Abstract

The classic numerical solution to the nonlinear differential equation predicting self-trapping of beams in space reveals the existence of a number of different bounded modes, each satisfying different initial conditions. Incidentally, similar equations also predict the nature of radially symmetric stationary traveling wave solutions for envelope propagation through cubically nonlinear dielectrics and with cylindrical (or spherical) symmetry. Later, approximate analytic solutions for the lowest-order mode was developed, employing variational methods and the Rayleigh-Ritz procedure. The lowest-order mode, which resembles a bell-shape curve, was approximated by the sum of two Gaussians, with different amplitudes and widths, suitably chosen to meet the best-fit criteria. In this presentation, we propose a simple straightforward power series approach to the problem, which is extremely useful if we are interested in paraxial behavior of beams or pulses in space. Representation for the different nonlinear stationary modes is discussed. Convergence criteria for the power series are addressed, and the results are compared with previous solutions obtained numerically and by variational methods.

© 1986 Optical Society of America