## 2 Electron Cyclotron Resonance, Cyclotron Autoresonance, and Brillouin Diagrams
### Cyclotron Autoresonance and CARM
Using a polar coordinate system \((r,\phi,z)\) and complex notations:  
\[E_{\perp}=E_{r}+iE_{\phi}\quad;\quad H_{\perp}=H_{r}+iH_{\phi}\quad, \tag{25}\]  
the relations between the azimuthal and radial components of the electric and magnetic fields (marked by indices \(\phi\) and \(r\), respectively) can be written in the following scalar form:  
\[Z=\frac{E_{r}}{H_{\phi}}=-\frac{E_{\phi}}{H_{r}}. \tag{26}\]  
The impedances of the TE and TM modes are given by the following relations:  
\[Z^{TE}=Z_{0}\frac{k}{k_{z}}=Z_{0}\frac{v_{\rm ph}}{c}=\mu_{0}v_{\rm ph}\quad, \tag{27}\]  
\[Z^{TM}=Z_{0}\frac{k_{z}}{k}=Z_{0}\frac{c}{v_{\rm ph}}=\frac{\mu_{0}c^{2}}{v_{ \rm ph}}\quad, \tag{28}\]  
where \(Z_{0}=\sqrt{\mu_{0}/\epsilon_{0}}\) is the characteristic impedance of the free space, \(\epsilon_{0}\) is permittivity of vacuum, and the following:  
Figure 11.0  
is the phase velocity of the waveguide mode (compare with the phase velocity of FCW given in Equation (7)). For the case of TE modes, using Equations (24) and (27), we express the right-hand side of (22) in terms of the transverse electric field and rewrite this equation in the following form:  
Figure 11.1  
Canceling the term \(\vec{v}_{\perp}\cdot\vec{E}_{\perp}\) in (23) and (30), we obtain immediately the following important integral (an exact constant of motion):  
Figure 11.2  
where \(\gamma_{0}\) and \(\beta_{z0}\) are the initial values of \(\gamma\) and \(\beta_{z}\). If \(v_{ph}\gg 1\), as is the case with gyrotrons operating at frequencies near the cut-off frequency of the resonant cavity (when \(k\approx k_{\perp}\)), this invariant reduces approximately to the conservation of the axial momentum, i.e., \(p_{z}=m_{0}\gamma v_{z}={\rm const}\). The above constant of motion applies both to TE modes in waveguides or cavity resonators and to TEM waves in free space. In the latter case, the analysis is analogous but takes into account that the phase velocity of TEM waves is equal to the speed of light in vacuum. Next, we consider electron motion in the field of TM modes. In addition to the difference in wave impedance, TE and TE modes also differ in that the TM waves have an axial component of the electric field \(E_{z}\), and instead of Equations (22) and  
Physically, this means that the axial momentum changes not only as a result of the recoil effect, but also due to the direct interaction with the axial component of the electric field of the TM wave. Following the same approach as before, we express the transverse magnetic field of the TM mode through the corresponding transverse electric field using (24) and (28). In such a way, we rewrite (32) in the following form:  
Figure 11.3  
From the comparison of (33) and (34), we obtain the following:  
Figure 11.4  
It is easy to see that the right-hand side of this equation is small and even disappears altogether at grazing incidence (group synchronism). Moreover, the axial electric field (as seen by an electron propagating with an axial velocity significantly different from the phase velocity of the wave) is a rapidly changing time-harmonic function, and as shown in [60], its contribution averaged over the period is zero. Therefore, from (34), we obtain the following _approximate_ constant of motion (invariant):  
Figure 11.5  
The invariants derived above (also known as autoresonance integrals) can be represented in the following unified (general) form, which is valid to both types of modes:  
Figure 11.6  
where \(Z=Z^{\rm TE}\) or \(Z=Z^{\rm TM}\), but bearing in mind that, in the first case, it is an exact constant, while in the second case, it is an approximate constant. Returning to the resonance condition (20), one can see that the denominator coincides with the constant of motion for TE modes only if \(\beta_{\rm ph}=1\) (when \(I^{\rm TE}=\gamma(1-\beta_{z})=\text{const}\)) and hence \(\omega=\frac{\delta\Omega_{c}}{\gamma(1-\beta_{z})}\). In contrast, for the TM modes, the synchronism is observed for any phase velocity and \(\omega=\frac{s\Omega_{c}}{\gamma(1-\frac{\beta_{z}}{\beta_{\rm ph}})}\).