# Radio-Frequency Undulators, Cyclotron Auto Resonance Maser and Free Electron Lasers
## 3 CARM, RFU and VUV Soft X-ray FEL Operation  
In the introductory section of this paper, we set out the general conditions to be fulfilled by a RF wave in the millimeter region, to sustain the operation of a RFU. The requirements in terms of power, power density, wavelength...of the RF field ensuring a FEL SASE operation have shown some criticalities, which relax with increasing FEL wavelength. The conclusions we have drawn apply to any RF device (Gyrotron, Gyro-klystron, Magnetron, etc.) CARM is suited to operate in the region above 100 GHz (\(\lambda_{w}<3\)) mm, where the other sources exhibit a breakdown in terms of maximum achievable power. We will therefore discuss a few specific issues for the use of 1-3 mm CARM-RFU. In the introductory section, we have a RF power density larger than \(I_{rf}\approx 10^{15}\) W/m\({}^{2}\) in order to achieve values of the strength parameter (\(K_{w}\) see Figure 2b) not far from unity (the subscript \(w\) has been replaced by \(rf\) to underscore that it refers to radio frequency intensity). The CARM power, namely, the amount of power transferred from the electron beam to the RF field, is expected to be about \(P_{rf}\approx 5.4\) MW, which should accordingly be transported along a pipe with a radius \(r=1/\pi(\sqrt{P_{rf}/I_{rf}})\approx 22\) um, small to transport radiation in the microwave range and for any realistic mechanical handling. In Figure 14, we have reported the electron beam energy, the electron beam current density, the electron beam transverse section and the peak current vs. the RF wavelength, assuming a significantly reduced value of the RFU power density (\(I_{rf}\approx 5.3\cdot 10^{13}\) W/m\({}^{2}\)) and FEL wavelengths in the VUV-X range (\(10\sim 20\)) nm. We should underscore that, even with these relaxed numbers, it is not an easy task to transport this amount of power density. In order to enhance the RF power density we can proceed by increasing CARM output power, using, e.g., a suitable compression of the associated pulse. We remind that the CARM is driven by an e-beam sustained by a pulse forming system, with a flat top of the order of microseconds [44]. The RF pulse has accordingly a comparable length. The power can be enhanced by compressing the pulse, by reducing it to a length comparable with the saturation length, namely tens of ns. Figure 13: Gain vs. the normalized detuning \(\Delta\) at different z-position for the \(\omega_{-}\) resonance calculated using the analytical solution in the SSA regime with the parameters reported in Table 1.  
According to the previous discussion, the CARM pulse compression to a duration of tens of ns (few meters, approximately the saturation length foreseen for RFU operation) is sufficient to ensure CARM peak power level near the GW level. The possibility of achieving these results has been suggested in refs. [45; 46], in which the use of a dispersive corrugated metal waveguide was proposed. The crucial idea put forward in these articles is that of exploiting a swept frequency modulated train of pulses that propagate inside the waveguide, with a monotonically increasing group velocity. Accordingly, the tail of the pulse will overtake its head, thus providing either a shortening of its duration and an increase (in absence of significant losses) of the relevant amplitude. Most of the compression process occurs at the end of the waveguide, where all the frequency components are present at the same time and the compression ratio (C) can be expressed as [46]  
\[C=\frac{\Delta f}{f_{0}}\frac{L}{\lambda_{0}}\bigg{(}\frac{1}{k_{1}}-\frac{1}{ k_{2}}\bigg{)}e^{-\gamma_{0}L} \tag{23}\]  
being \(k_{1,2}=v_{g,1,2}/c\), with \(v_{g,1,2}\) the group velocities at the beginning and at the end frequencies of the RF pulse, \(\Delta f/f_{0}\)--the fractional bandwidth, \(L\)--the waveguide length, \(\lambda_{0}=c/f_{0}\)--the CARM wavelength and \(\gamma_{0}\)--the waveguide losses. The compression factor, summarized by Equation (23), is one of the central point of the discussion.