Study on the asymptotic behavior of the interplay of stimulated Brillouin scattering and Brillouin-enhanced four-wave mixing in standard single-mode fibers

We theoretically study stimulated Brillouin scattering (SBS) in a standard single-mode fiber (SMF), taking Brillouin-enhanced four-wave-mixing (BEFWM) effects into account. In particular, we investigate the case when there is non-negligible back-reflection of the forward-pump field at the rear fiber end although such reflection is typically weak and undesired. We first justify that BEFWM can be treated as a steady-state process under an undepleted pump approximation as long as the nominal SBS gain remains as low as 40 dB unless the pump, Stokes, anti-Stokes fields interact under near-perfect phase-matching condition, which hardly happens in normal circumstances with a standard SMF. Under the steady-state and undepleted-pump condition, we find analytical solutions to the Stokes and anti-Stokes fields generated by the forward and backward-pump fields, and also derive their asymptotic formulae in both infinitesimal and infinite limits in terms of the phase-mismatch parameter of vertical bar Delta kL vertical bar, assuming that both seeding Stokes and anti-Stokes fields arise from white background noise components. When vertical bar Delta kL vertical bar I << 1, the acoustic fields driven by SBS and BEFWM tend to interfere destructively, and thus, SBS and BEFWM are anti-resonant to each other, thereby eventually resulting in both Stokes and anti-Stokes scatterings minimized at Delta k = 0. When vertical bar Delta kL vertical bar >> 1, all the asymptotic curves for the amplification ratios and extra gain factor obey the inverse square law with respect to vertical bar Delta kL vertical bar, irrespective of the level of the back-reflection at the rear fiber. In particular, when vertical bar Delta kL vertical bar is in the intermediate range where the FWM gain remains relatively large, SBS and BEFWM can be cooperative via the phase-pulling effect by the FWM gain, thereby leading to quasi-resonant growths of both Stokes and anti-Stokes fields. However, the extra gain by BEFWM reduces significantly if the level of the back-reflection remains below one percent, irrespective of the value of vertical bar Delta kL vertical bar. Since the interplay between SBS and BEFWM is inherently phase-dependent whilst it can still happen with white noise seeding with random phases, the related mechanism can further be exploited for all-optical switching functionality. We expect our theoretical modeling and formulation will be useful for designing and analyzing a variety of fiber systems that incorporate high-power narrow-linewidth light undergoing non-negligible back-reflection under various conditions.