Spectral approach with iterative clarification of a radiation boundary conditions for modeling of quasimodes of a gyrotrons open cavities

Abstract

Purpose. The article presents a new method for numerical simulation of quasi-eigenmode oscillations in open resonators of gyrotrons — powerful vacuum generators of electromagnetic waves in the millimeter and submillimeter ranges. The gyrotron cavity has the shape of a weakly inhomogeneous hollow circular metal waveguide. Methods. The proposed approach uses the inhomogeneous string equation with radiation boundary conditions to formulate a nonlinear spectral boundary value problem describing oscillations in a resonator, neglecting the couplings of waves with different radial indices. By linearizing with respect to frequency the radiation boundary conditions, the boundary value problem is reduced to a linear boundary value problem. To discretize this boundary value problem, the finite difference method is used and a linear generalized matrix eigenvalue problem is formulated. This problem is solved by the Arnoldi method with eigenvalues calculation in a shift-invert mode. An iterative algorithm is proposed that makes it possible to sequentially calculate a given number of frequencies and quality factors of quasi-eigenmodes of oscillations. Results. The computer program was developed written in the Wolfram Language and Fortran using the algorithms proposed in the work. The results of test calculations for real gyrotron resonators are presented, which demonstrate the high accuracy of the obtained values of frequencies, quality factors and field distributions of quasi-eigenmode oscillations in the studied resonators. Conclusion. The methods, algorithms and created program proposed in the article can significantly facilitate the process of developing gyrotrons intended for various practical applications and operating in new frequency ranges. The method of iterative refinement of boundary conditions can be generalized to the case of equations of the linear theory of a gyrotron and used to develop new methods for analyzing the starting conditions for the soft self-excitation in gyrotrons — generators.