Nonlinear Fiber Optics - 4 ed. Agrawal

156 Chapter 5. Optical Solitons
pair with an initial separation q 0 = 3.5 for several values of parameters r and θ. In the
case of equal-amplitude solitons (r = 1), the two solitons attract each other in the inphase
case (θ = 0) and collide periodically along the fiber length, just as predicted by
perturbation theory. For θ = π/4, the solitons separate from each other after an initial
attraction stage in agreement with the results shown in Figure 5.15. For θ = π/2, the
solitons repel each other even more strongly, and their spacing increases with distance
monotonically. The last case shows the effect of slightly different soliton amplitudes
by choosing r = 1.1. Two in-phase solitons oscillate periodically but never collide or
move far away from each other.
The periodic collapse of neighboring solitons is undesirable from a practical standpoint.
One way to avoid the collapse is to increase soliton separation such that L col ≫
L T , where L T is the transmission distance. Because L col ≈ 3000z 0 for q 0 = 8, and
z 0 ∼ 100 km typically, a value of q 0 = 8 is large enough for any communication system.
Several schemes can be used to reduce the soliton separation further without inducing
the collapse. The interaction between two solitons is quite sensitive to their relative
phase θ and the relative amplitude r. If the two solitons have the same phase (θ = 0)
but different amplitudes, the interaction is still periodic but without collapse [162].
Even for r = 1.1, the separation does not change by more than 10% during each period
if q 0 > 4. Soliton interaction can also be modified by other factors such as higher-order
effects [164], bandwidth-limited amplification [165], and timing jitter [166]. Several
higher-order effects are discussed in the following section.
5.5 Higher-Order Effects
The properties of optical solitons considered so far are based on the NLS equation
(5.1.1). As discussed in Section 2.3, when input pulses are so short that T 0 < 5 ps, it is
necessary to include higher-order nonlinear and dispersive effects through the generalized
NLS equation (2.3.43). If Eq. (3.1.3) is used to define the normalized amplitude
U, this equation takes the form
∂U
∂z + iβ 2 ∂ 2 U
2 ∂T 2 − β 3 ∂ 3 (
U
6 ∂T 3 = iγP 0e −αz |U| 2 U + i
ω 0
∂
∂T (|U|2 U) − T R U ∂|U|2
∂T
)
.
(5.5.1)
5.5.1 Moment Equations for Pulse Parameters
In general, Eq. (5.5.1) equation should be solved numerically. However, if we assume
that higher-order effects are weak enough that the pulse maintains its shape, even
though its parameters change, we can employ the moment method of Section 4.3.1 to
gain some physical insight. In the anomalous-GVD regime, we employ the following
form for U(z,T ):
( ) [
T − qp
(T − q p ) 2
U(z,T )=a p sech exp −iΩ p (T − q p ) − iC p
T p 2Tp
2 + iφ p
], (5.5.2)