# Synchronised TeraHertz Radiation and Soft X-rays Produced in a FEL Oscillator
## 3 The THz FEL Oscillator: Numerical Results
With the electron beam parameters in the range shown in Table 1, assuming the values of Q = 250 pC, \(I_{peak}=20\) A, \(\epsilon_{n}=1.4\) mm mrad, energy spread \(\Delta E=100\) keV and varying energies with a jitter of \(2\times 10^{-4}\), five reference cases have been studied: two based on an undulator with a period \(\lambda_{w}=3.5\) cm and three with \(\lambda_{w}=4.5\) cm. The data and the results  
Figure 5: Radiation size \(\sigma_{r}ad\) (m) along the undulator coordinate z(m) for \(\lambda=15.2\)\(\upmu\)m, emittance \(\epsilon=1\) mm mrad, loss = 6% for different round trip numbers. Figure 6: Intra-cavity radiation energy E(J) vs. time delay \(c\Delta t\) (m) between electrons and radiation for radiation wavelength \(\lambda=15.2\)\(\upmu\)m, emittance \(\epsilon=1\) mm mrad, loss = 6%. are summarised in Table 2. THz radiation energies larger than 1 mJ are always obtained for wavelengths between 13 and 50 \(\upmu\)m, with undulator lengths larger than 1.5 m. Figure 7 shows the energy growth as a function of the round trip number inside the cavity for the five analysed cases, in the case of cavity losses at 2%. The radiation power profiles at saturation are instead shown in Figure 8. In some cases, the power shape is multi-spiky, resulting in a broad and polychromatic spectrum. In these cases, the long wavelength and the relatively longer undulator lead to a high total slippage, with the involvement of more supermodes [39, 40, 41]. Regarding the efficiency of the Compton scattering, these cases are characterised by larger energy levels and present advantages in terms of output flux, but their wider bandwidths limit the spectral density of the X-ray yield. We also varied the reflection coefficient of the mirrors in order to study different cavity options. Table 3 summarises the intra-cavity energy obtained with different values of cavity losses and electron emittances in the case of a wavelength at 15 \(\upmu\)m. Figure 9 is relevant to the less advantageous case of Table 3, with total losses at 6% (2% of outcoupling and 4% of cavity dissipation) and an emittance of 1.4 \(mmmmrad\) and reports the gain \(G=\left(E(L_{und})-E(0)\right)/E(L_{und})\) as function of the round trip number. At saturation, gain and losses are exactly balanced, while before saturation, the gain reaches 30%. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\lambda_{w}\) (cm) & 3.5 & 3.5 & 4.5 & 4.5 & 4.5 \\ \(a_{w}\) & 2.33 & 2 & 2.95 & 2 & 2 \\ \(L_{und}\) (m) & 2 & 2 & 1.71 & 1.71 & 1.71 \\ \hline \(\gamma\) & 55.5 & 81.5 & 65 & 55.5 & 86 \\ \hline \(\lambda_{THz}\) (\(\upmu\)m) & 36 & 13.2 & 51.6 & 36 & 15 \\ \(E_{IC}\) (mJ) & 3 & 3.38 & 1 & 2.1 & 2.8 \\ \(E_{EC}\) (\(\upmu\)J) & 60 & 67 & 20 & 42 & 56 \\ \(bw\) (\%) & 4 & 2.6 & 1 & 2.5 & 2.15 \\ size (mm) & 2.7 & 1.6 & 3 & 3 & 2.2 \\ div (mrad) & 3.5 & 1.8 & 6 & 4 & 3.7 \\ \(P_{EC}\) (kW) & 3 & 3.4 & 1 & 2.1 & 2.8 \\ \hline \(\lambda_{X}\) (nm) & 3 & 0.5 & 3 & 3 & 0.5 \\ \hline \end{tabular}
\end{table} 
Table 2: THz radiation: undulator specifics, electron energy, and results (repetition rate at \(5\times 10^{7}\) Hz). \(E_{IC}\) and \(E_{EC}\) indicate the intra-cavity and extra-cavity radiation energy, respectively; size, div and \(P_{EC}\) are the output radiation size, divergence and the extra-cavity radiation power level, respectively. Mirror losses at 2%.