# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 2 Accelerator Lattice Overview
The laser modulator had 10 undulator periods, and the laser wavelength used was 1 \(\upmu\)m. The laser modulation of the beam could be described by the energy modulation contrast ratio, defined as the ratio of the energy variation \(\Delta\gamma\) to the uncorrelated energy spread \(\sigma_{\gamma,0}\). The energy modulation contrast ratio varied sinusoidally along the beam, and its amplitude was referred to as the modulation amplitude, symbolized by \(A\) as in Equation (1) [21; 17]. \[A=\frac{\Delta\gamma_{max}}{\sigma_{\gamma,0}} \tag{1}\]  
The modulation amplitude \(A\) was used to evaluate the strength of the laser modulation. Although the energy modulation \(\Delta\gamma\) could be preserved as the beam traverses the entire lattice, the uncorrelated energy spread \(\sigma_{\gamma,0}\) of the beam changed notably. At the position of the laser modulator, we used a laser power of 3.4 GW to generate a modulation amplitude at \(A_{L}=23.6\). Downstream of the laser modulator, various effects could cause the uncorrelated energy spread to increase, e.g., the incoherent synchrotron radiation (ISR) in the second bunch compressor (BC2), so the effective modulation parameter \(A\) of the entire LABC architecture would be less than \(A_{L}\). The modulated electron beam was accelerated to 12.0 GeV by the second section of the linac (L2). L2 also operated at 5.699 GHz. The off-crest acceleration and the short-range wakefield in the RF cavities evened out the chirp in the beam over the entire L2 acceleration  
Figure 2: The evolution of the Twiss parameters, along the beamline for (**a**) lattice 1 using a conventional C-chicane as BC1 and (**b**) lattice 2 using a double C-chicane as BC1. An overview of the lattice elements is provided at the top of the respective plot. Note that the profiles of the \(\beta\)-functions in lattice 2 inside the section of L2 appeared unaffected by the quadrupoles because only a small focusing strength was needed in this case. section. The energy modulation by the upstream LM was preserved after the acceleration by L2. The second bunch compressor reached a compression ratio of 6, compressing each 1-\(\upmu\)m period of the modulated electron beam into a current spike with a peak current at 3.0 kA. The entire electron bunch presented no macroscopic chirp; however, BC2 utilized the energy differential within each 1-\(\upmu\)m period of the modulated beam and converted it into the longitudinal electron density modulation, forming the microbunches (MB) with high current spikes. The normalized dispersive strength of BC2 was defined in Equation (2) [17], where the transfer matrix element \(R_{56}\) of BC2 is used, and \(k_{L}\) is the wave number of the laser used in the LM upstream. \[B=R_{56}k_{L}\frac{\sigma_{\gamma,0}}{\gamma_{0}} \tag{2}\]  
The modulation parameter \(A\) and the compression parameter \(B\) are the core parameters of an LABC architecture because they determine the longitudinal phase space distribution of the beam [17; 40]. The ideal parameter set of \(A\) and \(B\) for our LABC configuration was obtained by minimizing \(A\) while varying \(B\) in order to achieve the desired microbunch current profile, or the microbunched electron beam longitudinal phase space distribution, unique to our design of LABC [17]. Parameter \(A\) was minimized because a smaller \(A\) value led to a lower laser power required and a reduced energy spread within each individual microbunch, constituting a current spike. Instead of maximizing the microbunch peak current by pursuing a prominently spiky current profile for each microbunch, as in the configuration for ESASE, our configuration discretely overcompressed the laser-modulated electron beam for the purpose of forming a comparatively broad flat top at a reduced current level of 3.0 kA at the central section of the current distribution of each microbunch. This particular microbunch current profile had two advantages.