## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Beam-Wave Synchronism and Brillouin Diagrams  
In fast-wave gyro-devices, a helical electron beam interacts with an electromagnetic wave excited in an electrodynamical system, which is a cavity resonator in the case of gyrotrons and gyro-klystrons or a waveguide structure as is in the case of gyro-TWT or gyro-BWO. The cyclotron resonance takes place when the waveguide mode is properly synchronized with FCW. The possible operating points, where such synchronism exists, correspond to the intersections of the beam line (dispersion equation of FCW):  
\[\omega=\Omega_{\text{CH}}+v_{z}k_{z} \tag{10}\]  
and the dispersion curve of the waveguide mode:  
Figure 4: Normalized phase velocity (\(\beta_{\text{ph}}=v_{\text{ph}}/c\)) of slow and fast cyclotron waves. \[\omega^{2}=\omega_{c}^{2}+c^{2}k_{z}^{2}. \tag{11}\]  
Their coordinates on the dispersion (Brillouin) diagram (see Figure 5) are given by the following:  
\[k_{z_{+,-}}=\frac{\Omega_{\rm CH}\gamma_{z}^{2}}{c}\Bigg{(}\beta_{z}\pm\sqrt{1- \frac{\omega_{c}^{2}}{\gamma_{z}^{2}\Omega_{\rm CH}^{2}}}\Bigg{)}, \tag{12}\]  
\[\omega_{+,-}=\Omega_{\rm CH}\gamma_{z}^{2}\Bigg{(}1\pm\beta_{z}\sqrt{1-\frac{ \omega_{c}^{2}}{\gamma_{z}^{2}\Omega_{\rm CH}^{2}}}\Bigg{)}. \tag{13}\]  
Here, \(\beta_{z}=v_{z}/c\) is the axial velocity of the beam electrons \(v_{z}\), normalized to the speed of light in vacuum \(c\), and \(\gamma_{z}=\frac{1}{\sqrt{1-\beta_{z}^{2}}}\). An important parameter in these equations, which is sometimes called \"waveguide parameter\" is as follows [27]:  
\[\Psi=\frac{\omega_{c}^{2}}{\gamma_{z}^{2}\Omega_{\rm CH}^{2}}. \tag{14}\]  
Obviously, the case \(\Psi>1/\gamma_{z}^{2}\) (and thus \(\omega_{c}>\Omega_{\rm CH}\)) corresponds to a forward operation which we consider first. When \(\frac{1}{\gamma_{z}^{2}}<\Psi<1\), there are two intersection points, as illustrated in Figure 5. The CARM interaction takes place at the upshifted frequency \(\omega_{+}\), while the intersection at the lower frequency \(\omega_{-}\) corresponds to a Gyro-TWT or gyrotron operation. In the latter case, the term of the Doppler shift is small, and therefore, the gyrotrons operate at frequencies close to the cyclotron frequency or its harmonics (\(\omega\approx\Omega_{\rm CH}=s\Omega_{\rm C}\)). A special case is the so-called _grazing condition_ (coalescence between the field and the beam), shown in Figure 6. It is realized when \(\Psi=1\), i.e., when the following holds:  
\[\omega_{c}=\gamma_{z}\Omega_{\rm CH}. \tag{15}\]  
In this particular case, the coordinates of the intersection point are as follows:  
Figure 5: Brillouin diagram of the beam–wave synchronism at resonance. Figure 4:0  
Note that, at the grazing intersection (which is also called _group synchronism_) of the dispersion relation of the cyclotron resonance mode and the transverse waveguide/cavity mode, the velocity of the electrons is also equal to the group velocity, i.e., \(v_{z}=v_{gr}=\frac{c^{2}}{v_{\rm ph}}\). Figure 7 shows how the normalized frequencies at the lower and upper intersections of the Brillouin diagram change with increasing the magnetic field intensity (and thus \(\Omega_{\rm CH}\)). At \(\Psi=\frac{1}{\gamma_{z}^{2}}\), \(\omega=\omega_{c}=\Omega_{\rm CH}\) and the beam line passes through the vertex of the hyperbola of the waveguide mode. If \(\Psi<\frac{1}{\gamma_{z}^{2}}\), however (when \(\Omega_{\rm CH}>\omega_{c}\)), \(k_{z-}<0\) and the electron beam interacts with a backward propagating wave (see Figure 8) for which both the group and phase velocity are negative. For a given mode, the operating point can be controlled by varying separately or simultaneously two parameters of the electron beam, namely, the axial velocity \(v_{z}\) and the harmonic cyclotron frequency \(\Omega_{\rm CH}\). Varying \(v_{z}\), the slope of the beam line is changed, while by varying \(\Omega_{\rm CH}\), the beam line is translated parallel to itself.