## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM
In both cases, the electron beams (either polihelical or uniaxial) then propagate in an increasing magnetic field where the transverse velocity of the electrons gradually increases at the expense of their axial velocity. This process is habitually called \"pumping\" since the energy related to the transverse motion is being transferred to the electromagnetic field during the beam-wave interaction that takes place in the electrodynamical system of gyrotrons (here we do not consider CARM in which, as was shown, energy is extracted also from the axial momentum). Before going into detail, we introduce some basic relationships and notations that will be used below. The gyrating electrons move in the transverse plane on circular orbits with a cyclotron angular velocity \(\Omega_{C}\) and drift in the axial direction on helical trajectories with a pitch \(l=T_{c}v_{z}\), where \(T_{c}=2\pi/\,\Omega_{C}\) is the cyclotron period. The instantaneous radius of the circular orbit (also called Larmor radius) is \(r_{L}=v_{\perp}/\Omega_{C}\). As mentioned earlier, the relativistic dependence of \(\Omega_{C}\) on the energy (\(\gamma\)) of the electrons forming the beam leads to their nonisochronous motion and is responsible for one of the most essential mechanisms, namely, azimuthal bunching, which takes place at a cyclotron resonance as described above. \[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]1 Types of helical electron beams used in gyro-devices: (**a**) polihelical electron beam of a conventional gyrotron; (**b**) uniaxial (axis-encircling) electron beam of a large orbit gyrotron. (\(r_{c}\)—cavity radius, \(r_{g}\)—guiding center radius, \(r_{L}\)—Larmor radius). In a cylindrical coordinate system (\(r\), \(\varphi\), z), the relativistic equation of motion of an electron under the action of an electric field \(\vec{E}\) and a magnetic field \(\vec{B}\) is as follows:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]2  
Taking the vector product of both sides with \(\vec{r}\) and considering only the axial projection (component), we readily obtain the following:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]3  
The forces acting on the electron are shown in \[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]4 In an axially symmetric EOS, the axial \(B_{z}\) and the radial \(B_{r}\) components of the static magnetic field \(\vec{B}=\mathrm{curl}(\vec{A})\) can be expressed through the azimuthal component \(A_{\varphi}\) of the vector potential \(\vec{A}\) (since, for such symmetry, \(A_{r}=A_{z}=0\) and \(B_{\varphi}=0\)) as follows:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]5  
Furthermore, by virtue of the assumed symmetry, the azimuthal component of the static electric field in the EOS is absent. However, the change of the magnetic flux \(\Phi=2\pi rA_{\varphi}\) through a circle of radius \(r\) and centered on the axis gives rise to an azimuthal electric field component that can be obtained from [PERSON]'s law:  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]6  
\[\frac{d\gamma}{dt}=-\frac{e}{m_{0}c^{2}}\vec{v}_{\perp}\cdot\vec{E}_{\perp} \tag{23}\]7 Nominator of the coupling factor \(G\) for different values of \(m-s\).