# 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
The physical mechanism that led to a significantly smaller URWW effect (when using the LABC method) was that the total beam length of an LABC electron bunch was much longer than the length of the bulk compression bunch, as shown in the comparison of the abscissa scales in Figure 13. When using the LABC method, the bunch compression of BC2 did not shorten the total length of the electron beam. The longer beam length caused the longitudinal resistive-wall wakefield inside the LABC bunch to be much weaker than that of the bulk compression beam [14,17]. After the imprint of the URWW effect on the beam energy distribution, the average fractional energy spread in the individual LABC microbunches was increased to \(7.6\times 10^{-5}\), and the total fractional energy spread of all microbunches as an ensemble was \(8.0\times 10^{-5}\). Therefore, the URWW degradation of the microbunch properties of the LABC beam was minimal. In comparison, after applying the URWW energy depression, the central section of the bulk compression beam that produced a fractional energy spread less than \(1.5\times 10^{-4}\) constituted 44% of the entire beam charge, which was similar to the portion of the LABC beam charge in the microbunches that participated in the X-ray lasing. However, the charge portion of the central section of the bulk compression beam that contributed to a fractional energy spread, up to \(8.0\times 10^{-5}\), was only 19%. As a result, the LABC method was favorable in terms of achieving a better performance and reducing the negative effects of URWW. In Tables 8 and 9,when using our LABC method, the portion of the beam charge that participated in the X-ray lasing in the undulator was around 40%. This was smaller than the X-ray lasing beam charge portion typical in XFEL architecture using exclusively bulk beam compression schemes. Nonetheless, with the presentation of the capabilities of mitigating the beam instabilities, of preserving the ultra-low beam emittance, and of producing a beam with very low energy spread, the proposed XFEL architecture using an LABC method provides a competitive alternative to the conventional XFEL architecture. In addition, due to the overcompression of the microbunch in our LABC configuration, the lasing portion of the beam charge, as shown in Tables 8 and 9, was greater than in the ESASE scenario [17], where the microbunch current spike profile was much sharper. quadrupole arrangement was used to focus the beam for the entire length of the undulator. The length and strength of the quadrupoles, together with the length of the undulator segments, were used as variables in order to determine the maximal saturation lasing power. Because these preliminary simulations were not time dependent, the electron beam properties were not able to evolve over the course of the undulator. The simulations demonstrated 42 keV photon energy X-ray lasing with more than \(2.0\times 10^{10}\) photons per pulse. A refined undulator design, e.g., with carefully engineered tapering, would be sufficient to enhance the X-ray photon generation for the MaRIE requirement of \(5.0\times 10^{10}\) photons per pulse. However, the beam properties resulted in an estimated fractional X-ray spectral bandwidth of \(8.7\times 10^{-4}\), which was greater than the MaRIE requirement. This relatively large bandwidth was caused by the extremely small transverse beam size that had been employed to minimize the gain length and create margins for the design of the undulator beam focusing. In order to reduce the X-ray spectral bandwidth, a revised MO4 in lattice 1 was then tested to focus the electron beam at the entrance of the undulator with a horizontal and vertical rms beam size of 25 \(\upmu\)m. In this scenario, the estimated fractional spectral bandwidth of the X-ray pulse was reduced to \(1.9\times 10^{-4}\), satisfying the MaRIE requirement but increasing the saturation length.