Approximate solutions to the cylindrical scalar wave equation using the decomposition method

Abstract

Exact solutions can be obtained for electromagnetic wave propagation in a medium with a simple uniform refractive-index distribution. For more-complex distributions, approximate or numerical methods have to be utilized. We have previously used an elegant approximation scheme called the decomposition method for nonlinear differential equations[1] to obtain solutions for the wave equation with planar geometry. Both step and parabolic refractive index profiles were considered. The results were compared with those obtained by JWKB and modified Airy function methods [2]. However, because of the essential singularity present in the radial equation at r=0, we could not obtain solutions for the cylindrical waveguide. Here we describe a procedure to get around this problem and as an example work out the solution for a parabolic index fiber.