Analytical theory of low-frequency space charge oscillations in gyrotrons

Low-frequency oscillations attributed to reflected electrons bouncing adiabatically between the electron gun and the interaction space have been observed in many gyrotrons. An analytical model is considered which allows one to apply space-charge wave theory to the analysis of these oscillations. In the framework of the small-signal theory, the regions of low-frequency oscillations, the oscillation frequency and the temporal and spatial growth rates of low-frequency oscillations are determined in the relevant parameter space. The mode frequency is determined not only by the particle travel time, but by the travel time of charge waves on the reflected electron beam. This explains the existence of modes with noncommensurate frequencies.