Competing instabilities in the circular free-electron laser

A small signal theory of the circular free-electron laser (FEL) is developed. A matrix dispersion relation, which includes coupling between the transverse magnetic (TM) and transverse electric (TE) waveguide modes, is derived from a Eulerian fluid model. The full dispersion equation is then expanded around the TM and TE mode resonant frequencies of the circular coaxial waveguide. The growth rate for frequencies near the TM mode resonance agrees with previous results obtained from a nonlinear pendulum model of the circular FEL, and becomes the negative mass growth rate as the wiggler field strength approaches to zero. It is shown that the dispersion relation expanded near the TE mode resonance has a coupling with the wiggler field that is different from the usual FEL mechanism. In the limit of a weak wiggler field, the dispersion relation for frequencies near a TE resonance reduces to that of the cyclotron maser. Numerical calculations of the growth rate and the ratio of the amplitudes of TE and TM modes are presented.