Kinetic theory of a cyclotron-undulator hybrid fel in circular waveguide

In this paper, we model the cyclotron-undulator hydrid instability using the kinetic theory in circular waveguide geometry, following the generalized method of the field expansion in the guiding center coordinate system; and derive in detail the coupling coefficient for a cyclotron-undulator hybrid FEL operating at the fundamental harmonic of gyration and undulation. In our discussion, the effect of the transverse structure of the radiation fields is taken into account, but the undulator is considered as an ideally longitudinal one.