# Synchronised TeraHertz Radiation and Soft X-rays Produced in a FEL Oscillator
## 4 The TeraHertz Optical Cavity  
The TeraHertz optical cavities of FEL oscillators in operation are usually stable cavities constituted by a front and a rear mirror, with the undulator in between. The example described here, instead, is a four-mirror unstable cavity in a bow-tie configuration sketched in Figure 10. The mirrors are spherical and constitute a telescopic system, surrounding the Compton IP, which provides the focusing needed for increasing the luminosity of the system. This cavity has been inspired by those currently used in Thomson scattering sources, for instance in BriXSinO [23] or ThomX [24]. Other solutions could, however, be exploited. The radiation exits the undulator (point 1) with an assigned waist size \(w_{1}\) and curvature radius \(R_{1}\) provided by the GENESIS 1.3 calculations.  
The THz radiation must hit the holed front mirror with a size \(w_{2}(>>w_{1})\) large enough to guarantee a fraction of a few percent of extra-cavity radiation with a hole with a radius of a few mm, namely \(w_{2}\simeq 10\) cm. For the THz pulse propagation, we follow the ABCD  
Figure 8: Intra-cavity radiation power profile P(W) vs. coordinate s(m) = t(s)c. Window (**a**): undulator period \(\lambda_{w}=3.5\) cm; red curve: \(\gamma=55.5\), \(\lambda=36\) μm; blue curve: \(\gamma=81.5\), \(\lambda=13.2\) μm. Window (**b**): undulator period \(\lambda_{w}=4.5\) cm; red curve: \(\gamma=65\), \(\lambda=51.6\) μm; magenta curve: \(\gamma=55.5\), \(\lambda=36\) μm; blue curve: \(\gamma=86\), \(\lambda=15\) μm.  
Figure 9: Radiation gain vs. number of round trip for the case with \(\lambda=15\) μm, losses at 6% and emittance of 1.4 mm mrad.  
model of the Gaussian beams. The length between the end of the undulator and the mirror (point 2) is then chosen in such a way that  
\[w_{2}=\sqrt{(1+\frac{l_{1}+l_{2}}{R_{1}})^{2}w_{1}^{2}+(\frac{\lambda(l_{1}+l_{2} )}{\pi w_{1}})^{2}}\simeq 0.1\,\mathrm{m}\]  
Considering, for instance, the case with \(\lambda=15\) um and using the values of Table 2 for \(w_{1}\) and \(R_{1}\) (w = 2\(\sigma\) and R = \(\sigma/\theta_{div}\)), the second term \(\propto\lambda\) is negligible with respect to the first one, giving \((l_{1}+l_{2})\simeq(w_{2}/w_{1}-1)R_{1}\simeq 13\) m. A waist at the IP (point 6) of the order of \(1.5\times 10^{-4}\) m entails a focusing distance \(l_{4}\simeq 3.14\) m. The radiation reflected on mirror 1 should encounter a successive electron bunch in the IP, leading to \(l_{2}+l_{3}+l_{4}=m\Delta l\), where m is an integer number and \(\Delta l\) the bunch-to-bunch separation. With an angle of collision of \(2^{\circ}\), \(l_{2}\simeq 2.6\) m and \(l_{3}\simeq 0.24\) m. On the other hand, the total round trip length should also be an integer multiple n of \(\Delta l\), namely \(l_{5}+l_{6}+l_{7}+l_{u}+l_{1}\simeq n\Delta l\). The radiation should arrive consistently matched with the undulator (point 11). This condition is guaranteed by values of \(w_{11}\simeq 7.510^{-3}\) m and \(R_{11}=-1\) m, leading to \(l_{5}=0.27\) m, \(l_{6}=14.72\) m and \(l_{7}=0.24\) m. The resulting round trip length is 36 m and the linear dimension of the cavity is approximately 18 m. The mirrors turn out to have the following curvature radii: \(r_{1}\simeq 27.3\) m, \(r_{2}\simeq 3.14\) m, \(r_{3}\simeq 0.27\) m and \(r_{4}=0.39\) m. Radiation with wavelengths up to 50 um can be transported and focused with these same curvature lengths, by operating slight adjustments of the position of the mirrors. The considerable value of the power stored inside the Fabry-Perot cavity could produce relevant thermal effects on the mirrors, inducing the deformation of their surfaces with an insurgence of transverse mode degeneracy. Methods for overcoming this issue are based on inducing losses to higher-order modes and have been studied in [24; 42].