## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM
In general, for CARM interaction (when \(\beta_{\rm ph}\to 1\)), the constants of motion for the TE and TM modes coincide, and we have \(I^{\rm TE}=\gamma(1-\beta_{z})=I^{\rm TM}\). The above considerations could be \"translated\" in the language of quantum physics. For a single electron emitting a photon, the changes in its energy and axial momentum are \(\Delta E=\hbar\omega\) and \(\Delta p_{z}=\hbar k_{z}\), respectively (\(\hbar\) being [PERSON]'s constant), and therefore, \(\frac{\Delta E}{\Delta p_{z}}=\frac{\omega}{k_{z}}=c\beta_{\rm ph}\). Thus, when an electron is decelerated in a forward wave (\(k_{z}>0\)), its axial momentum decreases. On the contrary, when it is decelerated in a backward propagating wave, \(p_{z}\) increases. Near the cut-off frequency, where \(\beta_{ph}\to 0\), the momentum does not change. In the case of \(\beta_{ph}=1\) (as follows from (31) and (36)), momentum and energy change in such a way that \(\gamma(1-\beta_{z})=\text{const}\). Besides the importance of these invariants for the existence of CAR, they also provide a deeper insight on the relation between the physical parameters affecting the beam-wave interaction and allow some important properties of electron motion to be deduced. For example, using the constants of motion, it is possible and instructive to express the axial and the transverse velocities of the electron as functions of the energy only (or \(\gamma\)). For the TE modes, the relations are as follows:Figure 11.7  
In addition, analogously, for TM modes, we have the following:  
Figure 11.8  
Another observation, which gives physical insight, is that these constants of motion are satisfied even for a completely spent electron beam, i.e., when \(\gamma=1\) and \(\beta_{z}=0\) provided that the initial parameters satisfy the following conditions:  
Figure 11.9  
for TE modes, and  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]0  
for TM modes. It should be mentioned, however, that the latter conditions are necessary but not sufficient in order to extract the whole energy of the electron beam. During the interaction, the energy of an electron:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]1  
changes according to the following equation:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]2  
where (as above) \(p_{\perp}\), \(p_{z}\), \(v_{\perp}\), and \(v_{z}\) are the transverse and axial moments and velocities, respectively. From this equation and the autoresonance integral for the TE modes, it follows that the following holds:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]3  
In the case of gyrotron operation, \(1/\beta_{\rm ph}\ll 1\), and therefore, the second term can be neglected. However, in the case of CARM operation, energy extraction from the axial momentum also takes place. In general, the ratio between the second and the first term in Equation (43) is as follows:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]4  
where the factor \(\Gamma^{\rm TE}=\frac{\omega}{\Omega_{\rm CH}}=\left[\gamma\left(1-\frac{ \beta_{z}}{\beta_{\rm ph}}\right)\right]^{-1}\)gives the ratio between the upshifted resonant frequency and the nonrelativistic cyclotron frequency. Analogously, for TM modes, we have the following:  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]5  
and  
\[\frac{d}{dt}(m_{0}\gamma v_{z})=-e(\vec{v}_{\perp}\times\vec{B}_{\perp})_{z} \tag{22}\]6where \(\Gamma^{\text{TM}}=\left[\gamma\Big{(}1-\frac{\beta_{z}}{p_{\text{gr}}}\Big{)} \right]^{-1}\). Note that these relations are similar to (44), but everywhere the phase velocity is replaced by the group velocity of the wave in order to preserve their structure. Of course, they can also be written using the phase velocity and the relation \(\beta_{\text{ph}}\beta_{\text{gr}}=1\).