# Two-Color TeraHertz Radiation by a Multi-Pass FEL Oscillator
## 2 Materials and Methods  
We describe here the behaviour and the potentiality of an FEL oscillator conceived as two undulator modules separated by a drift, where optical elements and diagnostics can be possibly allocated, and embedded into an optical cavity equipped with mirrors suitable to the frequency range. The two modules can have different periods for permitting a more versatile two-color operation. The wavelength of the produced radiation pulses is given by the FEL resonance condition \(\lambda_{1,2}=\lambda_{w_{1,2}}(1+a_{w_{1,2}}^{2})/(2\gamma^{2})\), where \(\lambda_{w_{1,2}}\) and \(a_{w_{1,2}}=0.657\lambda_{w_{1,2}}(cm)B_{1,2}(T)\) are the periods and adimensional parameters of the two undulator modules, with \(B_{1,2}(T)\) being their peak on-axis magnetic field. Therefore, for a certain period of the two undulators, the wavelengths can be tuned either simultaneously by varying the electron Lorentz factor, or independently by changing the undulator magnetic fields. With \(\lambda_{w_{1}}=2.8\) cm being constant, the achievable ratio \(\lambda_{2}/\lambda_{1}\) as a function of the period \(\lambda_{w_{2}}\) of the second module is shown in Figure 1 for different magnetic field values \(B_{1,2}\) between 0.75 and 1.2 T. As shown in this plot, a period larger than 4 cm for the second module allows to efficiently generate with the same electron beam two colours with a wavelength ratio larger than 5. The darkest region in the plot corresponds to a large undulator magnetic field for both undulators, thus leading to higher FEL emission. Figure 2 shows the scheme of the source. It is composed of two undulator modules (a) and (b), separated by a quadrupole (c) for electron-beam matching and embedded in a cavity. Given the repetition rate of 92.86 MHz of the electron beam, the cavity length \(L_{c}=12.9\) m corresponds to 4 times the bunch separation. The two undulator modules, each 3 m long, have variable gaps for flexible tuning of the radiation. A further degree of versatility may be guaranteed by the use of modules constituted by subsections which can be tuned independently. The radiation beam is amplified and propagates within the undulator and the resonator optical line. Considering a Gaussian beam, its radius evolves along the propagation axis z according to \(\sigma_{r}(z)=\sigma_{r,0}\sqrt{1+(z/z_{R})^{2}}\), where \(\sigma_{r,0}\) is the beam waist and \(z_{R}=\pi\sigma_{r,0}^{2}/\lambda\) is its Rayleigh length. Therefore, strong diffraction affects long wavelength radiation pulses, and the operation at THz wavelengths requires gaps of about 1 cm or larger, for permitting the transport of a transversely large electron beam and containing the propagation of the highly diffracted radiation. This can limit the minimum value of periods and gaps that can be used; if necessary, propagation in waveguides could be foreseen, as, for instance, conceived in other devices [4]. In our scheme, the electron beam is driven in a beam pipe, whose vertical size should be of the order of 1 cm within the undulator gaps. The horizontal dimension of the pipe (orthogonal to the undulator magnetic field) could be larger and allow the free expansion of the radiation pulse along this direction. Moreover, the different velocity of light and electrons causes their detachment as they propagate within the undulator. The slippage length of the radiation with respect to the electron beam is estimated to be \(L_{s}=N\lambda\), N being the number of undulator periods. Long wavelength radiation pulses lead to larger slippage lengths. In order to compensate this effect and synchronize the radiation and the electron beam at the undulator entrance, the radiation needs to be delayed with respect to the electron bunches at each round trip. Furthermore, for maintaining this synchronization along the whole undulator, the electron bunch length \(\sigma_{e}\) should be at least two times the slippage length, leading to the condition: \(\sigma_{e}\geq 2L_{s}\).