# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 4 Bunch Compressors
### Second Bunch Compressor
#### 4.2.1 BC2 Mechanism, Design, and Performance  
Figure 8 illustrates the bunch compression mechanism of BC2. As stated above, BC1 performed a bulk compression of the beam. BC2, however, considered the incoming electron beam as a sequence of modulated beam sections with a period equal to the laser wavelength (1 \(\upmu\)m) of the laser modulator (Figure 8a); BC2 compressed each individual modulated period utilizing the energy differential within that laser wavelength, converting it to the longitudinal electron density modulation (Figure 8b) and shaping the beam current into a sequence of 3-kA current spikes. In addition, the modulation amplitude decreased at the BC2 exit, as compared to the results at the L2 exit, and this was due to the uncorrelated energy spread of the beam increasing from \(\sigma_{\gamma,0}=0.118\), at L2 exit, to 0.170, at BC2 exit, by the ISR effect in BC2. The major advantage of performing the second stage of the bunch compression by enhancing the peak current in each modulation period created by laser modulation, instead of simply by conventional bulk compression, was that the absolute value of \(R_{56}\) of BC2 could be greatly reduced. A smaller \(R_{56}\) reduced the energy spread increase caused by MBI. Meanwhile, however, as shown in Figure 8b, the beam current did not decrease to zero between the spikes. As a result, a portion of the charge in the beam could not participate in the X-ray lasing of the undulator. The compression ratio of BC2 was 6, as measured by the ratio of the downstream current spike peak of 3 kA to the upstream average beam current of 500 A. A triplet  
Figure 8: Longitudinal phase space of the electron beam plotted in the form of the energy modulation contrast ratio \(\Delta\gamma/\sigma_{\gamma,0}\) vs. the relative longitudinal coordinate, and beam current plotted within one laser wavelength (1 \(\upmu\)m), (a) before and (b) after the bunch compression by BC2. Within each energy modulation period by the laser modulator, BC2 converts the energy modulation into the longitudinal electron density modulation, enhancing the average beam current of 500 A prior to BC2 to a current spike of 3 kA downstream of BC2. Note that the modulation amplitude after BC2 decreased due to the increase in the uncorrelated energy spread by the ISR effect in BC2. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{3}{c}{**Value**} \\ \hline  & \(\alpha_{x}\) & \(-0.4179\) & \\  & \(\beta_{x}\) & \(426.1\) & m \\  & \(\alpha_{y}\) & \(-0.3318\) & \\  & \(\beta_{y}\) & \(458.6\) & m \\ Horizontal norm. emittance & \(\epsilon_{n,x}\) & \(93.3\) & nm rad \\ Vertical norm. emittance & \(\epsilon_{n,y}\) & \(86.8\) & nm rad \\ Horizontal rms beam size & \(\sigma_{x}\) & \(41.4\) & \(\upmu\)m \\ Vertical rms beam size & \(\sigma_{y}\) & \(41.2\) & \(\upmu\)m \\ Unco. frac. energy spread & \(\sigma_{\delta,0}\) & \(5.04\times 10^{-6}\) & \\ Co. frac. energy spread & \(\sigma_{\delta}\) & \(7.76\times 10^{-5}\) & \\ rms beam length & \(\sigma_{z}\) & \(18.6\) & \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Twiss parameters and other beam properties at L2 exit, as in lattice 1. Beam energy 12 GeV. using quadrupoles, 0.15 m in length (MO3 in Figure 1), was used between the L2 exit and BC2, so the emittance dilution due to CSR in BC2 was minimized. Two 0.05 m long weak quadrupoles were used between the first and the second bending magnets as well as between the third and the fourth bending magnets, to reduce the dispersion functions. The design parameters of BC2 are shown in Table 6.  
Downstream of BC2, the projected horizontal normalized emittance of the beam increased to 98.3 nm rad, likely due to CSR, while the vertical normalized emittance remained unchanged at 86.8 nm rad. The slice analysis of the beam divided the beam into 20,000 slices of equal lengths, and the results are provided in Figure 9. When comparing Figure 9a.i and b.i, we noted that the periodic 3 kA current spikes were formed in the beam.