Nonlinear Fiber Optics - 4 ed. Agrawal

140 Chapter 5. Optical Solitons
makes the use of solitons feasible in practical applications. However, it is important to
realize that, when input parameters deviate substantially from their ideal values, a part
of the pulse energy is invariably shed away in the form of dispersive waves as the pulse
evolves to form a fundamental soliton [91]. Such dispersive waves are undesirable
because they not only represent an energy loss but can also affect the performance of
soliton-based communication systems. Moreover, they can interfere with the soliton
itself and modify its characteristics. In the case of an input pulse with N close to 1,
such an interference introduces modulations on the pulse spectrum that have also been
observed experimentally [92].
Starting in 1988, most of the experimental work on fiber solitons was devoted to
their applications in fiber-optic communication systems [93]–[95]. Such systems make
use of fundamental solitons for representing “1” bits in a digital bit stream. In a practical
situation, solitons can be subjected to many types of perturbations as they propagate
inside an optical fiber. Examples of perturbations include fiber losses, amplifier noise
(if amplifiers are used to compensate fiber losses), third-order dispersion, and intrapulse
Raman scattering. These effects are discussed later in this chapter.
5.3 Other Types of Solitons
The soliton solution given in Eq. (5.2.8) is not the only possible solution of the NLS
equation. Many other kinds of solitons have been discovered depending on the dispersive
and nonlinear properties of fibers. This section describes several of them, focusing
mainly on dark and bistable solitons.
5.3.1 Dark Solitons
Dark solitons correspond to the solutions of Eq. (5.2.2) with sgn(β 2 )=1 and occur
in the normal-GVD region of fibers. They were discovered in 1973 and have attracted
considerable attention since then [96]–[114]. The intensity profile associated with such
solitons exhibits a dip in a uniform background, hence the name dark soliton. Pulselike
solitons discussed in Section 5.2 are called bright to make the distinction clear.
The NLS equation describing dark solitons is obtained from Eq. (5.2.5) by changing
the sign of the second-derivative term and is given by
i ∂u
∂ξ − 1 ∂ 2 u
2 ∂τ 2 + |u|2 u = 0. (5.3.1)
Similar to the case of bright solitons, the inverse scattering method can be used [97]
to find dark-soliton solutions of Eq. (5.3.1) by imposing the boundary condition that
|u(ξ ,τ)| tends toward a constant nonzero value for large values of |τ|. Dark solitons
can also be obtained by assuming a solution of the form u(ξ ,τ)=V (τ)exp[iφ(ξ ,τ)],
and then solving the ordinary differential equations satisfied by V and φ. The main
difference compared with the case of bright solitons is that V (τ) becomes a constant
(rather than being zero) as |τ|→∞. The general solution can be written as [114]
√
u(ξ ,τ)=η[Btanh(ζ ) − i 1 − B 2 ]exp(iη 2 ξ ), (5.3.2)