Nonlinear Fiber Optics - 4 ed. Agrawal

146 Chapter 5. Optical Solitons
5.4 Perturbation of Solitons
Fiber-optic communication systems operating at bit rates of 10 Gb/s or more are generally
limited by the GVD that tends to disperse optical pulses outside their assigned
bit slot. Fundamental solitons are useful for such systems because they can maintain
their width over long distances by balancing the effects of GVD and SPM, both of
which are detrimental to system performance when solitons are not used. The use of
solitons for optical communications was proposed as early as 1973 [76], and their use
had reached the commercial stage by 2000 [77]. This success was possible only after
the effects of fiber losses on solitons were understood and techniques for compensating
them were developed [129]–[136]. The advent of erbium-doped fiber amplifiers fueled
the development of soliton-based systems. However, with their use came the limitations
imposed by the amplifier noise. In this section, we first discuss the method used
commonly to analyze the effect of small perturbations on solitons and then apply it
to study the impact of fiber losses, periodic amplification, amplifier noise, and soliton
interaction.
5.4.1 Perturbation Methods
Consider the perturbed NLS equation written as
i ∂u
∂ξ + 1 ∂ 2 u
2 ∂τ 2 + |u|2 u = iε(u), (5.4.1)
where ε(u) is a small perturbation that can depend on u, u ∗ , and their derivatives. In
the absence of perturbation (ε = 0), the soliton solution of the NLS equation is known
and is given by Eq. (5.2.13). The question then becomes what happens to the soliton
when ε ≠ 0. Several perturbation techniques have been developed for answering this
question [137]–[144]. They all assume that the functional form of the soliton remains
intact in the presence of a small perturbation but the four soliton parameters change
with ξ as the soliton propagates down the fiber. Thus, the solution of the perturbed
NLS equation can be written as
u(ξ ,τ)=η(ξ )sech[η(ξ )(τ − q(ξ ))]exp[iφ(ξ ) − iδ(ξ )τ]. (5.4.2)
The ξ dependence of η,δ,q, and φ is yet to be determined.
The perturbation techniques developed for solitons include the adiabatic perturbation
method, the perturbed inverse scattering method, the Lie-transform method, and
the variational method [71]. All of them attempt to obtain a set of four ordinary differential
equations for the four soliton parameters. As an example, consider the variational
method discussed in Section 4.3.2; it was applied to solitons as early as 1979 [145]. In
this approach, Eq. (5.4.1) is obtained from the Euler–Lagrange equation using the Lagrangian
density [146]
L d = i 2 (uu∗ ξ − u∗ u ξ ) − 1 2 (|u|4 −|u τ | 2 )+i(εu ∗ − ε ∗ u). (5.4.3)