=1 over the rangeξ=0–10. The pulse
evolves toward the fundamental soliton by changing its shape, width, and peak power.
a certain peak power corresponding toN=1 in Eq. (9.1.4), it can be created even
when the pulse shape and the peak power deviate from the ideal conditions. Figure 9.2
shows the numerically simulated evolution of a Gaussian input pulse for whichN= 1
butu( 0 ,τ)=exp(−τ^2 / 2 ). As seen there, the pulse adjusts its shape and width in an
attempt to become a fundamental soliton and attains a “sech” profile forξ1. A
similar behavior is observed whenNdeviates from 1. It turns out that theNth-order
soliton can be formed when the input value ofNis in the rangeN−^12 toN+^12 [15].
In particular, the fundamental soliton can be excited for values ofNin the range 0.5
to 1.5. Figure 9.3 shows the pulse evolution forN= 1 .2 over the rangeξ=0–10 by
solving the NLS equation numerically with the initial conditionu( 0 ,τ)= 1 .2 sech(τ).
The pulse width and the peak power oscillate initially but eventually become constant
after the input pulse has adjusted itself to satisfy the conditionN=1 in Eq. (9.1.4).
It may seem mysterious that an optical fiber can force any input pulse to evolve
toward a soliton. A simple way to understand this behavior is to think of optical solitons
as the temporal modes of a nonlinear waveguide. Higher intensities in the pulse center
create a temporal waveguide by increasing the refractive index only in the central part
of the pulse. Such a waveguide supports temporal modes just as the core-cladding
index difference led to spatial modes in Section 2.2. When an input pulse does not
match a temporal mode precisely but is close to it, most of the pulse energy can still
be coupled into that temporal mode. The rest of the energy spreads in the form of
dispersive waves. It will be seen later that such dispersive waves affect the system
performance and should be minimized by matching the input conditions as close to
the ideal requirements as possible. When solitons adapt to perturbations adiabatically,
perturbation theory developed specifically for solitons can be used to study how the
soliton amplitude, width, frequency, speed, and phase evolve along the fiber.