## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Transverse Waves in Electron Beams  
In gyro-devices, helical electron beams are used in which the electrons gyrate in a strong magnetic field \(B\) with a relativistic cyclotron frequency \(\Omega_{C}\) given by the following:  
\[\Omega_{C}=\frac{\rm eB}{\gamma m_{0}}, \tag{5}\]  
Figure 2: Dispersion diagram of a hollow smooth-wall metallic waveguide.  
where \(e\) and \(m_{0}\) are the charge and rest mass of an electron. Here, \(\gamma\) is the relativistic Lorentz factor \(\gamma=\left(1-\frac{v^{2}}{c^{2}}\right)^{-1/2}=\left(1-\frac{v_{z}^{2}+v_{ \perp}^{2}}{c^{2}}\right)^{-1/2}\), and \(v_{z}\) and \(v_{\perp}\) are the axial and transverse components of the total velocity \(v\) of the electron, respectively.  
An electron beam of gyrating electrons drifting in the presence of an axial magnetic field supports two space-charge waves and two transverse cyclotron waves propagating along the beam axis [23; 24; 25]. Here, we consider only the latter since the former are not related to gyro-devices. The dispersion characteristics of the cyclotron waves (see Figure 3) are defined by the following:  
\[\omega=\mp\Omega_{\mathrm{C}}+v_{z}k_{z}. \tag{6}\]  
Here, the minus and plus signs correspond to SCWs (slow cyclotron waves) and FCWs (fast cyclotron waves), respectively. Their phase velocities depend on the axial velocity of beam electrons and on the angular cyclotron frequency and are given by the following:  
\[v_{\mathrm{ph}}=\frac{v_{z}}{1\pm\frac{\Omega_{\mathrm{C}}}{\omega}}, \tag{7}\]  
as shown in Figure 4. Therefore, the phase velocity of FCW can be greater than the axial velocity of the electron beam and, in the case of cyclotron resonance (\(\omega=\Omega_{c}\)), can even become infinite. Moreover, depending on the direction (sign) of the phase velocity, FCW can be either forward or backward propagating. In contrast, the phase velocity of SCW is always smaller than the axial velocity of the beam electrons and positive, which means that it propagates in the forward direction.  
The frequency \(\omega\) defined by the dispersion Equation (6) for FCW can also be considered as a Doppler-shifted cyclotron frequency of the gyrating beam electrons or one of its harmonics. In order to investigate the most general cases, from now on, in the derivations that follow, we will use \(\Omega_{\mathrm{CH}}=s\Omega_{\mathrm{C}}\)\((s=1,2,3,\ldots)\), i.e., the harmonics of \(\Omega_{c}\) rather than the fundamental cyclotron frequency \(\Omega_{\mathrm{C}}\). Before we proceed to that, however, it is important to mention that, here, we are considering an operation with the _normal Doppler effect_ for which the frequency \(\omega\) is larger than the Doppler term \(v_{z}k_{z}\) and the resonance condition (6) is satisfied for positive harmonic numbers. Contrary to this, in cyclotron resonance devices based on the _anomalous Doppler effect_[26], \(v_{z}k_{z}>\omega\) and the resonance  
Figure 3: Dispersion characteristics of fast and slow transverse cyclotron waves in beams of electrons gyrating with cyclotron frequency \(\Omega_{\mathrm{C}}\).  
condition occurs for negative harmonic numbers. Such a situation takes place if the phase velocity of the wave is smaller than the axial velocity of the electrons, i.e., in devices with slow-wave structures, which are beyond the scope of this paper.  
For the subsequent considerations, it would be instructive to recall that, at the beam-wave synchronism of the electromagnetic field with time dependence \(\sim\)\(e^{i\omega t}\), the phase of an electron with respect to the operating mode is as follows:  
\[\theta=\omega t-k_{z}z-s\varphi=\text{const}, \tag{8}\]  
where \(\varphi\) is the angular coordinate of the electron. The differentiation of this equation with respect to time gives the following:  
\[\omega-k_{z}v_{z}-s\varphi=0, \tag{9}\]  
which, taking into account that \(\phi=\Omega_{c}\), coincides exactly with the electron cyclotron resonance condition (6) written there for \(s=1\).