# Chirped-Pulse Amplification in an Echo-Enabled Harmonic-Generation Free-Electron Laser
## Appendix A The Influence of Collective Effects  
The collective effects can potentially result in a deterioration of the FEL performance in EEHG. In this Appendix, we conducted an investigation into the impact of coherent synchrotron radiation (CSR) through simulations, as well as the effect of incoherent synchrotron radiation (ISR) and intra-beam scattering (IBS) via theoretical calculation:  
Figure 10: The relative time delay (green), the peak power (blue), and the pulse duration (orange) of the compressed pulses when the azimuth angles are tuned within \(\pm 3^{\circ}\).  
* Coherent synchrotron radiation (CSR): The strong chicane in the first dispersion section makes the system vulnerable to the CSR effect. It imprints a long-wavelength energy modulation on the electron bunch. Such an energy modulation will interfere with the bunching process, which eventually leads to a decrease of the bunching factor [45]. In order to evaluate this influence on the proposed scheme, the simulation was performed with OCELOT. Each dipole has a length of 0.3 m. The drift lengths between the first and second dipoles in two chicanes are 4 m and 1.5 m, respectively. This leads to relatively small bend angles in the dipoles, which are about 0.0162 rad for the first chicane and 0.0059 rad for the second chicane. The influence of the CSR on the bunching factor can be found in Figure 11. The electron beam is slightly lengthened after modulation, and the bunching factor remains largely unaffected. This indicates a relatively minor influence of the CSR on the electron beam. However, it should be emphasized that the extent of the CSR's impact is subject to variation based on the specific configurations of the chicanes.
* Incoherent synchrotron radiation (ISR): Despite the CSR, the ISR effect induced by the strong chicane will also smear the fine structure of the longitudinal phase space. The ISR quantum-diffusion-induced energy spread growth can be written as [46] \[(\sigma_{\Delta E})_{ISR}=\left[\frac{7\hbar}{15 mc}Lr_{c}\gamma^{4}k_{\omega}^ {3}K_{\mu}^{2}F(K_{\mu})\right]^{-1/2},\] (11) where \(r_{c}\) is the classical radius of the electron and \(k_{\omega}\) and \(K_{\mu}\) denote the undulator wavenumber and the strength of the undulator, respectively. \(F(K_{\mu})=1.2K_{\mu}+1/(1+1.33K_{\mu}+0.4K_{\mu}^{2})\). According to the parameters specified in Table 1, the ISR-induced quantum diffusion effect was calculated to be about 13.2 eV, which is much smaller than the spacing of adjacent energy bands (-140 keV) and, therefore, negligible.
* Intra-beam scattering (IBS): The IBS describes multiple Coulomb scatterings in the electron beam, which leads to an increasing electron beam size and energy spread [47]. The IBS-induced energy spread growth can be expressed as [48] \[(\sigma_{\Delta E})_{IBS}\simeq\left(2\pi^{3/2}\ln\Lambda r_{c}\frac{I}{I_{A}} \frac{1}{4\pi\sigma_{\lambda}^{3}}\frac{\beta_{x}}{\gamma}s\right)^{1/2}E_{0},\] (12)where \(I\) denotes the beam current, \(I_{A}\approx 17\) kA is the Alfven current, \(\sigma_{x}\) and \(\beta_{x}\) denote transverse beam size and transverse \(\beta\)-function, \(s\) is the distance along the beamline, and \(E_{0}\approx 511\) keV is the static electron energy. In \(\Lambda\) denotes the Coulomb logarithm, which is calculated as  
\[\ln\Lambda=\ln\left(\frac{\sqrt{2}}{6}\frac{\epsilon_{N}^{3}}{r_{c}\sigma_{x} ^{2}}\left(\frac{I}{I_{A}}\right)^{-1/2}\gamma^{1/2}\right)-\ln\left(\frac{2 \epsilon_{N}}{\eta\sigma_{x}}\right), \tag{10}\]  
where \(\epsilon_{N}\) denotes the electron beam normalized emittance.  
The decreasing bunching factor can be estimated by \(b_{h}=b_{h}^{(0)}e^{-I/L}\)[49], where \(L=2\sigma_{E}^{2}/Dm^{2}B_{2}^{2}\). By utilizing the parameters listed in Table 1, the IBS-induced energy spread growth is about 8.6 keV, and the bunching factor decreases from 8.3% to 8.1%.