Nonlinear Fiber Optics - 4 ed. Agrawal

5.2. Fiber Solitons 133
Physically, the four parameters η, δ, τ s , and φ s represent amplitude, frequency,
position, and phase of the soliton, respectively. The phase φ s can be dropped from the
discussion because a constant absolute phase has no physical significance. It will become
relevant later when nonlinear interaction between a pair of solitons is considered.
The parameter τ s can also be dropped because it denotes the position of the soliton
peak: If the origin of time is chosen such that the peak occurs at τ = 0atξ = 0, one
can set τ s = 0. It is clear from the phase factor in Eq. (5.2.13) that the parameter δ represents
a frequency shift of the soliton from the carrier frequency ω 0 . Using the carrier
part, exp(−iω 0 t), the new frequency becomes ω ′ 0 = ω 0 + δ/T 0 . Note that a frequency
shift also changes the soliton speed from its original value v g . This can be seen more
clearly by using τ =(t − β 1 z)/T 0 in Eq. (5.2.13) and writing it as
|u(ξ ,τ)| = η sech[η(t − β ′ 1z)/T 0 ], (5.2.14)
where β ′ 1 = β 1 + δ|β 2 |/T 0 . As expected on physical grounds, the change in group
velocity (v g = 1/β 1 ) is a consequence of fiber dispersion.
The frequency shift δ can also be removed from Eq. (5.2.13) by choosing the carrier
frequency appropriately. Fundamental solitons then form a single-parameter family
described by
u(ξ ,τ)=η sech(ητ)exp(iη 2 ξ /2). (5.2.15)
The parameter η determines not only the soliton amplitude but also its width. In real
units, the soliton width changes with η as T 0 /η, i.e., it scales inversely with the soliton
amplitude. This inverse relationship between the amplitude and the width of a soliton
is the most crucial property of solitons. Its relevance will become clear later. The
canonical form of the fundamental soliton is obtained by choosing u(0,0)=1 so that
η = 1. With this choice, Eq. (5.2.15) becomes
u(ξ ,τ)=sech(τ)exp(iξ /2). (5.2.16)
One can verify by direct substitution in Eq. (5.2.5) that this solution is indeed a solution
of the NLS equation.
The solution in Eq. (5.2.16) can also be obtained by solving the NLS equation directly,
without using the inverse scattering method. The approach consists of assuming
that a shape-preserving solution of the NLS equation exists and has the form
u(ξ ,τ)=V (τ)exp[iφ(ξ ,τ)], (5.2.17)
where V is independent of ξ for Eq. (5.2.17) to represent a fundamental soliton that
maintains its shape during propagation. The phase φ can depend on both ξ and τ. If
Eq. (5.2.17) is substituted in Eq. (5.2.5) and the real and imaginary parts are separated,
one obtains two equations for V and φ. The phase equation shows that φ should be
of the form φ(ξ ,τ) =Kξ − δτ, where K and δ are constants. Choosing δ = 0 (no
frequency shift), V (τ) is found to satisfy
d 2 V
dτ 2 = 2V (K −V 2 ). (5.2.18)