Computationally efficient spectral analysis of an FEL oscillator using a Green function analysis

Previous work on a Green function approach to the evolution of the small gain linear FEL oscillator assumed a long electron pulse. Here this restriction is eliminated and a new expression for the complex amplitude of the optical field at the nth pass is obtained. The fully causal formalism includes arbitrary temporal profiles of the electron and optical beams and cavity detuning and losses. In our formalism, the solution for the field at the nth pass requires exponentiating a single matrix rather than sequentially evaluating the field at each pass. Various numerical studies are presented.