Superradiant, single-supermode and nonlinear regimes of short pulse free electron laser oscillators

We study both small signal and nonlinear regimes of free electron laser oscillators driven by short electron bunches. This work extends and completes a previous work, focusing the analysis first on the spectrum of the eigenmodes of the linear problem (supermodes) and on the description of the weakly nonlinear regime in terms of these eigenmodes and second on the fully nonlinear dynamics. Using an orthogonality property of the supermodes, we derive expressions for the amplitudes of the fundamental and secondary supermodes and we discuss the single-supermode stable operation. Then we reconsider the superradiant regime in a quasiperfectly synchronized, high-quality optical cavity. We show that superradiance actually is an intrinsically multi-supermode regime, which occurs when the spectrum is nearly degenerate. Going next to the nonlinear regime, we find the nonlinear modes of the system (stationary regimes), which appear through successive Hopf bifurcations when the linear eigenmodes become unstable. We analyze the stability of the fundamental nonlinear mode and show that it gets unstable through a new supercritical Hopf bifurcation when dissipation is decreased, giving rise to a limit cycle. Finally, we reconsider the routes to chaos, showing that although the dynamical behavior of the system depends in a complicated way on the control parameters, it can be described to a large extent by the iterations of one-dimensional maps.