# Chirped-Pulse Amplification in an Echo-Enabled Harmonic-Generation Free-Electron Laser
## 3 Simulation Results
### Simulation Results of CPA-EEHG
The projections show the power profiles and spectra of the seed lasers. The Wigner distribution of the radiation pulse after the radiator is shown in Figure 5. Since the gain bandwidth of the modulator (\(\sim\)4.5%) is several times broader than the bandwidth of the seed laser, all frequency components of the seed pulses effectively contribute to the energy modulation of the electron bunch. Consequently, the linear frequency chirp in the seed laser remains well-preserved in FEL pulses, which enables the compression of the radiation pulses through an optical pulse compressor. As shown in Appendix A, the analysis of the collective effects indicates a slight decrease in the bunching factor (from 8.3% to 8.1%), which is not expected to have a significant impact on the FEL performance. The generated frequency-chirped FEL pulse propagates through a double-grating compressor, as shown in Figure 1. When the introduced GDD is equal to and opposite the intrinsic GDD of the FEL pulse, the correction of the second-order effects on the phase will lead to a reduction of the pulse duration. The OPM geometry of the grating was adopted for the higher transmission efficiency. The compressor parameters are listed in Table 2. The designed gratings were selected to be commercially available. For the narrow-bandwidth pulses, the GDD introduced by the compressor is about \(-160\) fs\({}^{2}\). Figure 4: The simulation results along the radiator (**left**) and electron bunch (**right**) of CPA-EEHG. Figure 3.0 The Wigner distribution of the FEL pulse before the compressor. The two ruthenium-coated mirrors of the compressor were employed to deflect the FEL beam in the same direction as the input. Figure 6 shows the diffraction efficiency of the gratings and the reflectance of the mirrors at 13.5 nm. Taking the surface roughness and the misalignment of the grating pitch into consideration, the total transmission efficiency of the compressor was up to 15% at a central wavelength of 13.5 nm. The compressor also introduces the spatial chirp, which could potentially cause the quality degradation of the compressed FEL pulse. For the compressor designed in the simulation, the spatial chirp calculated in the FWHM bandwidth is about 0.55 mm at the exit of the compressor according to [35]. The transverse spot size of the simulated FEL pulse is 140 um with a divergence of around 24 umad. The FEL pulse diameter at the compressor entrance is \(\sim\)4 mm when the compressor is installed 80 m after the FEL source. Therefore, the introduced spatial chirp is negligible. Moreover, the influence of spatial chirp on the spot size can be mitigated by using an appropriately shaped concave mirror to focus the radiation beam. Figure 7 shows the Wigner distribution of the FEL pulse after the compressor. Comparing with Figure 5, the pulse length is compressed by roughly 70 times, from 186.43 fs to 2.70 fs, while the peak power correspondingly increases by about 6 times, from 3.94 GW to 23.93 GW. The time-bandwidth product (\(\sim\)0.486) is only a factor of 1.1 above the transform limit for a Gaussian pulse, which is in satisfactory agreement with the experimental result in [31]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Central wavelength & 13.5 & nm \\ Bandwidth & 0.81\% & \\ Groove density & 3600 & gr/mm \\ Blaze angle & 28\({}^{\circ}\) & \\ Altitude angle & 3\({}^{\circ}\) & \\ G1–G2 distance & 2.2 & m \\ Coating & Gold & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The designed grating parameters. Figure 6: The diffraction efficiency of the gratings at different altitude angles (**top**) and the reflectance of the mirrors at different incidence angles (**bottom**) at 13.5 nm. By adjusting the rotations and translations of the two gratings, the azimuth angles \(\mu\) can be tuned to modify the GDD of the compressor according to Equation (5).