# Chirped-Pulse Amplification in an Echo-Enabled Harmonic-Generation Free-Electron Laser
## 3 Simulation Results  
### Simulation Results of CPA-EEHG  
In order to investigate the feasibility of CPA-EEHG, a simulation was carried out based on the parameters listed in Table 1. The electron beam energy is 2.5 GeV with a relative slice energy spread of about \(10^{-4}\). The electron bunch charge is around 510 pC, and the electron bunch length is about 600 fs. Two laser pulses at 270 nm have been stretched to 300 fs (FWHM) and employed as seed lasers. The peak powers of the stretched seed pulses are about 25 MW and 100 MW, respectively. The Wigner distributions of the seed laser pulses are shown in Figure 2. Two seed lasers have the same linear chirp with a bandwidth (\(\Delta\lambda/\lambda\)) of about 1.5%. The modulation section contains two 2 m-long modulators with a period length of 90 mm and two chicanes with total lengths of 10 m and 5 m, respectively. The radiator consists of two 4 m-long variable-gap undulators with a 43 mm period length. The gaps of the undulators are tuned to amplify 13.5 nm (20 th harmonic) radiation pulses. The FEL's performance and radiation pulse propagation were simulated with GENESIS [37] and OCELOT [38], while the design of the compressor was performed with GSolver and XOP [39]. \begin{table}
\begin{tabular}{c c c c} \hline
**Section** & **Parameter** & **Value** & **Unit** \\ \hline \multirow{4}{*}{Electron beam} & Beam energy & 2.5 & GeV \\  & Emittance & 0.4/0.4 & mm\(\cdot\)mrad \\  & Peak current & 800 & A \\  & Bunch length & 600 & fs \\ \hline \multirow{3}{*}{Seed laser} & Wavelength & 270 & nm \\  & Pulse length & 300 & fs \\  & Peak Power & \(\sim\)25/\(\sim\)100 & MW \\ \hline \multirow{2}{*}{Modulator} & Period length & 0.09 & m \\  & Total length & \(\sim\)2 & m \\ \hline \multirow{2}{*}{Dispersion} & Dipole Length & 0.3 & m \\  & Total length & 10/5 & m \\ \hline \multirow{3}{*}{Radiator} & Period length & 0.043 & m \\  & Undulator length & 4 & m \\ \cline{1-1}  & Resonant wavelength & 13.5 & nm \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. To match the resonance condition in the radiator, a 3.2% energy chirp was imprinted on the electron beam in the simulation. The distributions of the electron beam energy and current are demonstrated in Figure 3. It is noteworthy that there might be a quadratic component in the energy chirp due to the RF curvature in acceleration during the experiment, which will broaden the FEL pulse bandwidth. This effect will slightly degrade the quality of the compressed pulses, leading to the performance degradation of CPA-EEHG. Optimized with the analytical equations in [40], the dispersion strengths \(R_{56}\) of two chicanes are \(\sim\)2.20 mm and \(\sim\)0.12 mm, respectively. After the second dispersion section, the electron beam generates microbunching, which has frequency components at the high harmonics of the seed laser. The relative slice energy spread at the entrance of the radiator is about \(3.6\times 10^{-4}\). The utilization of such an electron beam enables the generation and amplification of coherent radiation pulses at 13.5 nm (\(n=20\)) in the radiator. The simulation results along the radiator and electron bunch are shown in Figure 4. The red and gray lines in the left panel demonstrate the evolution of the FEL pulse energy and electron beam bunching factor, respectively. The pulse energy reaches 651 uJ after two undulators. The optimized 20 th harmonic bunching factor (\(\sim\)8.30%) distribution along the electron bunch at the entrance of the radiator is shown in the right panel. The power profile and the spectrum of the radiation pulse at the end of radiator are also illustrated in Figure 4. The peak power of the output radiation is around 3.94 GW, and the FWHM pulse duration is about 186 fs. The spectral bandwidth of the radiation pulse is approximately 0.81%, which is close to the theoretical expectation of 0.55%. Figure 3: The distribution of the electron beam energy (green) and current (red). Figure 2: The Wigner distribution of seed lasers.