Eigenmodes of Photonic Crystals

Abstract

As the first step of the analysis of the radiation field in a photonic crystal, we will formulate the eigenvalue problem of the wave equation and give a general numerical method to solve it. Based on the complete set of the eigenfunctions, we will also derive the expression for the retarded Green's function related to the electric field. In addition to the three-dimensional (3D) case, we will treat the two-dimensional (2D) crystals, for which the vectorial wave equation reduces to two independent scalar equations and the relevant expressions are simplified. 2.1 Wave Equations and Eigenvalue Problems We begin with Maxwell's equations. Because we are interested in the eigen-modes of the radiation field, and the interaction between the field and matter will be discussed in later chapters, we assume here that free charges and the electric current are absent. In this case, Maxwell's equations in the most general form are given in MKS units as follows. ∇ · D(r, t) = 0, (2.1) ∇ · B(r, t) = 0, (2.2) ∇ × E(r, t) = − ∂ ∂t B(r, t), (2.3) ∇ × H(r, t) = ∂ ∂t D(r, t). (2.4) The standard notations for the electric field (E), the magnetic field (H), the electric displacement (D), and the magnetic induction (B) are used in these equations. In order to solve the wave equations derived from Maxwell's equations, we need so-called constitutive equations that relate D to E and B to H. Since we do not deal with magnetic materials in this book, we assume that the magnetic permeability of the photonic crystal is equal to that in free space, µ 0 : B(r, t) = µ 0 H(r, t). (2.5)