Free Electron Laser Sideband Instability Reconsidered

摘要

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 perturbing sidebands over the unperturbed orbits yields the sideband growth for both trapped and untrapped particles. The unperturbed distribution f sub o is in an equilibrium with the main signal field. It is found that upper and lower sidebands that are symmetric relative to the FEL frequency have opposite growth rates. There is no differentiation in the magnitude of the growth between upper and lower sidebands. The stability is determined by the sign of df sub o/d(omega sub b), i.e., the relative population of oscillation quanta (h-bar)(omega sub b), omega sub b = bounce frequency around resonance. The shear d (omega sub b)/dj, where J is the action variable, is stabilizing and distributions with gradients df sub o/dJ localized near the separatrix have the minimum growth rates. The structure and scaling of the unstable spectrum are different from previous results obtained for electrons localized at the bottom of the ponderomotive well.