Exact gain of Brillouin fiber amplifiers in the regime of pump depletion

Thanks to the LambertW function , the gain of a Brillouin fiber amplifier operating in the depleted pump regime is solved numerically using the Newton-Raphson method. Unlike traditional boundary value problem solvers, which rely on shooting or relaxation techniques and often pose challenges such as high computational complexity and convergence issues, the Newton-Raphson method offers a simpler and more efficient numerical approach. The result obtained can be used to optimize the performance of Brillouin fiber amplifiers.