# Radio-Frequency Undulators, Cyclotron Auto Resonance Maser and Free Electron Lasers
## 2 CARM as a Source of RF Undulator
The interaction can be viewed as an intra waveguide Compton backscattering, the associated frequency up-shift is written as  
\[\omega=s\Omega_{CH}+k_{z}v_{z}. \tag{16}\]  
and the matching to the waveguide conditions is ensured by the coupling with the dispersion relation  
\[\omega^{2}=\omega_{c}^{2}+c^{2}k_{z}^{2} \tag{17}\]  
where \(s\) and \(\omega_{c}\) are the harmonic index and cutoff frequency (see below), respectively. Unlike the free space FEL, the intersection between the curves of Equations (16) and (17) admits (see Figure 7, for a geometric interpretation in the Brillouin \((\omega,k)\) space) two solutions, corresponding to gyrotron (lower frequency) and CARM (higher frequency) operations modes  
\[ \begin{array}{rcl}k_{z_{+,-}}&=&\frac{s\Omega_{CH}\gamma_{z}^{2}}{c}\left( \beta_{z}\pm\sqrt{1-\Psi}\right),\\ \\ \omega_{+,-}&=&s\Omega_{CH}\gamma_{z}^{2}\left(1\pm\beta_{z}\sqrt{1-\Psi}\right),\end{array}  \tag{18}\]  
being  
\[\Psi=\frac{\omega_{c}^{2}}{s^{2}\gamma_{z}^{2}\Omega_{CH}^{2}} \tag{19}\]  
The cutoff frequency in Equation (18) (\(\omega_{c}=c\chi_{m,n}/R\)) is defined in terms of the eigenvalues \(\chi_{m,n}\) of the wavenumber characterizing the \(TE_{m,n}\) mode excited by the FEL CARM interaction. The relevant physical role is that of controlling the \"nature\" of the roots in Equations (18), determined by the intersection of the curves of Equations (16) and (17). The \"wave parameter\" \(\Psi\) is lower than 1, in order to ensure two distinct intersections and is assumed to be greater than \(1/\gamma_{z}^{2}\), to avoid a backward wave root. Before introducing more specific details, let us further comment on the presence of both gyrotron and CARM modes. This is a distinctive feature with respect to the magnetic undulator configuration. In principle, both modes have a chance to grow. Particular care should therefore be devoted to the suppression of the \(\omega_{-}\) counterpart. The theoretical analysis of CARM has been developed in the past in a number of authoritative papers [32; 33; 35; 36; 37; 38; 39; 40; 41] and will not be reported here. We note simply that the set of equations ruling the CARM interaction, even though characterized by a more complicated phenomenology than the ordinary magnetic FEL, can be viewed in a way not dissimilar from the pendulum-like equations, adopted in the description of FEL undulator devices. Figure 7: Brillouin diagram for the different conditions of electron cyclotron resonance selections corresponding to the intersections of the beam line (straight line) with the dispersion curve of the operating cavity mode (dotted line). +++ ==ERROR: No output for this page==
In Figure 8 we report the growth of the CARM power for the set of parameters specified in the caption. We have included a comparison with the (analytical) small signal approximation (SSA), obtained in [42]. The power growth exhibits the same S-shaped logistic behavior of high-gain SASE FEL devices (see the superimposed dot-dashed curve), which is analytically reproduced (till the onset of the saturation) by  
\[P_{sat}(z)=P_{0}\frac{\overline{P}(z)}{1+\frac{P_{0}}{P_{F}}\big{(}\overline{ P}(z)-1\big{)}}\] (20)  
where \(\overline{P}(z)=P_{l}(z)/P_{0}\) with \(P_{l}(z)\) being the analytical linear solution of the growth signal, \(P_{0}\)--the input signal power and \(P_{F}\) the final CARM power given by  
\[P_{F}=\eta P_{beam}\] (21)  
being \(\eta\) the efficiency of the device, which will be commented on later in this section. The inspection of Figure 8 confirms that the power growth consists of three distinct phases:  
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Figure 10 shows the associated power growth for two frequencies (153-154) GHz close to the CARM resonance (\(\omega_{+}\)), which displays a maximum output power of 4.8-6.4 MW and thus an efficiency of 19-25%, respectively.