## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM
and reads as follows:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]8  
This being performed, Equation (52) yields the following:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]9  
and therefore, the canonical angular momentum \(p_{\varphi}\) is conserved, i.e.,  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]0  
Even a more elegant, concise, and insightful derivation of the above invariant can be performed by simply analyzing the Hamiltonian equations of motion in a cylindrical coordinate system \((r,\ \varphi,z)\):  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]1  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]2  
where the Hamiltonian \(H\) for axially symmetric fields (\(\frac{\partial}{\partial\varphi}=0\)) is as follows:  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]3  
and \(U\) is the axially symmetrical electrostatic potential. By virtue of the fact that this Hamiltonian does not depend explicitly on \(\varphi\), we have the following:  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]4  
which proves that the canonical angular momentum is an invariant (constant of motion). At the same time, since the following holds:  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]5 On the conservation of the angular momentum and formulation of [PERSON]’s theorem: forces acting on one electron. \[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]6  
it follows that \(p_{\varphi}=\gamma m_{0}r^{2}\dot{\varphi}-er\,A_{\varphi}\), which, combined with (61), exactly coincides with (57). Another invariant is the Hamiltonian itself, which is equal to the total energy of an electron and is explicitly independent of time, i.e., \(H=\text{const}\) since \(\frac{dH}{dt}=\frac{\partial H}{\partial t}=0\). In a paraxial approximation, \(A_{\varphi}\approx\frac{1}{2}B_{0}(z)r\), and we have the following:  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]7  
Then, considering the point P (see Figure 15) on the orbit for which \(r=r_{g}+r_{L}\), \(v_{r}=0\), and \(v_{\varphi}=v_{\perp}=\Omega_{C}r_{L}\)(recall than \({v_{\perp}}^{2}={v_{r}}^{2}+{v_{\varphi}}^{2}\)), we find the following:  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]8  
Applying (63) to the cathode with the radius \(r_{c}\), where the magnetic field is \(B_{c}\) and \(\dot{\varphi}=0\), we obtain \(B_{c}{r_{c}}^{2}=B_{0}({r_{g}}^{2}-{r_{L}}^{2})\). In electron optics, neglecting the relativistic dependence of the mass of electrons, the above invariant is usually written in the following form:  
\[Z\vec{H}_{\perp}=\vec{z}_{\perp}\times\vec{E}_{\perp}, \tag{24}\]9  
and is known as [PERSON]'s theorem. Specifying the initial values at time \(t=0\) as \(r=r_{0}\), \(\varphi=\varphi_{0}\), and \(\Phi=\Phi_{0}\), respectively, gives the following:  
Figure 11:0  
Figure 11:1 Schematic of the kinematics of a gyrating electron. Therefore, [PERSON]'s theorem states that the change in angular velocity (an thus in angular momentum) between two axial positions depends only on the change in the total magnetic flux but is independent of the electron trajectory between these cross sections. The generalized formulation of [PERSON]'s theorem derived in [65] has the following form:  
Figure 11:2  
Here, the path integral of the stream of electrons with velocities \(\vec{v}\) is taken along a closed contour C confining a fixed set of possible particle trajectories, and \(\Phi\) is the magnetic flux through the area enclosed by C. This generalization extends the theorem to many particles in an electron beam without axial symmetry. It is important to underline that one should account for the directions of the magnetic fluxes.