Lie-optic matrix algorithm for computer simulation of paraxial self-focusing in a plasma

Propagation algorithm for computer simulation of stationary paraxial self-focusing laser beam in a medium with saturating nonlinearity is given in Lie-optic form. Accordingly, a very natural piece-wise continuous Lie transformation that reduces to a restricted Lorentz group of the beam results. It gives rise to a matrix method for self-focusing beam propagation that is constructed and implemented. Although the results use plasma nonlinearities of saturable type, and a gaussian initial beam, these results are applicable for other media like linear optical fibers and to more general situations.