Finite difference solution for graded-index cylindrical dielectric waveguides: a scalar wave approximation

A simple numerical method based on finite differences to solve the scalar wave equation encountered in optical fibers is presented. The method uses the Ricatti transformation to the scalar wave equation and is capable of analyzing fibers of arbitrary refractive index profiles. Achieving errors as low as 0.005% for propagation constants is possible for lower-order modes. However, the error seems to increase for frequencies closer to cutoff.