Nonlinear Fiber Optics - 4 ed. Agrawal

5.4. Perturbation of Solitons 151
tons can survive in spite of such large energy variations, we include the gain provided
by lumped amplifiers in Eq. (5.4.8) by replacing Γ with a periodic function ˜Γ(ξ ) such
that ˜Γ(ξ )=Γ everywhere except at the location of amplifiers where it changes abruptly.
If we make the transformation
u(ξ ,τ)=exp
(
− 1 ∫ ξ
˜Γ(ξ )dξ
2 0
)
v(ξ ,τ) ≡ a(ξ )v(ξ ,τ), (5.4.15)
where a(ξ ) contains rapid variations and v(ξ ,τ) is a slowly varying function of ξ and
use it in Eq. (5.4.8), v(ξ ,τ) is found to satisfy
i ∂v
∂ξ + 1 ∂ 2 v
2 ∂τ 2 + a2 (ξ )|v| 2 v = 0. (5.4.16)
Note that a(ξ ) is a periodic function of ξ with a period ξ = L A /L D , where L A is the
amplifier spacing. In each period, a(ξ ) ≡ a 0 exp(−Γξ /2) decreases exponentially and
jumps to its initial value a 0 at the end of the period.
The concept of the guiding-center or path-averaged soliton [152] makes use of the
fact that a 2 (ξ ) in Eq. (5.4.16) varies rapidly in a periodic fashion. If the period ξ A ≪ 1,
solitons evolve little over a short distance as compared with the dispersion length L D .
Over a soliton period, a 2 (ξ ) varies so rapidly that its effects are averaged out, and we
can replace a 2 (ξ ) by its average value over one period. With this approximation, Eq.
(5.4.16) reduces to the standard NLS equation:
i ∂v
∂ξ + 1 ∂ 2 v
2 ∂τ 2 + 〈a2 (ξ )〉|v| 2 v = 0. (5.4.17)
The practical importance of the averaging concept stems from the fact that Eq. (5.4.17)
describes soliton propagation quite accurately when ξ A ≪ 1 [71]. In practice, this
approximation works reasonably well for values up to ξ A as large as 0.25.
From a practical viewpoint, the input peak power P s of the path-averaged soliton
should be chosen such that 〈a 2 (ξ )〉 = 1 in Eq. (5.4.17). Introducing the amplifier gain
G = exp(Γξ A ), the peak power is given by
P s =
Γξ A P 0
1 − exp(−Γξ A ) = GlnG
G − 1 P 0, (5.4.18)
where P 0 is the peak power in lossless fibers. Thus, soliton evolution in lossy fibers
with periodic lumped amplification is identical to that in lossless fibers provided: (i)
amplifiers are spaced such that L A ≪ L D ; and (ii) the launched peak power is larger by
a factor GlnG/(G − 1). As an example, G = 10 and P in ≈ 2.56P 0 for 50-km amplifier
spacing and a fiber loss of 0.2 dB/km.
Figure 5.14 shows pulse evolution in the average-soliton regime over a distance of
10,000 km, assuming solitons are amplified every 50 km. When the soliton width corresponds
to a dispersion length of 200 km, the soliton is well preserved even after 200
lumped amplifiers because the condition ξ A ≪ 1 is reasonably well satisfied. However,
if the dispersion length reduces to 25 km, the soliton is destroyed because of relatively
large loss-induced perturbations.