## 3 Simulation Results  
In order to gain more insight into the underlying physics of the electron injection and acceleration process for the different plasma-density modulations, three-dimensional PIC simulations were performed with the FBPIC code [45,46]. The parameters used matched the experimental measurement results. The longitudinal and transverse window sizes were 50 and 120 um, respectively. The grid cell size was as \(\Delta z=40\) nm and \(\Delta r=120\) nm. A linearly polarized laser pulse with a Gaussian function with a wavelength of \(\lambda_{0}=0.8\) um,  
Figure 2: (**a**) The typical shadowgraphs of the baffle and the shock wave when the baffle moved backward (from right to left) at a step size of 50 μm. In each case, the shadowgraph ranged from the middle of the baffle. The position of the baffle at each case was marked aside. The right part shows the measured typical spectra of \(e\) beams that correspond to cases 1 to 8. (**b,c**) The characteristic properties of \(e\) beams at different \(z_{\text{b}}\); all the points are averages of values measured over 100 shots. The orange bar chart in (**b**) represents the reproducibility of the \(e\) beams. normalized amplitude of a\({}_{0}\) = 1.24, pulse duration of \(\tau\) = 25 fs, and spot size of \(\omega_{0}\) = 40 \(\upmu\)m was launched from the left boundary of the simulation box along the z axis and focused at 1.4 mm behind the entrance of the plasma. For better understanding, we focus on three different cases of baffle positions: z\({}_{\text{b}}\) = 450, 550, and 650 \(\upmu\)m, corresponding to the aforementioned cases 6, 4, and 2 in Figure 2a. The locally magnified density profiles in the measurements around the shock wave are plotted in Figure 3a, and the simulated plasma profile has beenchosen to fit the measured one. For these three cases, the abrupt density transitions induced by shock waves can all help injecting electrons into the wakes once the laser focuses to reach the injection threshold near a\({}_{0}\) = 2.4. In fact, the different plasma-density modulations, as well as the evolution of the self-focusing of the laser, contribute to the various durations of the injected \(e\) beams in the PIC simulations, which may actually be contributed by both the density transition and conventional wavebreaking. As the density bump moved forward, a stronger density singularity at the rear of the bubble took shape and more electrons were injected with a higher charge and a larger energy spread. A relatively earlier electron injection at the down-ramp will make the injected electrons move away from the rear of the bubble to the center part more quickly, and they will thus experience a shorter dephasing length and relatively smaller gradient field. Meanwhile, \(e\) beams with a larger charge of \(\sim\)100 pC caused by the later injection at the subsequent down-ramp, which further reduces the local accelerating field for the non-negligible beam-loading effect [47, 48], usually have a lower peak energy due to the lower energy gain, as shown in Figure 2a. For this term, the \(e\) beams can thus be well tuned and optimized according to the actual demands. As seen in Figure 3b,c, the trends of the duration, peak energy, RES, and charge for these cases demonstrated the robust optimization process and its own advantages in this scheme. As shown in Figure 3, the \(e\) beams have different longitudinal characteristics and durations, as the injection periods vary among the cases, which are presented in the shaded areas at different injection positions. For the simulation case of \(z_{\text{b}}\) = 450 \(\upmu\)m, the bright  
Figure 3: (**a**) The measured density distributions (dashed line) and the evolutions of the normalized laser intensity (solid line) in the PIC simulations for baffle positions of z\({}_{\text{b}}\) = 450 (red), 550 (blue), and 650 \(\upmu\)m (black). As the baffle moved toward the nozzle, the shock wave also moved forward and had a higher peak.