Maximization of FEL gain for a hole-coupled resonator

For a hole-coupled resonant cavity without FEL gain, it is possible to have amplitude cancellation of the optical Gaussian modes at the hole location and, as a result, a null is generated to give very low cavity loss due to the presence of the hole. However, in a warm cavity, i.e. when the electron beam is present and the optical power starts to build up, the FEL gain tends to amplify the modes unequally, and the unbalanced gain for different optical modes results in significant power radiated through the hole. In this paper we present a guideline, based upon variation of the Rayleigh length, for mode selection that maximizes net gain.