Transverse mode coupling and supermode establishment in a free-electron laser oscillator

A three-dimensional study of transverse mode evolution in a free-electron laser (FEL) oscillator is presented. The total electromagnetic field circulating in the resonator is represented as a superposition of transverse modes of the cavity. Coupled-mode theory is employed to derive a generalized 3-D steady-state oscillation criterion, from which the oscillator supermode can be found analytically in the linear gain approximation. The oscillator supermode, which is the eigenmode solution of the oscillator at steady-state, keeps its transverse profile and polarization after each round-trip. Relations between the oscillator supermode and the amplifier supermode are discussed. It is shown that they are identical only when the feedback process is entirely non-dispersive and non-discriminating. We employ a 3-D nonlinear simulation code based on the same transverse mode expansion to demonstrate the evolvement of transverse modes in the oscillator towards formation of a supermode in a simple example. The simulation shows that the steady-state result of the oscillation buildup simulation is identical to the supermode predicted by the analytical approach.