Analytical approach to calculate the gain of Brillouin fiber amplifiers in the regime of pump depletion

We provide an accurate analytic expression for the Brillouin fiber amplifiers (BFAs) gain in the regime of pump depletion. The solutions are divided into three parts. In the first part, we review the weak pump regime where the highly accurate second-order corrected undepleted-pump-approximation-based solution is adopted; in the high-gain regime where both the fiber loss and nonlinear term are nonnegligible effects and need to be accounted for along with the pump depletion; and in the saturation regime where the Stokes power is at the rear end exceeds critical power. We provide an accurate analytic expression for the BFA gain and show results of experimental validation. The presented analysis can be used to accurately predict the performance of Brillouin fiber amplifiers working in any regime.