# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 3 Laser Modulator
According to Table 1, the Rayleigh length of the laser beam was much greater than the total length of the laser modulator. For the entire interaction duration between the electron \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{3}{c}{**Value**} \\ \hline Undulator length & \(L_{u}\) & 1.50 & m \\ Undulator number of periods & \(N_{u}\) & 10 & \\ Undulator period & \(\lambda_{u}\) & 0.15 & m \\ Undulator peak field & \(B_{0}\) & 0.53 & T \\ Undulator parameter & \(K\) & 7.45 & \\ \(\xi=K^{2}/(2+K^{2})\) & & 0.965 & \\ \(\mathcal{J}=\int_{0}(\xi/2)-f_{1}(\xi/2)\) & & 0.708 & \\ Laser wavelength & \(\lambda_{L}\) & 1.0 & \(\mu\)m \\ Laser beam waist & \(w_{0}\) & 2.0 & mm \\ Laser beam Rayleigh length & \(z_{R}\) & 12.6 & m \\ Laser peak power & \(P_{L}\) & 3.4 & GW \\ Laser pulse length & \(\tau_{L}\) & 1.2 & ps \\ LM modulation amplitude & \(A_{L}\) & 23.6 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the laser modulator. beam and the laser beam, the transverse rms electron beam sizes were below 100 \(\upmu\)m, i.e., below 5% of the 2.0-mm laser beam waist size. In addition, the 1.2-ps laser pulse length was much longer than the electron beam rms duration of \(\sigma_{t}=63\) fs. As a result, the laser fields that the electron beam witnessed were close to those of a linearly polarized plane wave, and the absolute energy modulation distribution within the beam was estimated using Equation (4) [40; 41]. \[\Delta\gamma(s)=\sqrt{\frac{P_{0}}{P_{0}}}\frac{2 KL_{u}\gamma}{\gamma w_{0}} \cos{(k_{L}s)}, \tag{4}\]  
where \(P_{0}\) is a constant with the same dimension as that of power, which is defined as \(P_{0}=I_{A}\cdot mc^{2}/e=8.71\) GW with \(I_{A}=4\pi\epsilon_{0}\cdot mc^{3}/e=17.0\) kA. The maximal energy modulation estimated by Equation (4) was \(\Delta\gamma_{max}=3.37\), while the calculation using Elegant was 2.74. The 20% discrepancy between Equation (4) and the Elegant result was due to the imperfect assumption of the laser field profile as a plane wave with a constant amplitude in space as well as neglecting the slippage effect. The simple form of Equation (4) could be used to roughly estimate the laser peak power required to achieve the desired energy modulation, which was to be refined and determined by the Elegant simulations. To describe the effectiveness of the energy modulation, the energy modulation contrast ratio \(\Delta\gamma/\sigma_{\gamma,0}\) for the beam downstream of the laser modulator in lattice 1 is shown in Figure 3. The modulation amplitude was achieved at \(A_{L}=23.6\), with the uncorrelated energy spread \(\sigma_{\gamma,0}=0.116\) before the beam entered the laser modulator. After laser modulation, the uncorrelated energy spread of the beam increased to 0.118. This increase in the uncorrelated energy spread could have been due to the differential of the laser electric field strength in the horizontal direction, resulting in slightly inconsistent beam horizontal momentum modulations at different horizontal coordinates. In addition, the uncompensated transverse dynamics induced by the laser modulation could have been a contributor, as well. Note that this modulation amplitude \(A_{L}\) was different from the LABC modulation parameter \(A\) of the entire accelerator architecture, which was discussed (at BC2 exit) in Section 4.2.2. In addition, because the energy modulation was achieved by the horizontal electric field coupling with the electron undulator motion, the horizontal momentum of the electron beam was also modulated by the laser, as shown in Figure 4, w  
Figure 3: Energy modulation contrast ratio of the entire beam downstream of the laser modulator. The central section of the plot that is shaded in yellow is expanded and provided at the top, showing the details of the electron beam energy modulation. lattice 1 was used as an example.