# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 4 Bunch Compressors
### First Bunch Compressor
#### 4.1.2 BC1 Using a Double C-Chicane  
The double C-chicane version of BC1 consisted of two C-chicanes (hereafter referred to as C-1 and C-2), with the the offset of the central dipoles in the respective C-chicanes oriented in opposite directions, as illustrated in Figure 1. This was the BC1 configuration used in lattice 2. The purpose of investigating a double C-chicane configuration for BC1 was to have the electron beam present a minimal horizontal momentum shift profile along the beam as it exited BC1, which was likely to be achievable from the horizontal kicks related to the CSR effects of the individual C-chicanes canceling each other [45]. Minimized transverse momentum shift profile would result in a reduced transverse motion of the beam, which was favored for the beam acceleration in L2. In order to reduce the dispersion functions, two weak quadrupoles with 0.05 m lengths were used in each C-chicane and inserted between the two central and the two side bending magnets. In order to perform the optimization of the double C-chicane configuration of BC1 and to reduce the complexity of the optimization by limiting the number of variants involved, the length and the magnetic field of the dipoles in C-1 were set as constants, at \(l_{d,1}=0.600\) m and \(B_{d,1}=0.24\) T. The compression ratio of C-1, the length \(l_{d,2}\) of the dipoles in C-2, and the magnetic field \(B_{d,2}\) of the dipoles in C-2 were used as the variables. Therefore, the optimization was carried out as a 3D parameter scan. The Elegant optimizer was used for each individual 3D parameter scan, with the optimization goal set to minimize the transverse projected emittances of the beam by using the MO1 triplet quadrupole strengths and the two inner drift space lengths as the optimizer variables. When each optimizer scan finished, the resulting beam was divided into 1000 equal-length slices in the longitudinal direction, and the current-weighted standard deviation of the slice-average horizontal kick of all the slices, symbolized as \(\sigma_{\overrightarrow{x}_{slice}}^{\prime}\), was calculated. In other words, the transverse projected emittances were minimized in each 3D parameter scan as the first step, and the value of \(\sigma_{\overrightarrow{x}_{slice}}^{\prime}\) was calculated afterwards based on the optimizer results. The goal of the described 3D parameter scan was set to search for the minimum value of \(\sigma_{\overrightarrow{x}_{slice}}^{\prime}\). As an example, a 2D contour plot is provided in Figure 6, showing the calculated value of \(\sigma_{\overrightarrow{x}_{slice}}^{\prime}\) varying according to the C-1 compression ratio and to the C-2 bending magnet dipole length \(l_{d,2}\), in a C-2 dipole magnetic field of \(B_{d,2}=0.10\) T. In the figure, the white marker is placed at the position of the minimum value of \(\sigma_{\overrightarrow{x}_{slice},min}^{\prime}=0.135\)\(\mu\) rad with \(l_{d,2}=0.600\) m and a C-1 compression ratio of 7.50. This 2D scan was performed for each \(B_{d,2}\) value investigated, with the corresponding minimal value of \(\sigma_{\overrightarrow{x}_{slice}}^{\prime}\) recorded, constituting a 3D parameter scan. A summary plot of the entire 3D parameter scan is shown in Figure 7. The value of \(\sigma_{\overrightarrow{x}_{slice},min}^{\prime}\) increased monotonically with the increase in \(B_{d,2}\), and no global minimum of \(\sigma_{\overrightarrow{x}_{slice}}^{\prime}\) could be identified. Therefore, the determination of the double C-chicane design parameters needed to take into account additional factors, e.g., the electron beam properties at the bunch compressor exit and the lattice's total length. A shorter lattice length is \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{2}{c}{**Value**} \\ \hline Compression ratio & \(C_{1}\) & 25 & \\  & \(R_{56}\) & \(-41.2\) & mm \\ Dipole magnetic field & \(B_{d}\) & 0.60 & T \\ Dipole length & \(l_{d}\) & 0.315 & m \\ BC1 total length & & 8.11 & m \\ \hline \hline \end{tabular}
\end{table}