# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression
## 3 Laser Modulator  
The laser modulator was implemented by a planar undulator and a high-power laser co-propagating with the electron beam. The laser modulator was 1.5 m long with 10 undulator periods. The laser wavelength was 1 \(\upmu\)m, and the laser power used was 3.4 GW, with a pulse length of 1.2 ps (4 mJ). The parameters regarding the undulator and the laser beam used in the laser modulator are listed in Table 1, where we abided by the notations used in the references [17; 40; 41]. The reasons that led to the determination of the laser wavelength being 1 \(\upmu\)m were as follows. On the one hand, we pursued a laser wavelength as small as possible. As mentioned in Section 2, the LABC parameters \(A\) and \(B\) determined the distribution of the longitudinal phase space of the electron beam. For a determined LABC compression parameter \(B\) that included a given uncorrelated fractional energy spread, as indicated in Equation (2), a greater laser wave number \(k_{L}\), i.e., a smaller laser wavelength \(\lambda_{L}\), enabled the usage of a smaller \(R_{56}\) value for the second bunch compressor, which was beneficial in suppressing the gain of MBI [13]. On the other hand, there was a lower limit for the value of the laser wavelength. As mentioned above, it was the microbunches formed inside the electron beam that would participate in the X-ray lasing. Meanwhile, we assume that the X-ray photons stemmed from the center of the electron microbunches. As a result, the half-width at the half-maximum (HWHM) of the microbunches (\(w_{z,\text{MB,HWHM}}\)) in the electron beam had to be large enough to cover the X-ray radiation coherence length [42; 43] for the slippage considered, so that the radiation coherence was assured for the entire X-ray lasing duration in the undulator. This criterion was acquired by Equation (3). \[w_{z,\text{MB,HWHM}}\geq l_{coh,X-ray} \tag{3}\]  
On condition that the desired LABC beam current profile was attained, a smaller laser wavelength would lead to a smaller value of \(w_{z,\text{MB,HWHM}}\). Equation (3) thus set the lower limit for the laser wavelength to be used. We defined the microbunches by the sections of the electron beam that contributed to a beam current above 2.0 kA, resulting in a microbunch HWHM of \(w_{z,\text{MB,HWHM}}=33.6\) nm from the simulation results, where 1-um laser wavelength was presumed. In the MaRIE XFEL, a reasonable estimate of 3000 undulator periods resulted in an X-ray coherence length of \(l_{coh,X-ray}=28.7\) nm, and thus the relationship in Equation (3) was satisfied. Therefore, our selection of a laser wavelength of 1 um was appropriate and justified. In the discussion above, we defined the lower limit of the microbunch beam current as 2.0 kA. As compared to the beam current level of 3.0 kA, 2.0-kA beam current resulted in a 30% longer gain length, according to our Ming-Xie parameterization analysis [44] (see Section 6.3 for basic undulator parameters). Moreover, the X-ray lasing power per unit charge for 2.0-kA beam current was 40% lower than that in the case of the 3.0-kA beam current. Therefore, the beam portion with a beam current below 2.0 kA had a significantly smaller contribution to the X-ray photon generation. In addition, we noted that the 2.0-kA lower limit of the microbunch beam current led to a microbunch HWHM sufficient for accommodating the slippage effect during the X-ray lasing process. The choice of the 2.0-mm laser waist size was made with the purpose of guaranteeing a homogeneous laser field strength that the electron beam witnessed on a certain transverse plane, with a reasonable laser power. It was not optimized for the best laser modulation efficiency with the electron beam parameters provided. A carefully optimized laser configuration by 3D laser-electron matching analysis with an effectively smaller Rayleigh length should be able to reduce the laser power required by the laser modulation of the electron beam by a significant amount.