The structure of the sideband spectrum in free electron lasers

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Abstract
The one‐dimensional, fast time averaged Hamiltonian is derived in a free electron laser (FEL) for electrons passing through a constant parameter wiggler and a radiation field. The exact unperturbed orbits without sidebands are obtained for all particles and arbitrary strength of the main signal. Integration, in action‐angle variables, of the linearized kinetic equation with perturbing sidebands over the unperturbed orbits yields the sideband growth rate including both trapped and untrapped particles. The structure and scaling of the unstable spectrum are different from previous results obtained for electrons localized at the bottom of the ponderomotive well. 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 gain between upper and lower sidebands. The stability is determined by the sign of df0/dωb, the relative population of quantized oscillators with energy quantum ℏωb, where ωb is the synchrotron frequency in resonance with the sideband. The shear dωb/dJ, where J is the action variable, is stabilizing and distributions with gradients df0/dJ localized near the separatrix have the minimum growth rates.