Multipole Methods for Photonic Crystal Calculations

Abstract

There are a variety of general theories for computing band structures of photonic crystals, including finite difference methods[1], plane wave methods[2], and multipole methods[3]. While the finite difference and plane wave methods are more general in terms of the range of geometries to which they can be applied, the multipole (or Rayleigh) methods are computationally more efficient and superior in terms of their analytic tractability, coming into their own in derivations of theoretical modelling in areas such as long-wavelength localization and homogenization, and in predicting numerically-difficult outcomes (e.g. anomalously large absorption in crystals with fine metallic wires). In this paper, we outline the development of a multipole theory in a manner applicable to both finite and infinite, periodic structures. The former is applied in the computation of the local density of states (which gives emission probability as a function of position), while the latter is used in the computation of the band structure (including complex gap states) of photonic crystals and studies of disordered media, localization and homogenization.