Nonlinear Fiber Optics - 4 ed. Agrawal

160 Chapter 5. Optical Solitons
turbative solution is not appropriate. A shape-preserving, solitary-wave solution of Eq.
(5.5.11) can be found by assuming u(ξ ,τ)=V (τ)exp(iKξ ) and solving the resulting
ordinary differential equation for V (τ). This solution is given by [178]
u(ξ ,τ)=3b 2 sech 2 (bτ)exp(8ib 2 ξ /5), (5.5.12)
where b =(40δ 4 ) −1/2 . Note the sech 2 -type form of the pulse amplitude rather than
the usual “sech” form required for standard bright solitons. It should be stressed that
both the amplitude and the width of the soliton are determined uniquely by the fiber
parameters. Such fixed-parameter solitons are sometimes called autosolitons.
5.5.3 Self-Steepening
The phenomenon of self-steepening has already been discussed in Section 4.4.1. It introduces
several new features for solitons forming in the anomalous-dispersion regime
[179]–[183]. The effects of self-steepening appear in Eqs. (5.5.3)–(5.5.6) through the
terms containing ω 0 . The most important feature is that self-steepening can produce
spectral and temporal shifts of the soliton even when T R = 0. In fact, we can integrate
Eq. (5.5.6) to obtain the spectral shift in the form
Ω p (z)= γE ∫ z
0 C p (z)
3ω 0 Tp 3 (z) e−αz dz, (5.5.13)
0
where E 0 = 2P 0 T 0 is the input pulse energy. As soon as the pulse becomes chirped, its
spectrum shifts because of self-steepening. If the chirp is negligible, spectral shift is
relatively small. Even when Ω p = 0, self-steepening produces a temporal shift because
of the last term in Eq. (5.5.5). If we assume that soliton maintains its width to the first
order, the temporal shift of the soliton peak for a fiber of length L given by
q p (L)=γP 0 L eff /ω 0 = φ max /ω 0 , (5.5.14)
where L eff is the effective length and φ max is the maximum SPM-induced phase shift
introduced in Section 4.1.1. Noting that ω 0 = 2π/T opt , where T opt is the optical period,
the shift is relatively small even when φ max exceeds 10π. However, when Ω p ≠ 0, the
temporal shift is enhanced considerably.
The preceding analysis is valid for relatively small values of the parameter s. When
pulses are so short that s exceeds 0.1, one must use a numerical approach. To isolate
the effects of self-steepening governed by the parameter s, it is useful to set δ 3 = 0 and
τ R = 0 in Eq. (5.5.8). Pulse evolution inside fibers is then governed by
i ∂u
∂ξ + 1 ∂ 2 u
2 ∂τ 2 + |u|2 u + is ∂
∂τ (|u|2 u)=0. (5.5.15)
The self-steepening-induced temporal shift is shown in Figure 5.18 where pulse
shapes at ξ = 0, 5, and 10 are plotted for s = 0.2 andN = 1 by solving Eq. (5.5.15)
numerically with the input u(0,τ)=sech(τ). As the peak moves slower than the wings
for s ≠ 0, it is delayed and appears shifted toward the trailing side. Although the pulse
broadens slightly with propagation (by about 20% at ξ = 10), it nonetheless maintains