Nonlinear Fiber Optics - 4 ed. Agrawal

298 Chapter 8. Stimulated Raman Scattering
Figure 8.11: Evolution of (a) pump and (b) Raman pulses over three walk-off lengths in the
specific case for which L D /L W = 1000, L W /L NL = 24, and L W /L G = 12.
For numerical purposes, it is useful to introduce the normalized variables. A relevant
length scale along the fiber length is provided by the walk-off length L W . By
defining
z ′ =
z , τ = T , U j = A j
√ , (8.3.15)
L W T 0 P0
and using Eq. (8.3.4), Eqs. (8.3.1) and (8.3.2) become
∂U p
∂z ′ + iL W ∂ 2 U p
2L D ∂τ 2
∂U s
∂z ′ − ∂U s
∂τ + irL W ∂ 2 U s
2L D ∂τ 2
= iL W
L NL
[|U p | 2 +(2 − f R )|U s | 2 ]U p − L W
2L G
|U s | 2 U p ,
= irL W
L NL
[|U s | 2 +(2 − f R )|U p | 2 ]U s + rL W
2L G
|U p | 2 U s ,
(8.3.16)
(8.3.17)
where the lengths L D , L W , L NL , and L G are given by Eq. (8.3.5). The parameter r =
λ p /λ s and is about 0.95 at λ p = 1.06 μm.
Figure 8.11 shows the evolution of the pump and Raman pulses over three walk-off
lengths using L D /L W = 1000, L W /L NL = 24, and L W /L G = 12. The pump pulse is
taken to be a Gaussian. The Stoles pulse is seeded at z = 0 as indicated in Eq. (8.3.12)
such that its power is initially smaller by a factor of 2 × 10 −7 compared with the pump
pulse. The results shown in Figure 8.11 are applicable to a wide variety of input pulse