## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM  
In this subsection, we consider a special case of synchronization between the electron beam and the electromagnetic wave known as _electron cyclotron autoresonance_ (CAR) (or self-sustained gyro-resonance). This phenomenon was independently discovered by [PERSON] [37] and [PERSON] with [PERSON] [38]. Soon thereafter, studies on this subject attracted considerable interest, and new concepts for devices generating electromagnetic radiation [39; 40; 41; 42; 43; 44] and particle accelerators [45; 46; 47; 48; 49] based on such resonance were formulated and demonstrated. These studies are well presented in comprehensive review articles [50; 51]. Figure 10: The variety of possible resonances in a gyrotron cavity with a dense spectrum of modes. From the analysis of the cyclotron resonance condition (Equation (10)), it appears that, during the interaction between the beam electrons and the wave, both the cyclotron frequency and the Doppler shift term change, but in opposite directions. As pointed out by the discoverers of CAR (and then confirmed in corresponding derivations and experiments by many other authors (in addition to [39; 40; 41; 42; 43; 44], see also the following list of selected references [52; 53; 54; 55; 56; 57; 58; 59]), at large axial velocities (and hence large Doppler up-shift), the operating point asymptotically approaches the intersection point, where the phase velocity is close to the speed of light and the changes in the two terms completely compensate (cancel) each other. Therefore, the beam and the wave remain synchronized, and the resonance is self-sustaining, as illustrated in Figure 11. In other words, at such a specific operating point, if the cyclotron resonance condition is initially fulfilled, it will be maintained despite the changes in energy during the beam-wave interaction. The scientists who discovered CAR envisaged the application of this remarkable phenomenon to particle acceleration--the so-called CARA (cyclotron autoresonance acceleration), which has been studied by many researchers (see, for example, [45; 46; 47; 48; 49]). This topic, however, is outside the scope of the present work, and we discuss here only the underlying importance of CAR for CARM devices. Cyclotron autoresonance is possible due to the existence of a fundamental invariant--a constant of electron motion, which we describe below. Its derivation is very straightforward, concise, and insightful. That is why we reproduce it briefly here following [60]. First, we consider the axial component of the relativistic equation of electron motion in a TE wave:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]  
and the equation that describes the change in its energy:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]  
where \(\vec{v}_{\perp}\) is the transverse velocity of an electron and \(\vec{E}_{\perp},\vec{B}_{\perp}\) are the transverse electric and magnetic fields of the wave, respectively. The magnitudes of these fields are related by the wave impedance \(Z\) through the following equation (written in vector notations):  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]  
Figure 11: Bundle of beam lines and their intersection with the dispersion curve of the cavity mode at cyclotron autoresonance. where the magnetic field strength \(H=\mu_{0}B\) and \(\mu_{0}\) is the permeability of free space.