Analysis of electron trajectories in a gyrotron resonator

A detailed mathematical analysis of the equation describing the electron interaction with the high-frequency field in a gyrotron resonator is presented. Electron trajectories in the phase space are classified. It is proven that in the cold-cavity approximation when the high-frequency field is represented by a Gaussian-type function, the solutions of the gyrotron equation are asymptotically equal to the solutions of the corresponding unforced equation. This means that chaos, which, in principle, can develop in a resonator for some values of control parameters, can be only transient, i.e., electrons again follow regular trajectories once they leave the interaction space. Understanding of the distribution of electron trajectories is important, both from the theoretical and the practical point of view. As an example, detailed numerical computations of electron trajectories are performed for those values of control parameters which correspond to the maximum efficiency.