Nonlinear Fiber Optics - 4 ed. Agrawal

346 Chapter 9. Stimulated Brillouin Scattering
the effects of GVD are ignored. This is justified by noting that pulse widths typically
exceed 1 ns, and the dispersion length is so large that GVD plays little role in the SBS
process. The frequency difference between the pump and Stokes waves is also so small
(about 10 GHz) that numerical values of γ and α are nearly the same for both waves.
The effective mode area appearing in Eq. (9.4.7) is defined as [32]
A eff = 〈|F p(x,y)| 2 〉〈|F s (x,y)| 2 〉〈|F A (x,y)| 2 〉
|〈F p (x,y)Fs ∗ (x,y)FA ∗ ≈ [〈|F p(x,y)| 2 〉] 2 〈|F A (x,y)| 2 〉
(x,y)〉|2 |〈|F p (x,y)| 2 FA ∗ , (9.4.9)
(x,y)〉|2
where the angle brackets denote integration over the entire x–y domain and we use F p ≈
F s in view of the small wavelength difference between the pump and Stokes waves.
When the acoustic mode occupies a much larger area than the fundamental fiber mode,
this expression reduces to the conventional definition of the effective mode area given
in Eq. (2.3.30). As a final simplification, we note that the z derivative in Eq. (9.4.7) can
be neglected in practice because of a much lower speed of an acoustic wave compared
with that of an optical wave (v A /v g < 4 × 10 −5 ).
For pump pulses of widths T 0 ≫ T B = Γ −1
B
, Eqs. (9.4.5) through (9.4.7) can be simplified
considerably because the acoustic amplitude Q decays so rapidly to its steadystate
value that we can neglect both derivatives in Eq. (9.4.7). The SPM and XPM
effects can also be neglected if the peak powers associated with the pump and Stokes
pulses are relatively low. If we define the power P j = |A j | 2 , where j = p or s, the SBS
process is governed by the following two simple equations:
∂P p
∂z + 1 ∂P p
v g ∂t
− ∂P s
∂z + 1 ∂P s
v g ∂t
= − g B(Ω)
A eff
P p P s − αP p , (9.4.10)
= g B(Ω)
A eff
P p P s − αP s , (9.4.11)
where g B (Ω) is given in Eq. (9.1.4) and g B (Ω B )=4κ 1 κ 2 /Γ B reduces to the expression
given in Eq. (9.1.5). These equations reduce to Eqs. (9.2.1) and (9.2.2) under steadystate
conditions in which the intensity I j = P j /A eff does not depend on time for j = p
or s.
9.4.2 SBS with Q-Switched Pulses
Equations (9.4.5)–(9.4.7) can be used to study SBS in the transient regime applicable
for pump pulses shorter than 100 ns. From a practical standpoint, two cases are of
interest depending on the repetition rate of pump pulses. Both cases are discussed
here.
In the case relevant for optical communication systems, the repetition rate of pump
pulses is >1 GHz, while the width of each pulse is