Beam propagation in an active nonlinear graded-index fiber

A theoretical model is developed by exploiting the variational technique to investigate the evolution of an optical beam inside an optically pumped graded-index fiber amplifier. The variational analysis is a semi-analytical method that provides us with a set of coupled ordinary differential equations for the beam's four parameters. Numerical solution of these equations is much faster compared to the underlying multidimensional nonlinear wave equation. We compare the results of the variational and full numerical simulations for the two pumping schemes used commonly for high-power fiber amplifiers. In the clad-pumping scheme, the use of a relatively wide pump beam results in a nearly uniform gain all along the fiber. In the case of edge pumping, a narrower pump beam provides gain that varies both radially and axially along the fiber's length. In both cases, the variational results are found to be in good agreement with time-consuming full numerical simulations. We also derive a single equation for the beam's width that can predict amplification-induced narrowing of the signal beam in most cases of practical interest.