Self-consistent calculation of the starting pitch factor of TE modes in gyrotron cavities

The starting pitch factor of a transverse electric (TE) mode in a gyrotron cavity is defined as the minimum pitch factor (electron velocity ratio) that an electron beam of given current must have, in order for the mode to be excited by the beam. In high-power gyrotrons with dense spectra of competing modes, the knowledge of the starting pitch factor of several modes at different operating parameters would provide a valuable insight into the expected mode competition. Therefore, a numerical code is developed for the calculation of the starting pitch factor, based on the mathematical model of the beam-wave interaction in the gyrotron cavity. Self-consistency is achieved by solving simultaneously the equations for the electron motion and for the axial profile of the high-frequency field. Numerical results for the calculation of the starting pitch factor as well as the starting current of TE modes in various cavities are presented. The code validation, involving comparisons with the time-dependent large-signal code EURIDICE as well as with the small-signal spectral code TWANGlinspec, is also presented. Using an example of a challenging coaxial cavity design for second harmonic MW-class operation, it is shown that, in high-power gyrotrons, the calculation of the starting pitch factors of the competing modes is much more advantageous than that of their starting currents because it can immediately identify practical gyrotron start-up scenarios with a triode-type electron gun that mitigate mode competition.