General formula of the energy transfer rate in gyrotron, CARM, and nonwiggler FEL ocscillators

A general formula of the energy transfer rate of the transverse-electric-mode gyrotron, CARM, and nonwiggler FEL oscillators is rigorously derived within the Vlasov framework. The present result can be used to simultaneously deal with the influences of the finite cavity length, spreads, higher harmonics, and nonresonance. It is examined to be in agreement with the Madey theorem of these devices for a cold electron beam.