## 3 Simulation Results  
To understand this interesting phenomenon and gain insight into the experimental results, 2D PIC simulations were performed using the Vorpal code. The simulations showed that after the laser pump reached the depletion length, the acceleration mode changed from LWFA to PWFA gradually, and the wakefield driven by the high-charge \(e\)-beam dominated the following electron acceleration, so the low-charge \(e\)-beam could be accelerated sustainably. In the simulation, a linearly polarized Gaussian laser pulse, with wavelength \(\lambda_{0}=0.8\) um, normalized amplitude \(a_{0}=1.6\), FWHM pulse duration \(\tau=33\) fs, and FWHM spot size \(\omega_{0}=30\) um, was inserted from the left boundary of a moving window. The window size was \(60\times 76.8\) um\({}^{2}\) in the longitudinal and transverse directions, and the grid cell size was determined as \(\Delta x=0.04\) um, \(\Delta y=0.064\) um. The laser pulse propagated in a 7.04 mm-long plasma, with the density profile of the plasma set to fit the measured density. The phase variation of the plasma wave is affected both by the density distribution and the variation of the laser pulse intensity, and the phase velocity of the base of the bubble can be approximated as \(\beta_{p}\approx-2/\left(\text{ck}_{p}\sqrt{a}\right)\times\frac{\text{d}a}{ \text{d}t}+1/\left(1-\pi/k_{p}^{2}\times\text{d}k_{p}/\text{d}x\right)\)[26], where \(k_{p}\) is the plasma wave number and c is the light speed in vacuum. The evolution of the laser intensity \(a_{0}\) is shown in Figure 3b. At the first density plateau (\(\text{d}k_{p}/\text{d}x=0\)), the laser intensity \(a_{0}\) increased from 1.6 to 3, owing to the laser self-focusing, and the phase velocity \(\beta_{p}\) decreased rapidly, which caused the self-injection of the first \(e\)-beam at \(x=0.88\) mm. After that, the laser intensity decreased and the phase velocity \(\beta_{p}\). began to increase at \(x=1.04\) mm; therefore, the following injection was prohibited. Compared with our previous work [7; 21], the density bump was introduced later due to the larger diameter of the first gas nozzle. Although the laser experienced further intense self-focusing because of the special plasma density profile, the upward region of the density bump from \(x=2.0\) mm to 2.2 mm could counteract the influence of the increased laser intensity [7], and \(\beta_{p}\) still remained higher than the light speed in a vacuum. What followed was a downward density ramp (\(\text{d}k_{p}/\text{d}x<0\)), taking the effects of the decreasing density and the increasing laser intensity into account, and the density-gradient injection of the second \(e\)-beam occurred at \(x=2.4\) mm. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & **E\({}_{\text{h}}\) (MeV)** & \
\begin{tabular}{c} **FWHM** \\ \(\Delta\)**E\({}_{\text{h}}\)/**E\({}_{\text{h}}\)** \\ \end{tabular} & **Q\({}_{\text{h}}\) (pC)** & **E\({}_{\text{1}}\) (MeV)** &
\begin{tabular}{c} **FWHM** \\ \(\Delta\)**E\({}_{\text{l}}\)/**E\({}_{\text{l}}\)** \\ \end{tabular} & **Q\({}_{\text{1}}\) (pC)** \\ \hline \#1 & 367.3 & 10.6\% & 87.6 & 298.8 & 6.23\% & 15 \\ \#2 & 436.3 & 1.4\% & 9.5 & 323.2 & 19.3\% & 81 \\ \#3 & 640.7 & 1.1\% & 11.7 & 230.5 & 50.3\% & 92.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of \(e\)-beams corresponding to Figure 2a. The subscript “h” represents the high-energy \(e\)-beam, and the subscript “l” represents the low-energy \(e\)-beam. The evolution of the transverse laser beam profile along the laser propagation is shown in Figure 3a. Figure 3a,b indicates that the laser pump depletion length was approximately 5.44 mm. Figure 3d-f shows three snapshots of the electron density distribution and the wakefield structure.