# Synchronised TeraHertz Radiation and Soft X-rays Produced in a FEL Oscillator
## 5 The Inverse Compton Scattering Process: Numerical Results  
The interaction between the electron beam and the THz radiation propagating backward was investigated with the classical model based on the Lienard-Wiechert potentials [43]. The double differential energy spectrum with respect to frequency \(\omega\) and solid angle \(\Omega\) is given by:  
\[\frac{d^{2}W}{d\omega d\Omega}=\frac{e^{2}}{4\pi^{2}c}\left|\int_{-\infty}^{ \infty}\!dt\,\mathrm{e}^{\mathrm{i}(wt-\underline{u}\cdot\frac{\mathrm{e}(l)}{ c})}\left(\underline{u}\times\frac{(\underline{u}-\underline{\beta})\times \dot{\underline{\beta}}}{(1-\underline{u}\cdot\underline{\beta})^{3}}\right)_{ ret}\right|^{2} \tag{2}\]  
Figure 10: Dual source of THz and X-ray radiation. The cavity is constituted by four mirrors, one of them (\(r_{1}\)) holed for THz and X-ray extraction. Point 1: undulator exit. Point 2: arrival on mirror 1. Point 3: departure from mirror 1. Point 4: arrival on mirror 2. Point 5: departure from mirror 2. Point 6: interaction point. Point 7: arrival on mirror 3. Point 8: departure from mirror 3. Point 9: arrival on mirror 4. Point 10: departure from mirror 4. Point 11: undulator entrance. Electron beam and THz radiation interact at an angle \(\alpha\) of few degrees. where \(\underline{r}(t)\) is the position of the electron, \(\underline{u}\) is the direction of the observer position, the index ret indicates that all the factors are evaluated at the retarded time, \(\underline{\beta}\) and \(\underline{\beta}\) are the electron velocity and acceleration normalised to the speed of light, with  
\[\underline{\dot{\beta}}=\frac{-e}{m_{e}c\gamma}\Big{[}\underline{E}_{L}\Big{(}1 -\underline{\beta}\cdot\underline{e}_{k}\Big{)}+\underline{\beta}\cdot \underline{E}_{L}(\underline{e}_{k}-\underline{\beta})\Big{]}. \tag{3}\]  
\(E_{L}(x,y,z,t)\) is the electric field obtained by propagating the THz pulse generated by the FEL calculation shown in Figure 8 and \(\underline{e}_{k}\) is the direction of the interacting radiation. The crucial need for focusing the THz radiation on the Compton IP derives from the dependence of the final X-ray yield on the rms pump transverse size \(\sigma_{\text{THz}}\) exhibited in Formula (1) and can be solved by the installation of the telescopic system based on mirrors in the scattering region. As shown in the previous paragraph, the radiation in the IP can be focused down to \(\sigma_{\text{THz}}\simeq 65-75\)\(\upmu\)m in the case of \(\lambda_{\text{THz}}\simeq 13-15\)\(\upmu\)m and to about 150-250\(\upmu\)m for \(\lambda=36\)-51\(\upmu\)m. Figure 11 presents the dependence of the total X-ray flux on the THz radiation size \(\sigma_{\text{THz}}\) for 3 nm of X-ray wavelength (window(a)) and 0.5 nm (window (b)). As can be seen, a suitable focus allows us to exceed \(10^{5}\) X-ray photon per shot. Figure 12 shows the number of photons \(N_{\text{X}}\) (solid lines, left axis) and the bandwidth \(b_{w}\) (dotted lines, right axis) obtained with different collimation angles \(\theta_{coll}\) for the five cases analysed and for a THz focusing reference value \(\sigma_{\text{THz}}\simeq 5\)\(\lambda_{\text{THz}}\). The X-ray radiation bandwidths of the case analysed are all very similar except for a slight difference in the initial value, due to the different bandwidths of the THz pulses. The number of photons, instead, strongly depends on the THz intensities. The cases obtained with the shorter undulator period generate more intense X-rays. Table 4 collects information about the X-rays. The results were obtained by collimating the output yield in a proper angle \(\theta_{coll}\simeq 10\) mrad for a bandwidth of approximately 10%. If broader or narrower spectra are required or tolerated, they can be achieved by varying the collimation angle, as presented in Figure 12. Usually, it is not possible to operate in the Fourier transform limit with Compton sources.