# Radio-Frequency Undulators, Cyclotron Auto Resonance Maser and Free Electron Lasers
## 3 CARM, RFU and VUV Soft X-ray FEL Operation
It requires a careful design of the radiation transport in the corrugated waveguide, that follows the \"active\" waveguide where the CARM amplification and growth occurs. The physical mechanism beyond Equation (23) is fairly straightforward; the helically corrugated waveguide is designed with a dispersion relation ensuring a significant dependence of the group velocity on the frequency, accordingly, if a pulse is modulated from one frequency to a frequency with a higher group velocity the pulse will be compressed. The simulations and the experimental results of the aforementioned paper [46; 47] report compression ratios around 25, for tens of GHz RF with a bandwidth of 5%. In accordance with our simulation the bandwidth of the RF generated by the CARM amplifier is sufficiently large (around 5.3%), and can be be eventually increased by using an input seed in the CARM cavity with larger bandwidth in order to compensate any major losses in the compression factor. Finally, the cavity length must be adjusted to contain the highest flat power of the signal to be compressed. This could be a limitation for the pulse length which in principle cannot be more than hundreds of ns for a reasonable cavity length (a couple of meters assuming a group velocity value 1/10 of light speed). A long CARM RF pulse (\(\sim\) us) can be also used by modulating the input signal in the CARM cavity in order to produce a conveniently chopped signal. Figure 14: Electron beam energy (**a**), electron beam current density (**b**), the electron beam transverse section (**c**) and the peak current (**d**) vs. the RF wavelength \(\lambda_{w}\) for two FEL radiation \(\lambda_{r}=10\) nm (continuous line) and \(\lambda_{r}=20\) nm (dashed line). According to our preliminary calculations, we can foresee an analogous behavior for RF CARM around 150 GHz; we have been conservative regarding the maximum CARM output power and we would like to underscore that even a factor two larger is within the realm of the present technology. Considering a reduction of the pulse duration by two orders of magnitude and assuming a power loss of 50% we can foresee an amount of power, available for wave-undulator operation, 250 MW. The required power density is accordingly obtained by confining the RFU over a surface of \(0.7\sim 1\) mm radius (reasonably large to transport radiation in the wavelength region of our interest). The next step is to specify how the RF transport cavity should be designed. The conclusion we may draw, from the inspection of the plots, is that the CARM RFU FEL is conceivable, at least in the soft X-ray region. The RFU power should be confined over a small surface to ensure the required density power level, warranting \(K_{uv}\) values suitable for FEL SASE operation. The following analysis is limited to RF wavelength within 1-3 mm. The requests on the electron beam are not particularly challenging, with respect to those of the RFU power transport waveguide, which according to the prescription of ref. [24] requires the selection of \(TE-TM\) modes combination, characterized by a small transverse section, ensuring the required large intensity. Regarding the electron-radiation interaction schemes, several solutions have been foreseen, as illustrated in Figure 15 where the \"standard\" RFU standing wave undulator is reported on the left, while the flying undulator concepts on the right with phase and group velocity are directed in the opposite electron direction. There is a natural drawback affecting the counter propagating scheme, namely the effective interaction length. If the RF wave propagates inside the guide with group velocity \(v_{g}\) and if the RFU pulse has a duration \(\tau\), the effective interaction length is [24]  
\[L_{eff}=\frac{v_{g}\tau}{1+\frac{v_{g}}{c}} \tag{24}\]  
for \(v_{g}\approx c\). In other words, the effective length becomes a factor two lower than the pulse length itself, which means only half of the undulator is exploited. On the other side, the \"co-propagating\" scheme in Figure 16 (left) does not suffer the same problem.