# Radio-Frequency Undulators, Cyclotron Auto Resonance Maser and Free Electron Lasers
## 2 CARM as a Source of RF Undulator  
The CARM is a Free Electron device representing the transition element between microwave tubes and FELs, as nowadays conceived. The key elements to understand the CARM FEL dynamics are sketched out in Figure 6, which displays a moderately relativistic e-beam, with a relativistic factor \(\gamma\), moving inside the waveguide of a resonant cavity, along an axial static magnetic field \(B_{0}\), executing helical trajectories with a period  
\[\Lambda_{CH}=\frac{2\pi c}{\Omega_{CH}} \tag{9}\]  
where \(\Omega_{CH}=eB_{0}/(\gamma m_{e})\), the cyclotron relativistic frequency. Figure 6: Geometry of the CARM interaction. The \"behavioural\" paradigm of CARM is the same as for any free electron radiation generator, namely, the electrons are expected to lose energy in favor of a selected operating mode, if appropriate matching conditions are satisfied. The way in which the energy exchange occurs is traced back to the standard mechanisms of the e-beam energy modulation and bunching. The last effect is characterized by transverse and longitudinal contributions. Going back to Figure 6, we note that the underlying dynamics can be described as follows. The electrons with longitudinal velocity \(v_{z}\) are propagating in a longitudinal magnetic field in the presence of a co-propagating electromagnetic field characterized by a wavevector \(k_{z}\) which in terms of frequency and phase velocity (\(v_{p}\)) reads  
\[k_{z}=\frac{\omega}{v_{p}}. \tag{10}\]  
We can now establish an analogy between CARM and magnetic undulator FEL (U-FEL) using only kinematics arguments. The electron velocity inside the cavity is specified by its longitudinal and vertical components, linked to the relativistic factor by  
\[ \begin{split}\beta_{z}^{2}+\beta_{\perp}^{2}=1-\frac{1}{\gamma^ {2}}\\ \beta_{z,\perp}=\frac{v_{z,\perp}}{c},\end{split}  \tag{11}\]  
and  
\[\gamma_{z}=\frac{1}{\sqrt{1-\beta_{z}^{2}}}=\frac{\gamma}{\sqrt{1+(\gamma \beta_{\perp})^{2}}}. \tag{12}\]  
The electron and the radiation move, inside the cavity, at different speeds. We expect that after each helical path the following slippage is accumulated  
\[\delta=\big{(}v_{p}-v_{z}\big{)}\frac{\Lambda_{CH}}{c}. \tag{13}\]  
This is the phase advance of the electromagnetic wave after each helical path with respect to the electrons. Constructive interference of the wavefront of the emitted radiation at the next period is ensured if  
\[\delta=\lambda \tag{14}\]  
with \(\lambda\) being the wavelength of the field propagating with the electrons. Putting together Equations (9), (14) and (11), we end up with the matching condition. The transverse component of the velocity \(\beta_{\perp}\) is the key parameter allowing the coupling with the wave transverse field. In terms of the analogy we are suggesting it plays the same role as the undulator strength parameter being \(K\equiv\gamma\beta_{\perp}\). In the relativistic regime and assuming that \(v_{p}\simeq c\) Equation (13) reduces to  
\[\delta\cong(1-\beta_{z})\Lambda_{CH}\simeq\frac{\Lambda_{CH}}{2\gamma^{2}}(1+ K^{2}) \tag{15}\]  
where the trajectory helical path is understood to play the same role of the undulator period (for further comments see [18; 34]). The Equation (15), derived under the assumption of (ultra) relativistic regime, provides a brief idea of how CARM and U-FEL can be viewed within a common framework. In the following we will develop a more appropriate treatment valid for the nonrelativistic regime. The discussion might be misleading if not properly commented. Therefore, we underscore that the interaction occurs in a waveguide, whose dispersion relation needs to be included to derive the matching conditions, specifying the CARM operating wave length.