TABLE OF CONTENTS Electromagnetic phenomena are governed by Maxwell s equations. The topic of propagation of electromagnetic fields in general dielectric media is vast. We will only concern ourselves with periodic or otherwise symmetric dielectric structures. A description of a radiation state of a system can be found by solving Maxwell s equations and satisfying appropriate boundary conditions. It is generally true that if additional symmetries are present in a system there are conserved physical quantities that can be identified to describe the general behavior of such a system. Frequently, such conserved quantities can be used to label a particular radiation state unambiguously. We start by considering how the symmetries of a dielectric media in time and in space define the general properties of solutions. Our immediate goal is to characterize the behavior of radiation propagation in photonic crystals without the need of numerical solutions. In all the derivations we use cgs units and let the speed of light c = 1.
In the absence of free charges and currents, Maxwell s equations are:
where, respectively, E( r , t) and H( r , t) are the microscopic electric and magnetic fields and D( r , t), B( r , t) are the displacement and magnetic induction fields. Constitutive relations are taken as:
thus, assuming isotropic dielectric constant ?( r), no nonlinearity and nonmagnetic materials, ? = 1. Substituting (2.5) into (2.1 2.4) we arrive at:
In general, electromagnetic fields are complicated functions of time...
