Analytical theory of mode competition in start-up scenarios of high-power gyrotrons

Summary form only given. In high-power gyrotrons operating at high-order modes, when the voltage rises from 0 to its nominal value (80-100 kV), the cyclotron resonance conditions can be fulfilled for many modes. When the mode spectrum is rather dense, regions of self-excitation of some modes overlap and, therefore, the competition between such modes occurs. This problem, as a rule, is studied with the use of non-stationary, multi-frequency codes (such as MAGY 12, where it was shown that for successful excitation of the desired mode, the most important is to suppress a counter-rotating mode with the frequency close to that of the operating mode. In the present paper, we develop the analytical theory describing the competition between two modes with close frequencies. Such a theory can be applied to the case when the minimum start current of the first excited mode is larger than the minimum start current of the mode, which can be excited a little later. The intensity of these two modes start to grow from the noise level (which is on the order of 0.1 mW) and the modes do not interact until at least one of them reaches the kW power level. Clearly, when the voltage rises slowly (as it takes place in long-pulse gyrotrons), the first mode reaches the saturation level first and then suppresses the second mode. However, in short-pulse gyrotrons, the second mode, which, when the voltage approaches the region of small start currents, grows faster than the first one, can successfully compete with the first mode. Therefore, the results of the mode competition in gyrotron start-ups may depend on the voltage rise time. Correspondingly, one may envision the difference in gyrotron operation in short-pulse and long-pulse (CW) regimes. Our theory describes these phenomena and allows one to formulate criteria predicting the winner in such mode competition or, in other words, to define the critical voltage rise time, which separates the regions of final operation in the first and secon- modes