Electrodynamics of Inhomogeneous 2D Periodic Media

Abstract

A new method is proposed to represent electromagnetic field in inhomogeneous 2D periodic medium (PM) as a discrete set of amplitude vectors (AVs) each of which contains amplitudes of spatial harmonics of the field scattered by a particular particle. A scattering operator of a particle is introduced to establish a relationship of the amplitudes of harmonics of the scattered and excitation fields, and an external source is introduced to consider the scattering field that is not related to the excitation field of a particle. An exact nonlocal equation is derived for AVs to describe propagation of electromagnetic waves in PM, and an approximate localized difference equation (an analog of the wave equation in continuous medium) is obtained. Quadratic relationships are obtained for AVs: theorem on active power, Lorentz lemma, and reciprocity theorem. A problem of excitation of homogeneous PM by an external source is solved, and the Green function of PM is obtained. Boundary conditions for AV are introduced at the PM defects that disturb periodicity. A method to formulate and solve the boundary-value problem for AVs in PM with defects is proposed. Analytical solutions to canonical boundary-value problems of PM electrodynamics (eigenwaves of homogeneous PM, normal modes of a cavity, and eigenwaves of an infinite waveguide) are considered. Construction of efficient numerical algorithms for solution of boundary-value problems in electrodynamics of inhomogeneous PM is discussed.