Exact, implicit, integral solution of depletion and saturation in Raman and Brillouin fiber amplifiers

Stimulated Raman and Brillouin scattering (SRS-SBS) are well known nonlinear optical effects that found applications in optical fiber amplifiers. Analytical solutions of the nonlinear equations governing the pump-signal interaction were found under the undepleted pump approximation or with depletion but when the pump and the signal have equal loss coefficients. In the other cases the interaction is studied by means of numerical solutions; for forward (FPA) propagating amplifiers this means to solve an initial value problem while for the backward case (BPA) it is a nonlinear boundary value problem, where convergence and computation time become important issues. In this paper, exact, implicit, integral solutions are provided, in the pump depletion regime, with unequal loss coefficients; they are valid both for the FPA and BPA.