## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM
More specifically, if a reverse of the magnetic field is present between the cross sections where \(\Phi\) and \(\Phi_{0}\) are considered, then the following holds:  
Figure 11:3  
Therefore, in the latter case, the gyrating electrons acquire an additional \"twisting\". It is easy to show that the conservation of the canonical angular momentum in the presence of magnetic field reversal leads to the formation of an axis-encircling electron beam. Consider Equation (64) at two cross sections, namely, before and after the reversal (cusp), marked by indices 1 and 2:  
Figure 11:4  
If before the cusp, \(r_{g1}>r_{L1}\), then after the cusp, where the magnetic field changes its direction (from \(B_{1}\) to \(-B_{2}\)), \(r_{L2}>r_{g2}\) is required, which means that the electron encircles the axis (see Figure 12b). Moreover, if \(B_{1}=|B_{2}|\), the guiding center radius before the cusp becomes the Latmor radius after it, and the guiding center radius becomes the Larmor radius. The formation of an axis-encircling electron beam due to the reverse of the magnetic field is illustrated in Figure 16 for both an ideal and a realistic magnetic cusp [66; 67; 68]. If an electron is moving in the positive z direction at radius \(r_{i}\) (injection radius) in a uniform magnetic field upstream (z < 0) of a magnetic cusp, the conservation of kinetic energy and the canonical angular momentum dictate that, after the cusp transition, the Larmor radius and guiding center radius are as follows:  
Figure 11:5  
Figure 11:6  
In the considered ideal (step-function) cusp, the magnetic field changes its value abruptly from \(B_{1}\) to \(-B_{2}\). If \(B_{1}=-B_{2}\) (balanced cusp), we have \(r_{L2}=r_{g1}\) and \(r_{g2}=0\), resulting in circular electron orbits centered on the axis after the cusp. However, for any nonideal cusp (where the magnetic field does not change abruptly), the orbits have a finite radius of the guiding center; i.e., they are off-centered. This can be illustrated (see Figure 16b) by considering a balanced magnetic cusp (\(B_{1}=-B_{2}\)) with a finite transition and magnetic field distribution:  
Figure 11:7For such a cusp [66], the minimum downstream guiding center radius is as follows:  
Figure 11:8  
where \(\Omega=eB_{2}/m_{0}\) and \(\xi\) is a measure of the width of the field transition. The presence of a minimum guiding center radius is called \"coherent off-centering\" [66], since all gyrating electrons perform such an orbit in phase, resulting in a rippled (scalloped) beam envelope downstream of the cusp with a scallop length:  
Figure 11:9  
Another important constant of the electron motion in adiabatic approximation is related to the orbital degree of freedom and is thus called the orbital (transversal) adiabatic invariant, which reads as follows:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]0  
where \(p_{\perp}=mv_{\perp}\) is the oscillatory electron momentum, and reduces to \(I_{\perp}=v_{\perp}^{2}/B=\text{const}\) for nonrelativistic motion. In the latter case, taking into account that \(v_{\perp}=\text{e}\mathds{R}_{L}B/m_{0}\), this invariant gives the relation that connects the magnetic field intensity \(B(z)\) and the Larmor  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]1 Configuration of the magnetic field and electron orbits in an ideal (**a**) and in a realistic (**b**) cusp. radius \(R_{L}(z)\) in two arbitrary cross sections, \(z_{1}\)and \(z_{2}\) (for example, at the cathode \(z_{1}=z_{e}\) and at the cavity \(z_{2}=z_{c}\)):  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]2  
This means, in particular, that as the magnetic field is increased from an initial value of \(B_{1}=B(z_{1})\) to some value \(B_{2}=B(z_{2})\), the radius of an electron orbit will be decreased as the ratio \((B_{1}/B_{2})^{1/2}\).