## 2 Adler's Equation and Phase Locking and Synchronization of Gyrotrons  
For many applications (e.g., thermal processing of materials, plasma heating in fusion devices, etc. ), the output parameters of gyrotron radiation that are of interest are the power and frequency, while the coherence and phase stability of the generated microwaves are not essential. In other applications, however, the phase coherence and spectral properties are of critical importance. These include advanced spectroscopic methods such as ESR and DNP-NMR spectroscopy, high-resolution radar imaging and deep-space communication. The latter applications require radiation sources that provide precise amplitude, frequency, and phase in a specified bandwidth with low noise. Phase stability is also a key performance criterion for high-gradient linear accelerators that require multiple phase-locked RF sources to ensure synchronization between the charged particles and the accelerating field. The pioneering work on the synchronization of gyrotron oscillators was carried out at the Institute of Applied Physics in Gorky, USSR (now Nizhny Novgorod, Russian Federation) [32; 33; 34; 35]. The general theory of synchronization of oscillations in the gyrotron was developed by [PERSON] and [PERSON] [33]. They analyzed two possible ways of synchronization, namely the use of an external signal injected into the interaction cavity or by pre-bunching the electron beam in an input cavity [34]. Later, they attracted much attention and interest from researchers in the US and other countries around the world [36; 37; 38; 39; 40; 41]. In recent years, considerable progress has been made both in theory and in the implementation of various concepts for the synchronization of gyrotrons (see, e.g., [42; 43; 44; 45; 46; 47; 48; 49; 50]). Over the years, a variety of approaches and sophisticated physical models for the synchronization and locking of gyrotrons have been developed, but they all share a common genealogy, as the idea of phase locking goes back to the seminal work of [PERSON] (best known as the co-inventor of the television remote control [51]), who formulated the famous equation describing the phenomenon of injection locking in nonlinear oscillators [52]. That is why below we discuss the Adler's relation that was derived initially for oscillators with pentodes before applying it to many other classes of oscillators including the gyrotron. To begin with, recollect that oscillatory systems are usually susceptible to injection locking or frequency pulling. Injection locking is a nonlinear phenomenon that can be observed in many natural and technical oscillators. It occurs when an oscillator is perturbed by a weak signal whose frequency is close to the free-running frequency of the oscillator. The [PERSON]'s equation that describes the phenomenon of injection locking in nonlinear oscillators and allows us to estimate the frequency range around an oscillator's natural frequency, where injection locking (syncing to an external signal) occurs, can be written in the following form [51; 52]:  
\[\frac{d\theta}{dt}=\Delta\omega_{0}-\frac{E_{1}}{E}\frac{\omega_{0}}{2Q}\sin \theta, \tag{1}\]  
where \(\theta\) is the oscillator's phase, \(\Delta\omega_{0}=\omega_{0}-\omega_{1}\) is the undisturbed beat frequency, \(\omega_{0}\) is the frequency of the free running oscillator, \(\omega_{1}\) and \(E_{1}\) being the frequency and voltage of the applied signal, respectively, \(E\) is the voltage induced in the grid coil, and \(Q\) is the the quality factor of the load. Here the ratio \(E_{1}/E\) is proportional to the square root of the ratio of the the powers of the external (priming) signal and the output signal. This equation is derived for the case when the condition \(\frac{\Delta\omega_{0}}{\omega_{0}}\ll\frac{1}{Q}\) is satisfied. If the instantaneous beat frequency \(\frac{d\theta}{dt}\
eq 0\) the oscillator is perturbed but not locked.