Sidebands in free electron lasers

The problem of sideband growth in a free electron laser (FEL) is studied using canonical formalism. The exact unperturbed orbits, without sidebands, are obtained in action-angle variables for all particles. Integration of the linearized Vlasov equation with sidebands over the unperturbed orbits yields the sideband growth that includes both trapped and untrapped particles. It is found that the upper and lower sidebands that are symmetric about the FEL frequency have opposite growth rates, with equal magnitude. The stability of the upper or lower sidebands is determined by the sign of dfo/dΩb, i.e., the relative population of oscillation quanta hΩb, Ωb = bounce frequency around resonance.