# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 6 Beam Analysis at X-ray Undulator Entrance
### Beam Properties  
The last matching optics (MO4 in Figure 1) downstream of BC2 shaped and transported the beam for the final input into the X-ray undulator. The designed horizontal and vertical rms beam sizes at this position were approximately 2.5 \(\upmu\)m, with a minimal magnitude of \(\alpha\)-functions. This beam size had been determined as a preliminary selection for the X-ray lasing investigations according to the Ming-Xie parameterization method [44], which indicated that an average \(\beta\)-function smaller than 2 m at the entrance of the undulator was essential for yielding X-ray pulses with a sufficient number of photons to meet the MaRIE requirement. Because it was the microbunched portion of the electron beam that participated in X-ray lasing, the energy spread calculation considered only the sections inside the beam where the current was above 2.0 kA. We present the simulation results of the electron beam at the X-ray undulator entrance of lattice 1 (Table 8, Figure 12a.i-f.i) and lattice 2 (Table 9, Figure 12a.ii-f.ii).  
Figure 11: Illustration of the uncorrelated energy spread distribution along the beam, plotted for the beam at the photoinjector exit; at the L1 exit with IBS calculation included in the Elegant simulation; and at the L1 exit, without considering IBS.  
Microbunches carrying a current spike at 3.0 kA were formed in the beam for both lattices 1 and 2 (Figure 12a). The beam in lattice 1 had a comparatively more consistent transverse centroid distribution than the beam in lattice 2, the head and tail sections of which had notable horizontal centroid shifts acquired via BC1 bunch compression using a double C-chicane (Figure 12b). The beam in lattice 1 had a horizontal average momentum shift variation between \(-2\) and \(2\)\(\upmu\)rad, as compared to the negligible horizontal average momentum shift variation in the beam in lattice 2, except for the mild perturbations in the head and tail sections (Figure 12c). Figure 12d shows that both the beams in lattices 1 and 2 has their slice emittances preserved after traversing the lattices. The beam in lattice 2 has a comparatively more consistent profile of transverse rms sizes along the beam, while the beam in lattice 1 has greater vertical rms sizes at the head and tail sections (Figure 12e). Finally, the central sections of the electron beams from both lattices 1 and 2 exhibited a steady value of approximately \(1\)\(\upmu\)rad for the horizontal and vertical rms momentum.  
The Twiss parameters and the other properties of the beam at the entrance of the X-ray undulator, as shown here, have not ultimately been optimized for X-ray lasing performance in the undulator; therefore, the simulation results shown in this section serve as a nominal example of the beam preparation achieved by the end of the accelerator lattice. Prior research [17] indicated a projected horizontal normalized emittance of 200 nm rad and an uncorrelated fractional energy spread within the current spikes below \(1.5\times 10^{-4}\) in order for the MaRIE XFEL, when using the LABC method, to meet the requirements for X-ray lasing performance. In comparison, the beam properties provided in Tables 8 and 9 for both lattice 1 and 2 easily met these requirements.  
Figure 12: Comparison of the longitudinal distributions of the electron beam properties at the entrance of the X-ray undulator, after the beam traversed lattice 1 (**a.i–f.i**) and lattice 2 (**a.ii–f.ii**). The beams are divided into 20,000 slices with equal lengths, and the beam properties per slice are shown for (**a**) beam currents, (**b**) transverse centroid shifts, (**c**) transverse average momentum shifts, (**d**) normalized transverse emittances, (**e**) rms transverse beam sizes, and (**f**) rms transverse beam momentum profiles.