## 2 Adler's Equation and Phase Locking and Synchronization of Gyrotrons
The possibility to stabilize the gyrotron frequency under the influence of a wave reflected from a remote output window during the competition of the operating mode with two neighboring modes, which have frequencies equally spaced from the frequency  
Figure 1: Phase dynamics: (**a**) Free running and priming oscillators have the same frequency (synchrony); (**b**) the oscillators are phase locked (also called phase entrainment), and there is a stable and constant phase difference; (**c**) the oscillators are unlocked (phase drift ). of the operating mode, has been studied in [57]. In this study, the influence of the shift of the reflector within the limits of several wavelengths on the frequency stabilization is considered. The frequency and phase fluctuations of the radiation caused by low-frequency technical noise of the accelerating voltage of the electron beam are compared in a stand-alone gyrotron and a gyrotron with a reflection. The first experiments on frequency-locked operation of the 170 GHz/1 MW gyrotron have been reported in [50]. The \"driver\" gyrotron with a power of 20 kW was equipped with frequency stabilization by means of PLL control of the modulating anode voltage. The radiation was injected into the resonant cavity of the MW gyrotron via the specially designed transmission line and the quasi-optical mode converter for co- and counter-rotating modes. In the experiment, the frequency-locked operation of the 1 MW gyrotron was demonstrated in various operating regimes. The maximum output power was increased to 1.2 MW while the spectral width of the radiation of the powerful gyrotron was reduced to 30 kHz, limited by the pulse duration. A theory of peer-to-peer locking of high-power gyrotron oscillators coupled with delay and operated in the regime of hard self-excitation that provides maximal efficiency has been presented in [58]. The developed approximate theory of phase locking in the case of a small delay is based on the generalized Adler's equation, which allows for the treatment of in-phase and anti-phase locking modes. Also, in this paper, a more rigorous bifurcation analysis of phase locking is presented by using XPPAUT software 8.0. The simulations performed by the authors 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. In a recent experimental study [59] of a 170 GHz/1 MW gyrotron locked by a narrowband external signal, an extension of the operating zone by 2.4 times compared to a free-running gyrotron has been demonstrated. The locking signal with a power of less than 16 kW was provided by a gyrotron driver operating in a pulse mode (100 \(\upmu\)s/10 Hz) and stabilized by the phase-lock loop. The observed expansion of the operating zone of the operating mode (\(TE_{28,12}\)) into a lower magnetic field region (compared to the free-running regime) was accompanied by an increase in the output power by 10% and the improvement of efficiency from 32.1% to 35.2% The significant expansion of the operating zone in the region of the larger magnetic field demonstrated the suppression of the excitation of parasitic modes. In the frequency-locked regime, the width of the gyrotron radiation spectrum was reduced to 20 kHz, which is defined by the locking signal from the stabilized gyrotron driver, compared to 2-4 MHz in standalone mode. The significance of these results stems from the fact that gyrotron oscillator locking allows for improved mode stability and precise phase and frequency control of the generated millimeter-wave signal. In experiments for testing a medium-powered 170 GHz gyrotron, developed as a master oscillator for frequency locking of megawatt-level sources, output power of more than 25 kW with about 30% efficiency has been obtained in CW mode. Using a PLL system, the gyrotron frequency has been stabilized, and the spectrum width of less than 2 Hz has been measured, which corresponds to the relative value of about \(10^{-11}\)[44].