Nonlinear Fiber Optics - 4 ed. Agrawal

238 Chapter 7. Cross-Phase Modulation
provided frequency shifts are such that Ω j =(β 2 j v gj ) −1 and λ j = K j − δ 2 j /(2d j).
For the central j = 0 component, d 0 = γ 0 = 1. In the absence of XPM terms, this
component has the standard soliton solution U 0 (x)=sech(τ). We assume that, in the
presence of XPM, all components have the same “sech” shape but different amplitudes,
i.e., U n (τ) =a n sech(τ). Substituting this solution in Eq. (7.3.24), the amplitudes a n
are found to satisfy the following set of algebraic equations:
a 2 0 + σ ∑ γ n a 2 n = 1,
n≠0
γ n a 2 n + σ ∑ γ m a 2 m = d n . (7.3.25)
m≠n
As long as parameters d n and γ n are close to 1 for all components, this solution describes
a vector soliton with N components of nearly equal intensity. In the degenerate
case, d n = γ n = 1, the amplitudes can be found analytically [45] in the form
U n =[1 + σ(N − 1)] −1/2 , where N is the total number of components. The stability of
any multicomponent vector soliton is not guaranteed and should be studied carefully.
We refer to Ref. [52] for a detailed discussion of the stability issue.
7.4 Spectral and Temporal Effects
This section considers the spectral and temporal changes occurring as a result of XPM
interaction between two copropagating pulses with nonoverlapping spectra [53]–[59].
For simplicity, the polarization effects are ignored assuming that the input beams preserve
their polarization during propagation. Equations (7.1.15) and (7.1.16) then govern
evolution of two pulses along the fiber length and include the effects of groupvelocity
mismatch, GVD, SPM, and XPM. If fiber losses are neglected for simplicity,
these equations become
where
∂A 1
∂z + iβ 21 ∂ 2 A 1
2 ∂T 2 = iγ 1(|A 1 | 2 + 2|A 2 | 2 )A 1 , (7.4.1)
∂A 2
∂z + d ∂A 2
∂T + iβ 22 ∂ 2 A 2
2 ∂T 2 = iγ 2(|A 2 | 2 + 2|A 1 | 2 )A 2 , (7.4.2)
T = t − z
v g1
,
d = v g1 − v g2
v g1 v g2
. (7.4.3)
Time T is measured in a reference frame moving with the pulse traveling at speed v g1 .
The parameter d is a measure of group-velocity mismatch between the two pulses.
In general, two pulses can have different widths. Using the width T 0 of the first
pulse at the wavelength λ 1 as a reference, we introduce the walk-off length L W and the
dispersion length L D as
L W = T 0 /|d|, L D = T 2
0 /|β 21 |. (7.4.4)
Depending on the relative magnitudes of L W , L D , and the fiber length L, the two pulses
can evolve very differently. If L is small compared to both L W and L D , the dispersive
effects do not play a significant role and can be neglected. For example, this can occur