Application of Lie theory to optical resonators: The two dimensional master equation

Abstract

The goal of this dissertation is the derivation of a differential equation that describes the evolution of an electromagnetic field in a stable cavity that has no axial symmetry (a toroidal system). The approach uses concepts from the theory of Lie groups and Lie algebras. Since the mathematics may be unfamiliar to the general reader, before the derivation for toroidal systems is executed, the differential equation for an optical system with radial symmetry will be derived using the general mathematical approach. After some of the theorems and formalisms associated with toroidal systems are presented, a description of general toroidal systems and their actions on electromagnetic fields will be presented. The action of systems on electromagnetic fields will be shown to be a linear representation of a group (locally). Having established the preliminaries, the differential equation can be derived. The desired differential equation is derived in three steps. In the first step, a set of differential operators that appear in a simplified equation are derived by recognizing them as the basis of a Lie algebra representation associated with the local linear representation on electromagnetic fields. In the second step, coefficients for the reduced problem are derived. Finally, the complete differential equation is presented. Algorithms that allow one to implement the above results will be presented. These algorithms will be used to execute a computation in a numerical example. By way of verification, it will be shown that the results of this dissertation subsume previous work in several ways including the structure of modes in stable toroidal cavities and the prediction of angular momentum.