Linear Theory of an Electron Cyclotron Maser Operating at the Fundamental

We report the development of a general, linear theory of an electron cyclotron maser (ECM) operating at the fundamental. A set of analytic expressions, valid for all TE cavity modes, is derived for the starting current and frequency detuning using the Vlasov-Maxwell equations in the weakly relativistic limit. These results are applicable for an arbitrary electron velocity distribution as well as any longitudinal distribution of the RF field. It is shown that the starting current can be expressed in a simple form which contains the Fourier transform of the longitudinal field distribution. Analytic results are presented for specific longitudinal field variations, including uniform, sinusoidal, and Gaussian. It is found that the starting characteristics of an ECM are stongly influenced by the axial dependence of the RF field, but weakly affected by the velocity spread of the electron beam. The problem of multimode oscillation is treated rigorously in the linear theory by using a Slater expansion of the cavity field. Mode competition is investigated for higher order cavity modes and examples are presented. The results of this comprehensive analysis of ECM linear theory should be useful as a diagnostic of ECM performance and should facilitate comparison between theory and experiment.