High. accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces

A methodology is presented that allows the derivation of low-truncation-error finite-difference representations of the two-dimensional Helmholtz equation, specific to waveguide analysis. This methodology is derived from the formal infinite series solution involving Bessel functions and sines and cosines. The resulting finite-difference equations are valid everywhere except at dielectric corners, and are highly accurate (from fourth to sixth order, depending on the type of grid employed). Nonetheless, they utilize only a nine-point stencil, and thus lead to only minor increases in numerical effort compared with the standard Crank-Nicolson equations.