# Synchronised TeraHertz Radiation and Soft X-rays Produced in a FEL Oscillator
## 1 Introduction
Following the road map proposed in [29] and references therein, another condition of operation threshold can be determined, i.e., \(0.27(4\pi\rho)^{3}{N_{w}}^{3}>Loss\) where \(\rho\) is the Pierce parameter of the FEL [30; 31] and loss is the percentage of energy lost by the light during the round trip from the end to the entrance of the undulator. This condition holds when the electron beam is much longer than the cooperation length \(L_{c}=\lambda/(4\pi\rho)\), i.e., when \(\sigma_{e}>>L_{c}\) (long-bunch limit). The opposite limit features a different scaling law, leading to a slightly different threshold. If \(\sigma_{e}\) is about 5-10 times the cooperation length, the effective operative condition lays in an intermediate region between the two limiting cases ruled by the short and the long-bunch models. The radiation Rayleigh length \(Z_{R}\) should be of the order of half the undulator length and the dimension of the radiation beam turns out to be of the order of \(\sigma_{r}\simeq\sqrt{\frac{Z_{R}\lambda}{2\pi}}>>\sigma_{e}\). For this reason, since the radiation transverse size is much larger than the electron beam's size, even if the radiation losses due to the interaction with the mirrors  
Figure 2: Wavelength of the THz and X-ray radiation vs. electron Lorentz factor \(\gamma\). Solid lines: THz radiation produced with the FEL oscillator. Dashed lines: X-ray radiation generated by means of Thomson back-scattering. In black: \(\lambda_{w}=3.5\) cm, \(a_{W}=2\); similar values can also be obtained with \(\lambda_{w}=4.5\) cm, \(a_{W}=1.7\). In blue: \(\lambda_{w}=4.5\) cm, \(a_{W}=2.5\). In magenta: \(\lambda_{w}=5.5\) cm, \(a_{W}=3\). and to the pulse extraction along the trajectory outside the undulator are relatively small, and the electron beam-radiation coupling at the undulator entrance can be scarce. For instance, typical values of the radiation size are approximately 3 mm, while the electron radius is 100-200 \(\upmu\)m, leading to an estimate of loss as large as approximately 99%. The number of undulator periods \(N_{w}\) satisfying the aforementioned constraint is \(N_{w}>\frac{1.54}{4\pi\rho}\) in the case of a long bunch, or \(N_{w}>\frac{118}{4\pi\rho}\) with a short electron beam. The graph in Figure 3 presents the required undulator length \(L_{und}=\lambda_{w}N_{w}\) as a function of the beam peak current I(A) for five cases based on two different undulator periods, where three-dimensional and inhomogeneous effects were taken into account in the calculation of the Pierce parameter as in Ref. [32]. Both long (solid lines) and short (dotted lines) electron beam cases are reported in the graph. Sustainable undulator lengths and reasonable peak currents (7 A \(<I<\) 25 A) in the range attainable by a SC high repetition rate accelerator are needed. Electron beams with 150 pC of charge, 15-20 A of peak current, reasonable values of emittance (\(<\)2 mm mrad) and energy spread (\(<\)5 \(\times\) 10\({}^{-3}\)) could produce more than 1 m] of intra-cavity (IC) radiation energy at 10-50 \(\upmu\)m of wavelength in less than 2 m of undulator, generating extra-cavity (EC) pulses of tens of \(\upmu\)J. The intra-cavity THz pulse constitutes the scattering radiation in the inverse Compton process, generating \(N_{X}\) photons at \(\lambda_{X}\). The number of X-ray photons per shot \(N_{X}\) can be indeed roughly estimated as [33; 34]:  
\[N_{X}=\sigma_{Th}\frac{N_{e}N_{THz}}{2\pi(\sigma_{e}{}^{2}+\sigma_{THz}{}^{2})}. \tag{1}\]  
where \(\sigma_{Th}=6.65\times 10^{-29}\) m\({}^{2}\) is the Thomson cross-section, \(N_{e}=Q/e=1.25\times 10^{9}\), \(\sigma_{e}=100\)\(\upmu\)m, \(N_{THz}=E_{IC}/(h\
u_{THz})=E_{IC}\lambda_{THz}/(hc)\simeq 2\)-\(5\times 10^{17}\), an amount of approximately 10\({}^{5}\) X-ray photons per shot is delivered by the system on the whole spectrum, provided that \(\sigma_{THz}\leq 0.3\) mm. At a repetition rate of approximately 50 MHz, the source could deliver to the users \(5\times 10^{12}\) X-ray photons/s together with 2-5 \(\times 10^{23}\) THz photons/s.