## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Beam-Wave Synchronism and Brillouin Diagrams
It is instructive also to show that the equations of beam lines passing through an arbitrary intersection point with coordinates (\(k_{z0}\), \(\omega_{0}\)) can be written in the following two equivalent forms:  
Figure 4:1  
or  
Figure 4:2  
Here, \(\frac{\Omega_{\rm CH}}{\gamma}\) and \(c\beta_{z}\) are parameters different for each beam line of the bundle. From the resonance condition at the intersection point, we obtain the following:  
Figure 4:3  
Figure 4:4 Brillouin diagram at grazing condition. For the following analysis (in Section 2.4), however, it is convenient to express (using Equation (10)) the frequency at the resonance as follows:  
Figure 4:5  
Here, we use the nonrelativistic cyclotron frequency \(\Omega_{CS}=eB/m_{0}\) to show explicitly the dependence of the operating frequency on the harmonic number \(s\) and the relativistic factor \(\gamma\). In the dispersion diagrams above, the waveguide modes are represented by continuous curves, which is legitimate for the continuous spectrum of the hollow metallic waveguide. However, in the cavity resonators, the spectrum is discrete, and therefore, the corresponding resonances can be represented by points lying on the hyperbola and having a discrete set of the axial wavenumbers given by the following:  
Figure 4:6  
where the axial mode index \(q\) is an integer number and \(L_{\text{ef}}\) is the effective length of the resonator. This means that resonances are possible for a sequence of high-order (\(q>1\)) forward or backward axial modes (HOAM), as illustrated in Figure 4:7 Generally, the axial mode index \(q\) corresponds to the number of the field variations along the cavity axis. The axial field profile is described by the Helmholtz equation, which is supplemented by certain boundary conditions. The analysis of the solutions of this equation leads to a more detailed picture of the resonances with finite bandwidths and is discussed in [28]. According to this  
Figure 4:8 Normalized frequencies at the lower and higher intersection points of the Brillouin diagram vs. the magnetic field normalized to its value \(B_{\text{gc}}\) at grazing condition. This illustrative example is plotted for electron beam parameters \(\beta_{z}=0.6,\ \gamma_{z}=1.25\). approach, the resonances should be represented by zones (bands) rather than points. This concept is illustrated in Figure 4:9 It has been shown (and demonstrated experimentally [29; 30; 31; 32; 33; 34; 35]) that, by exciting a sequence of successive HOAM, continuous frequency tuning can be performed simply by changing the magnetic field intensity (and hence \(\Omega_{\text{CH}}\)). One further remark is in order at this point. The FCW line in the dispersion diagrams represents an idealized monoenergetic electron beam without velocity scattering (distribution). Therefore, such line should be considered as the result of averaging, or alternatively, one should consider different lines for the different fractions of beam electrons. For simplicity, in the above illustrative examples, we have considered only the possible operating points (resonances) of a single cavity mode. In reality, however, there is a dense spectrum of nonequidistant modes in the gyrotron resonant cavity, and therefore, many intersections of the beam lines (corresponding to different harmonics) and dispersion curves of various modes are possible. This is illustrated in \[\omega^{2}=\omega_{c}^{2}+c^{2}k_{z}^{2}. \tag{11}\]0 The density of the mode spectrum (which increases with frequency) and the finite bandwidth of the resonances are responsible for _mode competition_--a severe problem in selecting a suitable mode and  
\[\omega^{2}=\omega_{c}^{2}+c^{2}k_{z}^{2}. \tag{11}\]1 Brillouin diagram for the case of gyro-backward wave interaction. \[\omega^{2}=\omega_{c}^{2}+c^{2}k_{z}^{2}. \tag{11}\]2 Discrete spectrum of the resonances (**left** plot) and schematic representation of the concept of finite-bandwidth resonances of HOAM in a gyrotron cavity (**right** plot).