Abstract

A general recursive formulation suitable for the wave propagation problem in nonlinear optics is developed and applied to the problem of propagation in nonlinear optical fibers. This technique, which is based on the direct numerical solution of the wave equation, differs from other techniques in diffraction theory that are based on the Rayleigh-Sommerfeld diffraction integral. We calculate the transverse field distribution for all increments of the propagation distance, taking into account all other incremental changes of the waveguide or of the complex index of refraction of the optical system. This method employs the boundary conditions of the problem to generate two auxiliary functions that in turn generate the electric field profile by using a recursive equation. This method is simple, stable, and of second order in the transverse and in the propagation coordinates. It is also readily applicable to problems formulated in Cartesian and cylindrical coordinates. The extension of this method to three dimensions is also discussed. The technique is applied to the problem of propagation in nonlinear fibers. Two cases are studied. First, the fiber itself is assumed to be nonlinear while the second case, the cladding, is assumed to be nonlinear. The second case is especially interesting since optical bistability has been predicted in similar configurations. All the results presented are generated on a desk-top microcomputer.

© 1985 Optical Society of America