Nonlinear Fiber Optics - 4 ed. Agrawal

350 Chapter 9. Stimulated Brillouin Scattering
4
Stokes
Normalized Amplitude
3
2
1
Pump
Acoustic
0
0 1 2 3 4 5 6 7 8 9 10
Normalized Time
Figure 9.12: Temporal profiles associated with the pump, Stokes, and acoustic waves when the
three form a Brillouin soliton for μ = 0.1 and b = 0.95.
Figure 9.12 shows an example of the SBS-coupled solitons for μ = 0.1 and b =
0.95. In this case, the Stokes wave travels backward in the form of a bright soliton at a
velocity larger than v g by a factor of V . At the same time, the pump travels forward as
a dark soliton. Another solution of Eqs. (9.4.13)–(9.4.15) shows that Stokes can also
form a dark soliton, while the pump propagates as a bright soliton [96]. Such solitons
exist even in the absence of GVD (β 2 = 0) and XPM (γ = 0) because they rely on the
presence of a solitary acoustic wave. They are referred to as the Brillouin solitons and
constitute an example of the so-called dissipative solitons because they can exist in
spite of losses. Such solitons have been observed in a Brillouin-fiber ring laser [95].
9.4.3 SBS-Induced Index Changes
When the pump and Stokes pulses differ in their carrier frequencies by the Brillouin
shift exactly (Ω = Ω B ), the Stokes falls on the Brillouin-gain peak and experiences
the most gain. However, if Ω = ω p − ω s differs from Ω B by even a few MHz, the
gain is reduced but, at the same time, the refractive index changes by a small amount
because of the SBS-induced amplification. This can be seen from Eqs. (9.4.5)–(9.4.7)
as follows. If we use the steady-state solution of Eq. (9.4.7) in Eq. (9.4.6), the Stokes
wave satisfies
− ∂A s
∂z + 1 ∂A s
v g ∂t
+ α 2 A s = iγ(|A s | 2 +2|A p | 2 κ 1 κ 2 /A eff
)A s +
Γ B /2 + i(Ω − Ω B ) |A p| 2 A s , (9.4.20)
where the last term represents the contribution of SBS. This term is complex for Ω ≠
Ω B , and the imaginary part results from the SBS-induced index change by an amount