Theory and simulation of transverse supermode evolution in a free-electron laser oscillator

We derive the conditions for establishment of a transverse supermode in a free-electron laser (FEL) oscillator, and demonstrate the evolution of a supermode by means of a three-dimensional nonlinear code. Both the analytical formulation and the numerical code are based on coupled-mode theory. The oscillator supermode is a combination of transverse modes that keeps its field profile at any point along the oscillator intact after each round trip, and therefore it is the steady-state result of the oscillation buildup process. In the FEL, as in any laser, the oscillator supermode is identical with the amplifier supermode only if the feedback process is entirely nondispersive. If this is not the case, the steady-state supermode field profile varies along the oscillator axis. The simulations demonstrate that the transverse supermode evolution process is primarily a linear regime process and can be proceeded or even completed before saturation.