Embedding approach for rapid convergence of plane waves in photonic calculations

Abstract

The embedding method is developed for solving Maxwell's equations in a region of space, region I, joined onto region II, which may be a dielectric object or an extended substrate. In this method, region II is replaced by an embedding operator defined over its boundary with I, and Maxwell's equations are solved explicitly only in region I. It is shown how approximate solutions of Laplace's equation, which can corrupt the solutions of Maxwell's equations with finite frequency, may be suppressed in the general case. This is applied to lattices of dielectric spheres, and the method shows greatly improved convergence compared with standard plane-wave methods. Fewer than 100 plane waves of three polarizations with embedding give accurate energy bands and effective dielectric constants. The embedding method is also applied to determining the spectral density in a lattice of metallic spheres.