# Two-Color TeraHertz Radiation by a Multi-Pass FEL Oscillator
## 3 Working Points and Simulation Results  
This section contains the working points of the source and reports the simulation results in a wavelength range where the FEL efficiency is robust. The parameters of the electron beam used in the simulations are summarized in Table 1, being the typical output of a super-conducting accelerator suitable for this kind of application. Considering \(\lambda_{w2}=4.5\) cm for the second module, a possible choice of undulator parameters deduced by the resonance graph of Figure 4 is reported in Table 2. By using such undulator periods and parameters, this source has the best performance in correspondence of a range of relatively short wavelengths between 10 \(\upmu\)m and 40 \(\upmu\)m. Longer wavelengths up to 300 \(\upmu\)m would require longer undulator periods, being at the limit of our device. \begin{table}
\begin{tabular}{c c c} \hline Energy & MeV & 15–50 \\ \hline Bunch charge & pC & 100–250 \\ \hline Repetition rate & MHz & 100 \\ \hline Peak Current & A & 15–30 \\ \hline Slice normalized emittance & mm mrad & 1.5–2 \\ \hline Slice energy Spread & \% & 0.1 \\ \hline Bunch length & mm & 1–2 \\ \hline \end{tabular}
\end{table} 
Table 1: Electron beam parameters. Figure 3: Required undulator period number N for successful FEL oscillator operation as a function of electron beam current I(A) for two typical cases. The numerical modeling of the two-color source has been performed by using the three-dimensional, time-dependent FEL code GENESIS 1.3 [18]. Starting from the electron beam parameters listed in Table 1 and in order to simulate the fluctuations of the bunch train, we have injected into the undulators a sequence of randomly prepared electron beams, which are different from each other both microscopically and macroscopically. This was done by randomly changing the seeds of the Hammersley sequences loading the particle phase space from shot to shot. Furthermore, the values of the macroscopic electron beam parameters, such as energy, current, emittance, and energy spread, have been varied to reproduce the shot-to-shot jitters. Each radiation output result is obtained by maximizing the output power as a function of the delay time compensating the slippage, and cycling the radiation within the cavity, taking into account the details of the optical line that returns the radiation to the undulator entrance [19, 20]. Mirror reflectivities of the order of 95-97% have been assumed. A 40 MeV electron beam, propagating inside the long period undulator, emits radiation at \(\lambda=35\,\upmu\)m. At the end of this first module, the radiation has slipped with respect to the electron beam by a length \(N\lambda\) of a few mm. The radiation is synchronized to a fresh electron bunch at the beginning of the first undulator module after each round trip. The fraction of energy extracted from the electron beam is moderate, so its phase space is substantially not deteriorated, permitting considerable emission in the second short period undulator tuned at \(\lambda=13\,\upmu\)m. The delay between the two pulses is equal to the slippage after the first module, corresponding to about 10 ps. This temporal separation could be compensated or increased by suitable radiation transfer lines. Apart from this time delay, the two pulses generated by the same electron beam are naturally synchronized, while the shot-to-shot jitters of the accelerator affect the radiation. Figure 5 shows the generation of radiation at 35 \(\upmu\)m, with an undulator of a 4.5 cm period. Radiation energy growth along the undulator axis as a function of the number of successive shots generated by the superconducting accelerator is shown in Plot (a) for an