Free Electron Lasers with Variable Parameter Wigglers: A Strictly Nonlinear Oscillator with Slowly Varying Parameters

摘要

The first part of this dissertation gives a detailed mathematical modeling of free electron lasers (FEL), and the second part is a study of a number of general perturbation methods for solving FEL and other systems. A dimensionless model of the motion of electrons in FEL is derived and secular effects due to slow variations in the magnetic field amplitude of the wiggler, and the effect of the density of electrons on the rate of evolution of the signal field are systematically accounted for. The mathematical problem reduces to the analysis of a strictly nonlinear oscillator with slowly varying parameters and small perturbation terms. Two distinguished limits governing the rate of variation of the wiggler, and three distinguished limits governing the evolution of the signal field are identified. To analyze solutions of FEL and other strictly nonlinear systems, the method of Kuzmak-Luke and the method of averaging are reviewed and extended, then it is shown that both methods are equivalent. In order to implement the calculation of explicit solutions, two approximation schemes are developed. The first uses an approximation for the potential such that the first order solution is expressible in terms of elliptic functions. The second method involves use of a truncated Fourer series to represent either the leasing order solution or the transformation of the governing equations to a form which is solvable by standard methods. The discussion is concluded with several examples illustrating these ideas. (ERA citation 12:028992)