Nonlinear Fiber Optics - 4 ed. Agrawal

7.3. XPM-Paired Solitons 237
It is well known that period of any Jacobi elliptic function becomes infinite in the
limit p = 1. In this limit, the preceding periodic solutions reduce to the the bright–dark
soliton pairs discussed earlier in this section. We should stress that the mere existence
of a periodic or solitary-wave solution does not guarantee that it can be observed experimentally.
The stability of such solutions must be studied by perturbing them and then
propagating the perturbed field over long distances. Numerical simulations show that
all periodic solutions are unstable in principle, but the distance after which instability
sets in depends on the strength of perturbation [49]. In particular, a periodic solution
can survive over tens of dispersion lengths for relatively weak perturbations.
7.3.4 Multiple Coupled NLS Equations
The concept of XPM-paired solitons can be easily generalized to multicomponent
solitons in which multiple pulses at different carrier frequencies are transmitted over
the same fiber. In practice, such a situation occurs naturally in wavelength-divisionmultiplexed
(WDM) lightwave systems [51]. In this case, in place of Eqs. (7.1.15) and
(7.1.16), one needs to solve a set of multiple coupled NLS equations of the form
(
∂A j
∂z + 1 ∂A j
+ iβ 2 j ∂ 2 A j
v gj ∂t 2 ∂t 2 = i
γ j |A j | 2 + σ ∑ γ k |A k |
)A 2 j , (7.3.21)
k≠ j
where j = −M to M and 2M +1 is the total number of components. The dimensionless
parameter σ indicates the XPM strength and has a value of 2 when all waves are linearly
polarized. These equations have periodic as well as soliton solutions for certain
combinations of parameter values [45]–[48]. In this section, we focus on the soliton
solutions with multiple components. Such solitons are often referred to as multicomponent
vector solitons [52].
We normalize Eq. (7.3.21) using the central j = 0 component as a reference and
introduce the normalized variables ξ , τ, and u j as indicated in Eq. (7.3.10), where the
dispersion length L D = T0 2/|β
20| and β 20 is taken to be negative. The set (7.3.21) can
then be written as
( ∂u j
i
∂ξ + δ j
where δ j = v −1
gj
− v −1
g0
)
(
∂u j
+ d j ∂ 2 u j
∂τ 2 ∂τ 2 +
γ j |A j | 2 + σ ∑ γ k |A k |
)A 2 j = 0, (7.3.22)
k≠ j
represents group-velocity mismatch with respect to the central
component, d j = β 2 j /β 20 , and the parameter γ j is now dimensionless as it has been
normalized with γ 0 .
The solitary-wave solution of Eq. (7.3.22) is found by seeking a solution in the
form
u j (ξ ,τ)=U j (τ)exp[i(K j ξ − Ω j τ)], (7.3.23)
where K j is the propagation constant and Ω j represents a frequency shift from the
carrier frequency. It is easy to show that U j satisfies the ordinary differential equation
(
d j d 2 U j
2 dτ 2 +
γ j |U j | 2 + σ ∑ γ k |U k |
)U 2 j = λ j U j (K j − 1 2 β 2 jΩ 2 j)U j , (7.3.24)
k≠ j