Self-focusing effects in free electron lasers

Self-consistent equations describing the evolution of the transverse structure of the field in FEL's with a high- Q cavity have been derived. It is shown that in the region of mismatch of electron wave synchronism, where the self-focusing criterion is satisfied and the real part of permittivity introduced by an electron beam into a cavity increases with increasing wave amplitude, a plane wave is unstable to perturbations propagating at a small angle to this wave. The mirror transparency profiles ensuring the existence of spatially localized stationary electromagnetic beams with a flat phase front are found.