Closed-form solution for parametric second Stokes generation in Raman amplifiers

The efficiency of a first Stokes Raman amplifier can be seriously reduced by the generation of a second Stokes beam. The dominating process for this generation is often a Raman four-wave mixing interaction which couples the pump and second Stokes beams. A one-dimensional model for this effect has a closed-form solution, simply described in terms of an equivalent input, under conditions where high first Stokes conversion efficiency is possible. Excellent agreement with numerical integration results has been obtained. Use of the model for maximizing conversion efficiency of pump beams with large temporal or spatial intensity nonuniformity is described, with an example worked out for a triangular pulse shape.<>