# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 5 Linac Sections
### First Linac Section  
There were two goals for the first linac section: to accelerate the electron beam to an energy of 750 MeV and to prepare the longitudinal energy chirp inside the beam for the downstream bunch compression by BC1. To achieve these goals, L1 was divided into two  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameter** & \multicolumn{3}{c}{**Value**} \\ \hline  & \(\alpha_{x}\) & \(-14.78\) & \\  & \(\beta_{x}\) & \(201.69\) & m \\  & \(\alpha_{y}\) & \(-14.79\) & \\  & \(\beta_{y}\) & \(201.74\) & m \\ Horizontal norm. emittance & \(e_{n,x}\) & \(54.5\) & nm rad \\ Vertical norm. emittance & \(e_{n,y}\) & \(54.5\) & nm rad \\ Horizontal rms beam size & \(\sigma_{x}\) & \(191.4\) & \(\mu\)m \\ Vertical rms beam size & \(\sigma_{y}\) & \(191.4\) & \(\mu\)m \\ Unco. frac. energy spread & \(\sigma_{S,0}\) & \(1.34\times 10^{-5}\) & \\ rms beam length & \(\sigma_{z}\) & \(477.4\) & \(\mu\)m \\ \hline \hline \end{tabular}
\end{table}
Table 7: Twiss parameters and other beam properties at the photoinjector exit. Beam energy 153 MeV.  
sections. The first section was the 5.699 GHz linear accelerator, and the second section was the linearizer operating at 17.098 GHz (3 times the operating frequency of the linear accelerator section).  
The linear accelerator section boosted the electron beam energy by off-crest acceleration at a phase of 76.33 degrees from 153 MeV at the photoinjector exit to 851 MeV, over a distance of 6.00 m. The accelerator cavities were assumed to operate at an accelerating gradient of 120.0 MV/m. The evolution of the longitudinal energy distribution within the beam over the linear accelerator section is shown in Figure 10a, at the photoinjector exit, and extends to Figure 10b, at the end of the linear accelerator section.  
The linearizer section operated at a decelerating phase of \(-92.64\) degrees, and the beam energy was reduced from 851 MeV to 750 MeV at the exit of L1, over a distance of 2.68 m. The magnitude of the accelerating gradient of the linearizer was 38.1 MV/m. The function of the linearizer was to rectify the shape of the longitudinal energy chirp, increasing the linearity in the relationship between the energy and the relative longitudinal coordinate of the electrons in the beam. The finalized chirp is shown in Figure 10c, where the curve exhibited a very prominent linearity. However, the second order variation still remained in the chirp profile to cancel the nonlinear \(T_{566}\) effect in BC1.  
Figure 11 shows the increase in the uncorrelated energy spread along the beam, as it traverses L1. At the photoinjector exit, the uncorrelated energy spread was below 2.1 keV. By the end of L1, the uncorrelated energy spread grew, in general, by approximately 0.2 keV, with the intra-beam scattering (IBS) considered in the Elegant simulation. As also shown in Figure 11, if IBS was excluded from the simulation, the change of the uncorrelated energy spread was trivial. Therefore, other factors, e.g., the longitudinal space charge force, had negligible influence on the uncorrelated energy spread in L1.  
Figure 10: The evolution of the longitudinal energy distribution within the beam from (**a**) the photoinjector exit, (**b**) at the end of the linear accelerator section, and (**c**) at the end of the linearizer section. The final form of the longitudinal energy distribution within the beam is the chirp that is used in BC1.  
The horizontal and vertical normalized emittances increased from 54.4 nm rad at the photoinjector exit (Table 7) to 65.9 nm rad at the L1 exit (Table 2). This was due to the chromatic RF focusing by the linac cavities and was a correlated effect, which could be compensated for by the instances of the matching optics downstream.