Nonlinear Fiber Optics - 4 ed. Agrawal

5.2. Fiber Solitons 131
and write it in the form
i ∂U
∂ξ = sgn(β 2) 1 ∂ 2 U
2 ∂τ 2 − N2 |U| 2 U, (5.2.2)
where P 0 is the peak power, T 0 is the width of the incident pulse, and the parameter N
is introduced as
N 2 = L D
= γP 0T0
2
. (5.2.3)
L NL |β 2 |
The dispersion length L D and the nonlinear length L NL are defined as in Eq. (3.1.5).
Fiber losses are neglected in this section but will be included later.
The parameter N can be eliminated from Eq. (5.2.2) by introducing
u = NU = √ γL D A. (5.2.4)
Equation (5.2.2) then takes the standard form of the NLS equation:
i ∂u
∂ξ + 1 ∂ 2 u
2 ∂τ 2 + |u|2 u = 0, (5.2.5)
where the choice sgn(β 2 )=−1 has been made to focus on the case of anomalous GVD;
the other case is considered in the next section. Note that an important scaling relation
holds for Eq. (5.2.5). If u(ξ ,τ) is a solution of this equation, then εu(ε 2 ξ ,ετ) is also
a solution, where ε is an arbitrary scaling factor. The importance of this scaling will
become clear later.
In the inverse scattering method, the scattering problem associated with Eq. (5.2.5)
is found to be [71]
i ∂v 1
∂τ + uv 2 = ζ v 1 , (5.2.6)
i ∂v 2
∂τ + u∗ v 1 = −ζ v 2 , (5.2.7)
where v 1 and v 2 are the amplitudes of the two waves scattered by the potential u(ξ ,τ).
The eigenvalue ζ plays a role similar to that played by the frequency in the standard
Fourier analysis except that ζ can take complex values when u ≠ 0. This feature can
be identified by noting that, in the absence of potential (u = 0), v 1 and v 2 vary as
exp(±iζτ).
Equations (5.2.6) and (5.2.7) apply for all values of ξ . In the inverse scattering
method, they are first solved at ξ = 0. For a given initial form of u(0,τ), Eqs. (5.2.6)
and (5.2.7) are solved to obtain the initial scattering data. The direct scattering problem
is characterized by a reflection coefficient r(ζ ) that plays a role analogous to the Fourier
coefficient. Formation of the bound states (solitons) corresponds to the poles of r(ζ ) in
the complex ζ plane. Thus, the initial scattering data consist of the reflection coefficient
r(ζ ), the complex poles ζ j , and their residues c j , where j = 1toN if N such poles exist.
Although the parameter N of Eq. (5.2.3) is not necessarily an integer, the same notation
is used for the number of poles to stress that its integer values determine the number of
poles.