## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM
The initial transverse electron velocity at the cathode is given by the ratio of the crossed electric \(E_{\perp c}\) and magnetic \(B_{c}\) field there [18]:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]3  
From the conservation of the adiabatic invariant, it follows that the transverse velocity in the interaction region of the cavity, where the magnetic field reaches the value \(B_{0}\), is as follows:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]4  
Here, \(g=\frac{B_{0}}{B_{c}}\) is the magnetic compression ratio. Furthermore, as we have already explained, in the beam compression region (between the electron gun and the resonant cavity), in an increasing magnetic field, the axial momentum of the gyrating electrons is converted to a transverse momentum due to the conservation of energy and magnetic flux. The \"pumped\" electron beam is characterized by the pitch factor (velocity ratio) \(\alpha=v_{\perp}/v_{z}\). More specifically, the transverse component of the kinetic energy of the electrons increases in the proportion of \(B_{2}/B_{1}\). The pumping of the beam, however, is restricted by the reflection of electrons after their axial velocity becomes zero. Using the adiabatic invariant \(I\) and the conservation of electron energy, it is easy to show that a magnetic field \(B_{2}=B_{1}\Big{(}1+\alpha_{1}^{-2}\Big{)}\) will reflect all electrons with a pitch factor \(\alpha>\alpha_{1}=\alpha(z_{1})\). The maximum transverse electron energy \(E_{\perp 0}\), which is available for transfer to the wave, is only a part of he total kinetic energy \(E_{0}\) and depends on \(\alpha\) according to the following relation:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]5  
The transverse (orbital) efficiency is defined as the ratio of the energy extracted from the beam \(\Delta E_{\perp}=E_{\perp 0}-E_{\perp}\) (\(E_{\perp 0}\) and \(E_{\perp}\) being the kinetic rotational energy at the entrance and the exit of the cavity, respectively) and \(E_{\perp 0}\):  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]6  
Analogously, defining the total electronic efficiency \(\eta_{e}\) as the ratio of the energy \(\Delta E\) transferred from the beam to the wave in the cavity and the total kinetic energy \(E_{0}=E_{\perp 0}+E_{z0}\) (where \(E_{z0}\) is the energy of the longitudinal motion), one finds the following:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]7  
These relations show that beams with large pitch factors are required in order to achieve high efficiency. However, as underlined above, the value of the velocity ratio is limited by the reflections, and in EOS with MIG, usually, \(\alpha\leq 1.7\) (typically \(\alpha=1.2\)-\(1.5\)). Since some of the generated radiation is lost due to ohmic loses on the cavity walls, the total efficiency \(\eta\) is always less than \(\eta_{e}\) and is given by the following:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]8where \(P_{out}\) and \(P_{b}\) are the output power of the generated radiation and the total power of the electron beam. \(Q_{tot}=Q_{D}Q_{\Omega}/(Q_{D}+Q_{\Omega})\) is the total quality factor of the resonator, and \(Q_{D}\), \(Q_{\Omega}\) are the diffractive and ohmic quality factors, respectively. Here, \(\eta_{r}\) is the so-called circuit efficiency:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]9  
One design goal is always to keep the \(Q_{D}/Q_{\Omega}\) ratio as small as possible in order to increase the circuit efficiency (and thus the total efficiency) and reduce the thermal losses at the cavity walls. As follows from Equation (84) for \(Q_{\Omega}>>Q_{D}\), the circuit efficiency approaches unity.