New finite-difference formulas for dielectric interfaces

The finite difference method is often-used numerical simulation method in electromagnetics. In this paper a new methodology is presented that allows the derivation of finite difference formulas near dielectric interfaces with high accuracy. Derived finite difference formulas have been used in the electric field computations in electrostatics (the two-dimensional Laplace's equation is employed) and in full-vectorial waveguide simulations in photonics (the three-dimensional Helmholtz's equation and the beam propagation simulation technique in frequency domain are employed). The finite difference formulas derivation is made under a power series expansion of the transverse field components in the case for uniform rectangular discretization mesh. The resulting finite difference formulas provide highly accurate solutions, both for electrostatic and waveguide propagation problems even on coarse grids and thus enable a very cost-effective and rapid numerical field simulations. Reported methodology and derived formulas have not been used in finite difference method formulations in literature. Some results for the electric field computation and dielectric waveguide eigenmode and propagation analysis are presented. .