Mode competition and control in free electron lasers with one and two dimensional Bragg resonators

We present a time domain analysis of the transverse mode dynamics which occur in FEL oscillators using oversize microwave cavities. It is shown that both traditional one-dimensional and novel two-dimensional Bragg cavities can produce radiation at a single frequency with a one-mode azimuthal distribution, which corresponds to spatial synchronization of the electromagnetic radiation. In one-dimensional system, electronic mode selection occurs via nonlinear mode competition. In contrast, electrodynamic mode selection occurs in two-dimensional Bragg cavities, resulting in the production of a single azimuthally-symmetric mode after the linear stage of evolution. The two-dimensional Bragg cavity is shown to retain its selectivity when its radius greatly exceeds the radiation wavelength.