# Synchronised TeraHertz Radiation and Soft X-rays Produced in a FEL Oscillator
## 1 Introduction
The Compton IP is set in a position where the length \(L_{c}\) of the path the radiation covers for coming back to this same point after the reflection equals a multiple of the distance between two electron bunches, i.e., \(L_{c}=n^{\prime}\Delta l\) (\(n^{\prime}\) integer). The THz radiation, propagating backward, crosses a successive electron bunch and scatters X-ray radiation in the direction of the electrons at the same repetition rate. The X-ray and THz radiation that flow through the front mirror are generated in sequence by two successive bunches of the electron beam train and are therefore naturally synchronised within the characteristic temporal jitters of the SC accelerator. The possible combinations of THz and X-ray wavelengths produced by the device are analysed in Figure 2. The resonance condition of the FEL oscillator is \(\lambda_{THz}=\lambda_{w}(1+a_{w}^{2})\)\(/(2\gamma^{2})\), where \(\gamma=E/mc^{2}\) is the Lorentz factor of the electron beam, \(\lambda_{w}\) and \(a_{w}=eB\lambda_{w}/(\sqrt{2}2\pi mc)\) the period and the undulator parameter (\(B\) is the undulator peak on-axis magnetic field), being \(m\), \(e\) and \(c\) the electron rest mass, charge and the speed of light, respectively. \begin{table}
\begin{tabular}{c c} \hline
**Quantity** & **Before the Undulator** \\ \hline Energy & 20–100 MeV \\ Charge & 100–200 pC \\ Energy spread & 0.1–1\% \\ Slice emittance & 0.6–2 mm mrad \\ Transverse size \(\sigma_{x}\) & 0.1–0.25 mm \\ Length \(\sigma_{z}\) & 0.5–5 mm \\ Average current & \textless{}7–8 mA \\ \hline \end{tabular}
\end{table} 
Table 1: Electron beam characteristics at the undulator entrance. Figure 1: Dual source of THz and X-ray radiation. The cavity is constituted by four mirrors, one of them holed or translucent. The Compton interaction point is on the right, and the focusing is realised by the mirrors themselves. Electron beam and THz radiation interact at angle. Figure 2 shows the resonant values \(\lambda_{THz}\) as a function of the electron beam Lorentz factor \(\gamma\) ranging from 40 (20 MeV) to 110 (55 MeV), for various values of the undulator period \(\lambda_{w}\) between 3.5 cm and 5.5 cm at values of \(a_{w}\) between 2 and 3, respectively. As shown herein, the undulator can be tuned to cover a wide THz wavelength range with relatively low electron energy values. The intermediate value \(\lambda_{w}=4.5\) cm, for instance, allows to span the THz wavelengths from 100 to 35 \(\upmu\)m, only varying the electron energy from approximately 20 to 50 MeV. By changing \(a_{w}\), a wider range of wavelengths can be produced. In the same graph, the Compton radiation wavelength \(\lambda_{X}\simeq\lambda_{THz}/(4\gamma^{2})\) generated by the scattering between the electron beam and the THz pulses is also presented (dotted lines), thus showing the couples of THz/X-ray wavelengths simultaneously generated. Interesting wavelengths in the soft-tender X-ray regime are, for instance, the water window at \(\lambda=3\) nm [25] or the spectroscopy range around 0.5 nm [26]. THz radiation at 30-55 \(\upmu\)m can be paired with X-rays in the water window at 3 nm. With the same undulator, THz radiation in the range 13-25 \(\upmu\)m generates X rays at \(\lambda=0.5\) nm. Long wavelength FEL radiation pulses present strong diffraction, which limits the minimum value that undulator periods and gaps can assume. The slippage along the propagation direction is moreover large and causes the possible transition to the superradiant regime [27; 28], where the emitted energy is proportional to the square of the charge. In order to avoid the strong spectrum deterioration by deep radiation saturation, an electron beam rms length \(\sigma_{e}\) of the order of two times the slippage length \(L_{s}=N_{w}\lambda_{THz}\) (\(N_{w}\) being the number of periods of the undulator module) could individuate a suitable working point satisfying the condition: \(\sigma_{e}\geq 2L_{s}\). We can then determine the necessary undulator length.