Spatial beam dynamics in graded-index multimode fibers under Raman amplification: A variational approach

We investigate the spatial beam dynamics inside a multimode graded-index fiber under Raman amplification by adopting a semianalytical variational approach. The variational analysis provides us with four coupled ordinary differential equations that govern the beam's evolution and are much faster to solve numerically compared to the full nonlinear wave equation. Their solution also provides considerable physical insight and allows us to study the impact of important nonlinear phenomena such as self-focusing and cross-phase modulation. We first show that the variational results agree well with full numerical simulations over a wide range of operating parameters. We use them to investigate the signal beam's amplification for different initial widths of the pump and signal beams. This allows us to quantify the pumping conditions under which the quality of a signal beam could be improved at the fiber's output end. Our analysis may prove beneficial in practice for two reasons. First, it can be used to scan the design parameters of an experiment in a time-efficient fashion and determine the optimum values of these parameters. Second, it can provide physical insight into nonlinear phenomena involved in the Raman-amplification process.