## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM
In the case of TE modes, the limiting single-particle efficiency of the beam-wave interaction is given in [40] and reads as follows:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]7  
Using the constant of motion \(\Gamma^{\text{TE}}\), it can be expressed in the following more compact form:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]8  
Another important parameter is the so-called electron recoil parameter [16; 40; 54]:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]9  
which characterizes how the axial momentum varies with beam energy. The larger \(b\) is, the faster the axial momentum decreases with the decrease in \(\gamma\)[54]. For the gyrotron operation, \(b\approx 0\), while for CARM, the typical values are in the range 0.2-0.5 or greater [40]. Plots of optimal normalized efficiency are given by Fliflet (see Figures 1 and 2 in [54]). The knowledge of the parameter \(b\) allows one to evaluate the single-particle efficiency from these plots without solving again the equations of motion. Equations of Motion of Gyrating Electrons and Their Invariants (Integrals of Motion) Describing the Formation of Helical Electron Beams in the Electron-Optical System (EOS) of Gyrotrons  
In the previous sections, we considered some of the basic relationships relevant to the resonance conditions and beam-wave interaction in gyro-devices. In this section, we discuss some important analytical expressions (including the constants of motion) that describe the formation of helical electron beams used in these tubes. As mentioned above, the active medium in gyro-devices is an electron beam interacting with a fast wave (i.e., with a phase velocity greater than the speed of light in vacuum) in the electrodynamic system (resonant cavity or waveguide), where the beam is synchronized with the electromagnetic wave according to the cyclotron resonance conditions explained above. Usually, two types of beams are used (see Figure 12). The first, used in conventional gyrotrons, is a polyhedral hollow (tubular) electron beam in which the individual orbiting electrons follow helical trajectories whose guiding centers are at a certain distance (radius of the guiding center \(r_{g}\)) from the axis. The second type, used in the so-called large orbit gyrotron (LOG) (see, for instance, [61; 62; 63; 64]), is usually referred to as an axis-encircling (or uniaxial) electron beam since the individual orbits encircle the axis and their radius is comparable to the radius of the resonator, in contrast with conventional devices where the electrons gyrate around their axes, following helices with a much smaller radius. The helical electron beam is formed in an electron-optical system (EOS) using a magnetron injection gun (MIG), in which the electrons are accelerated and their orbits twisted in crossed electric and magnetic fields. More sophisticated EOSs are used to generate uniaxial electron beams, which include a cusp electron gun or a kicker (special magnetic coils). LOG operation is inherently high harmonic, since the uniaxial electron beam interacts only with TE\({}_{mn}\) modes whose axial index \(m\) equals the harmonic number (\(m=s\)) and can be significantly greater than unity. For the same reason, LOG offers an increased mode selectivity. This follows from the analysis of the coupling coefficient \(G\), which characterizes the strength of the beam-wave interaction as a function of the beam radius:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]0  
where \(R_{b}\) and \(R_{c}\) are the beam radius and the cavity radius, respectively. If \(m-s\
eq 0\), \(G=0\) for \(R_{b}=r_{g}=0\) (see Figure 13). Therefore, the electron beam in LOG couples effectively only with the corotating modes whose azimuthal indices are equal to the harmonic number.