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 |Δ kL| , assuming that both seeding Stokes and anti-Stokes fields arise from white background noise components. When |Δ kL|≪ 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 Δ k=0 . When |Δ kL|≫ 1 , all the asymptotic curves for the amplification ratios and extra gain factor obey the inverse square law with respect to |Δ kL| , irrespective of the level of the back-reflection at the rear fiber. In particular, when |Δ kL| 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 |Δ kL| . 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.