# Radio-Frequency Undulators, Cyclotron Auto Resonance Maser and Free Electron Lasers
## 3 CARM, RFU and VUV Soft X-ray FEL Operation
The cavity is indeed designed with corrugated helical ripples [24] which allow the pulse train to move with group velocity pointing in the same direction of the electrons and the phase velocity opposite. The effective interaction length is therefore  
\[L_{ceff}=\frac{v_{g}\tau}{1-\frac{v_{g}}{c}} \tag{25}\]  
greater than the head-on case by a factor \((1+v_{g}/c)(1-v_{g}/c)\). It might be thought that for the co-propagating scheme, the Compton backscattering up-shift does not occur. This is not true because the waveguide structure ensures the  
Figure 15: Compton backscattering with a standing wave (**left**). Compton scattering with a short pulse (**right**). The cavity filling time (lasting few ns) can be synchronized with repetition rate of the e-beam injection. up-shift, according to what is shown in Figure 16 (right) where the effective intra-guide RF propagation is displayed. The radiation follows a kind of reflection, at each ripple, allowing multiple reflections pushing its \"center of mass\" in the forward direction. The head-on interaction occurs in the region where electrons and radiation overlaps (brown disk in Figure 16 (left)). The use of a standing wave backscattering configuration is, however, feasible, at the price of operating the device with a longer waveguide. Both configurations require the necessity of a pulse compression and the transport through a waveguide supporting hybrid mode structures, whose detailed design will be presented elsewhere. The solution foreseen in Figure 16 seems to be more appealing for the architecture we consider. In Figure 17 we report a sketch of the whole device, indicating the RF CARM injected inside the corrugated waveguide (with a mode converter/mirror) and after the compression delivered in the second corrugated waveguide where the FEL SASE interaction occurs. In this paper, we consider the possibility of operating a FEL SASE device using a RFU pumped by a CARM device. The microwave CARM is a nearly mature technology, useful to develop \"compact\" FELs operating in the VUV and soft X-ray region. The intrinsic limitations of these devices are, as previously underscored, associated with the maximum CARM available power (limiting the \(K_{w}\) strength parameter) and with the physical dimensions of the radiation transport vacuum pipe radius, limiting the cutoff frequency of the propagated radiation. The combination of these two limiting factors has indicated a threshold of 10 nm as the minimum operating wavelength. It should  
Figure 16: Mirror model of the intra-guide RFU propagation (**left**) and helical ripple waveguide and flying RFU pulse (**right**). Figure 17: CARM-Undulator, SASE-FEL layout. Lower part: CARM generator and radiation transport system. Upper part: consecutive helical waveguides, the first acts as RF power compressor, the second, feeded with an external e-beam, realizes the SASE interaction region. furthermore be stressed that, although we have indicated a pulse compression method as a useful device to enhance the microwave power, we did not underscore enough that it is still at the level of experimental studies, not yet conducted in the frequency range above 100 GHz. This solution, if viable, would provide a real step towards the realization of compact X-ray FEL devices. The construction of this type of short FEL wavelength device is a challenging promising task, which foresees the use and merging of high-level technologies, within the present or next future capabilities. The E.D.P., S.C., G.L.R., E.S., I.S. and G.D. made the same contributions in terms of conceptualization, writing, data control and critical reading. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The Authors like to express their sincere appreciation to an anonymous Referee, whose challenging comments improved the quality of the paper. The authors declare no conflict of interest.