# Two-Color TeraHertz Radiation by a Multi-Pass FEL Oscillator
## 3 Working Points and Simulation Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Color 1** & \multicolumn{4}{c}{**Color 2**} \\ \hline \(\lambda_{w_{1}}\) & cm & 2.8 & \(\lambda_{w_{2}}\) & cm & 4.5 \\ \hline B & T & 0.5–1.2 & B & T & 0.5–1.2 \\ \hline \(\lambda_{1}\) & \(\upmu\)m & 3-90 & \(\lambda_{2}\) & \(\upmu\)m & 7–300 \\ \hline gap & cm & 1.0 & gap & cm & 1.7 \\ \hline Length & m & 3 & Length & m & 3 \\ \hline \end{tabular}
\end{table} 
Table 2: Two color set-up: undulator parameters. Figure 4: Radiation wavelength \(\lambda\) vs electron Lorentz factor \(\gamma\) for \(\lambda_{w}=2.8\) cm in blue, and 4.5 cm in red. B = 1.2 T and 0.5 T.  
undulator length \(L_{w}=2\) m, with the corresponding power and spectral distributions in Plots (b) and (c). The case with \(L_{w}=3\) m is shown in the right column. In Figure 6, the same quantities are shown for the short period module tuned at 13 \(\upmu\)m.  
These cases can be run with the same device by closing the undulator gaps according to the magnetic length needed. The operation with a short portion of the undulator (left columns of Figures 5 and 6) gives rise to single-spiked pulses both in the temporal and spectral domains, while deep-saturation frozen fringes, stable in time from shot to shot, appear when operating with the whole undulator length (right columns of the same figures). This second interesting regime, constituted by a statistically stable sequence of spikes, has been widely discussed in [21]. Since each spike of the pulse train is considerably narrower than the corresponding one in the single-spike situation and spectral and power distributions are connected by the Fourier Transform operator, the spectrum is conversely larger. Figure 5: Radiation at \(\lambda=35\upmu\)m from an electron beam of 40 MeV with an undulator of period \(\lambda_{w}=4.5\) cm. Left column: undulator length \(L_{w}=2\) m. Right column: undulator length \(L_{w}=3\) m. (**a,d**) Energy growth (J) as a function of the number N of shots; (**b,e**) power profile (W) vs. s (mm). (**c,f**) Spectral profile in arbitrary units vs. \(\lambda\left(\upmu\right)\). A two-color source of this kind, whose radiation properties are summarized in Table 3, would enable the generation of both pump and probe radiation that is either single- or multi-spiked. In the latter case, the excitation and stroboscopic measurements of processes lasting a few picoseconds could be performed with a statistically coherent sequence of spikes. The distance between two successive peaks is 8 ps for the longer wavelength and 4 ps for the shorter one. Figure 6: Radiation at \(\lambda=13\,\upmu\)m from an electron beam of 40 MeV with an undulator of period \(\lambda_{w}=2.8\) cm. Left column: undulator length \(L_{w}=2\) m. Right column: undulator length \(L_{w}=3\) m. (**a,d**) Energy growth (J) as a function of the number N of shots; (**b,e**) power profile (W) vs. s (mm). (**c,f**) Spectral profile in arbitrary units vs. \(\lambda\) (\(\upmu\)m). The stability and coherence degree of the oscillator pulses of the same color has been estimated through the cross-correlation between two generic consecutive pulses after saturation:  
\[\Gamma_{12}(\tau)=\left|\frac{\int dtE_{1}(t)E_{2}^{*}(t-\tau)}{\sqrt{\int dt{|E_{ 1}|}^{2}}\sqrt{\int dt{|E_{2}|}^{2}}}\right|,\]  
where \(E_{1,2}\) is the complex field of the radiation of the pulses. In Table 3, the equal time coherence degree \(\Gamma_{12}(\tau=0)\) has been reported, showing a large coherence for the single-spike case, and a considerable value also in the case of multiple peaks.