Theory of Peer-to-Peer Locking of High-Power Gyrotron Oscillators Coupled with Delay

Peer-to-peer locking is a promising way to combine the power of high-power microwave oscillators. The peer-to-peer locking of gyrotrons is especially important because arrays of coupled gyrotrons are of special interest for fusion and certain other applications. However, in case of coupled microwave oscillators, the effect of delay in coupling is very significant and should be taken into account. In this article, we present the model of two delay-coupled gyrotrons. We develop an approximate theory of phase locking based on the generalized Adler's equation, which allows for the treatment of in-phase and anti-phase locking modes. We also present a more rigorous bifurcation analysis of phase locking by using XPPAUT software under the limitation of small delay time. The structure of the phase-locking domains on the frequency-mismatch-coupling-strength plane of parameters is examined. Finally, we verify the results by numerical simulations in the case of finite delay time. The simulations reveal various regimes, including peer-to-peer locking, the suppression of one gyrotron by another, as well as the excitation of one gyrotron by another.