Low-truncation-error finite difference equations for photonics simulation. I. Beam propagation

A methodology is presented that allows the derivation of low-truncation-error finite difference equations for photonics simulation. This methodology is applied to the case of wide-angle beam propagation in two dimensions, resulting in finite difference equations for both TE and TM polarization that are quasi-fourth-order accurate even in the presence of interfaces between dissimilar dielectrics. This accuracy is accomplished without an appreciable increase in numerical overhead and is concretely demonstrated for two test problems having known solutions. These finite difference equations facilitate an approach to the ideal of grid-independent computing and should allow the simulation of relevant photonics devices on personal computers.