  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.
