# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 6 Beam Analysis at X-ray Undulator Entrance
### Achieved Mitigation of Beam Instabilities  
It was important to emphasize that in the XFEL architecture using the LABC method shown in this work, the desired beam properties at the X-ray undulator entrance were acquired without the application of a laser heater [48], which is typically applied for the purpose of MBI suppression by intentionally increasing the uncorrelated energy spread downstream of the photoinjector. In our accelerator lattice design, the uncorrelated energy spread of the electron beam was consistently maintained at a very low level. The energy spread increase caused by MBI was successfully constrained by using the small magnitude of the \(R_{56}\) of the second bunch compressor, and by placing the second bunch compressor at the end of the electron beam acceleration. A comparison of the URWW effect on the electron beam, processed with our LABC (in lattice 1) method and with the conventional bulk compression method for BC2, is shown in Figure 13. BC2 for the bulk compression of the beam was implemented at 2.5 GeV, producing a peak beam current of 3.0 kA, and the beam was fully de-chirped by the end of the ensuing third section of the linac (L3) to the final beam energy of 12 GeV. This bulk compression was developed in order to compare the URWW effect, and it was not a beamline that had been substantially optimized. The URWW calculations were performed using in-house Python scripts that had been composed based on the analytical theories in [14], assuming that the electron beam properties remained constant while the electron beam traveled though the undulator. The calculations included 3000 undulator periods, with individual undulator periods of 18.6 mm. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{3}{c}{**Value**} \\ \hline  & \(\alpha_{x}\) & \(5.087\times 10^{-5}\) & \\  & \(\beta_{x}\) & \(1.670\) & m \\  & \(\alpha_{y}\) & \(-2.145\times 10^{-4}\) & \\  & \(\beta_{y}\) & \(2.006\) & m \\ Horizontal norm. emittance & \(\epsilon_{n,x}\) & \(98.3\) & nm rad \\ Vertical norm. emittance & \(\epsilon_{n,y}\) & \(86.8\) & nm rad \\ Horizontal rms beam size & \(\sigma_{x}\) & \(2.64\) & \(\mu\)m \\ Vertical rms beam size & \(\sigma_{y}\) & \(2.72\) & \(\mu\)m \\ MB frac. energy spread & \(\sigma_{S,\text{MB}}\) & \(7.5\times 10^{-5}\) & \\ MB beam portion & & \(38.8\%\) & \\ LABC parameters & \(A\) & \(16.1\) & \\  & \(B\) & \(0.0927\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Lattice 1 electron beam Twiss parameters and other properties at X-ray undulator entrance. Beam energy 12 GeV. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{3}{c}{**Value**} \\ \hline  & \(\alpha_{x}\) & \(1.768\times 10^{-4}\) & \\  & \(\beta_{x}\) & \(2.158\) & m \\  & \(\alpha_{y}\) & \(1.885\times 10^{-3}\) & \\  & \(\beta_{y}\) & \(1.292\) & m \\ Horizontal norm. emittance & \(\epsilon_{n,x}\) & \(66.1\) & nm rad \\ Vertical norm. emittance & \(\epsilon_{n,y}\) & \(98.9\) & nm rad \\ Horizontal rms beam size & \(\sigma_{x}\) & \(2.47\) & \(\mu\)m \\ Vertical rms beam size & \(\sigma_{y}\) & \(2.33\) & \(\mu\)m \\ MB frac. energy spread & \(\sigma_{S,\text{MB}}\) & \(7.4\times 10^{-5}\) & \\ MB beam portion & & \(40.4\%\) & \\ LABC parameters & \(A\) & \(16.1\) & \\ \(B\) & \(0.0948\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Lattice 2 electron beam Twiss parameters and other properties at X-ray undulator entrance. Beam energy 12 GeV. The electron beam vacuum chamber was assumed to be made of copper with an inner radius of 3 mm and with the alternating current (AC) conductivity parameter [14] set as \(\Gamma=1.0\). As Figure 13 shows, the fractional energy depression by URWW is one order of magnitude smaller in our LABC method, as compared to the bulk compression method.