# Recirculated Wave Undulators for Compact FELs
## 6 Discussions on the Recirculated Wave Undulator FEL Scheme  
As already discussed and demonstrated in Ref. [27], a CO\({}_{2}\) laser (\(\lambda_{0}=10.6\)\(\upmu\)m), with an intensity \(I_{p}=4.2\times 10^{18}\) W/m\({}^{2}\), corresponding to an energy per pulse of 40 J delivered in 300 \(ps\) (\(P_{L}=130\) GW) over an effective area \(\Sigma_{0}=\pi w_{0}^{2}\sim\pi\times 10^{-8}\) m\({}^{2}\) would be enough to provide a wave undulator with sufficiently large \(K\) to support the FEL SASE operation [33; 34; 35]. The value for the strength parameter under these operational conditions is \(K\sim 0.186\). This \(K\) value is too low for standard FEL, but sufficiently high for the emission of coherent radiation in the wave undulator scheme. The energy for the electron beam we consider is around 35 MeV corresponding to a \(\gamma_{r}\sim 70\). We choose an electron current value \(I_{e}=j_{e}\Sigma_{e}=3\) kA, where \(\Sigma_{e}=0.075\Sigma_{0}\), which means that the electron beam is focused more tightly than the laser beam: this can allow a greater output flux of X-ray photons and eases the spatial overlap of the two beams practically speaking. Moreover, the initial relative energy spread is \(10^{-4}\). Even if the considered energy of the electron beam is quite low and the beam focusing rather tight, space charge effects can be neglected. The extension of the interaction region \(Z_{i}\) is evaluated as the minimum between two times the Rayleigh length \(Z_{R}=\pi w_{0}^{2}/\lambda_{0}\) and the laser pulse length \(c\tau\) where \(\tau\) is the FWHM pulse duration. In formulas:  
\[Z_{i}=min\big{\{}c\tau,2Z_{R}\big{\}} \tag{27}\]  
In our case \(2Z_{R}/c\tau<<1\), so the interaction region is determined by the Rayleigh length. The number of periods per wave undulator both in the waist \(w_{1}\) and \(w_{2}\) are \(2Z_{R}/\lambda_{0}\), which is many hundreds. As anticipated in the previous section, the working principle of the compact FEL based on the wave undulator is the following: the electron beam is injected into the ring cavity interacting at the laser waists \(w_{1}\) and \(w_{2}\). The electron bunches are microbunched at the radiation wavelength's period which is in our case \(\lambda_{r}\sim 1\) nm, falling in the soft X-ray region. When this occurs, coherent radiation power is emitted. A further and important aspect to discuss is the interaction between the electron beams and the wakefields in the machine. The laser can circulate in the ring cavity and interact with the electrons thanks to hollow parabolic mirrors, while the beam passing through the holes generates wakefields (only the first mirror encountered by the electron beam is significant). Given the shortness of the electron bunches considered here, a single bunch of electrons might be not able to efficiently interact with the wakefields. On the other hand, the wakefields might interact with trailing bunches for a certain time structure of the electron bunches in the machine. For facing this issue, the use of dielectric mirrors to suppress the effect of wakefields might be more favorable than metallic-coated optics. Furthermore, for the parameters considered here, the laser diameter at the hollow parabolic mirror is expected to be about 10 cm, therefore the hole would not need to be extremely small.