Nonlinear Fiber Optics - 4 ed. Agrawal

354 Chapter 9. Stimulated Brillouin Scattering
Figure 9.15: Temporal evolution of the Stokes (left column) and pump (right column) intensities
without (top row) and with (bottom row) feedback. Fiber losses correspond to αL = 0.15. (After
Ref. [102]; c○1985 OSA.)
However, a steady state is not reached because of the instability indicated in Figure
9.14. Instead, both the pump output at z = L and the Stokes output at z = 0 exhibit
steady oscillations. Interestingly enough, a steady state is reached if the feedback is
increased such that R 1 R 2 ≥ 2 × 10 −2 . This happens because b 0 for this amount of
feedback lies in the stable regime of Figure 9.14. All such dynamic features have been
observed experimentally [102].
9.4.5 Modulation Instability and Chaos
Another instability can occur when two counterpropagating pump waves are present
simultaneously, even though none of them is intense enough to reach the Brillouin
threshold [103]–[108]. The origin of this instability lies in the SBS-induced coupling
between the counterpropagating pump waves through an acoustic wave at the frequency
ν B . The instability manifests as the spontaneous growth of side modes in the pump
spectrum at ν p ± ν B around the pump frequency ν p [104]. In the time domain, both
pump waves develop modulations at the frequency ν B . The SBS-induced modulation
instability is analogous to the XPM-induced modulation instability discussed in Section
7.3 except that it occurs for waves propagating in the opposite directions. The
instability threshold depends on the forward and backward input pump intensities I f
and I b , fiber length L, and parameters g B , ν B , and Δν B .
Figure 9.16 shows the forward pump intensity I f (in the normalized form) needed
to reach the instability threshold as a function of the intensity ratio I b /I f for Δν B /ν B =
0.06 and several values of the normalized fiber length 4πnν B L/c. The instability
threshold is significantly smaller than the Brillouin threshold (g B I f L = 21) and can be
as small as g B I f L = 3 for specific values of the parameters. Numerical results show that