## 3 Simulation Results
As shown in Figure 3d, the second \(e\)-beam (circled in white) with a charge of 6.4 pC was injected into the wakefield as the laser propagated to \(x\) = 2.4 mm, and the first \(e\)-beam (circled in red) with a total charge of 30.6 pC preceded the second \(e\)-beam with a separation distance of 12.4 \(\upmu\)m. At \(x\) = 4.8 mm (Figure 3e), the first \(e\)-beam reached the front part of the bubble and entered into the deceleration region. When the laser propagated to \(x\) = 7.04 mm, the laser pump depleted. Figure 3f shows a new bubble structure without any laser which was excited by the first \(e\)-beam. Figure 3c shows the evolution of the energy of the two-color \(e\)-beams. When the first \(e\)-beam reached its dephasing limit, its peak energy was about 441.1 MeV and that of the second \(e\)-beam was 358.2 MeV. With the sustainable acceleration of the PWFA, the second \(e\)-beam finally gained a peak energy of 649.1 MeV with an energy spread of 1.2% On the other hand, since the first \(e\)-beam drove the wakefield, it lost energy up to 244.3 MeV with a large energy spread of 59.6%. Because the charge of both the \(e\)-beams was far below the beam-loading loaded charge, the beam-loading effect could be neglected [27]. The energy distribution of the e-beams along the x-axis with the on-axis wakefield \(E_{x}\) at different times is shown in Figure 4a-f. Figure 4a-d indicates that because of the decay of the driving laser intensity, the maximum longitudinal acceleration electric field \(E_{max}/E_{0}\) decreased from 0.048 (x = 2.4 mm) to 0.012 (x = 5.44 mm), where \(E_{max}\) is the maximum acceleration field and \(E_{0}\) is the cold nonrelativistic wave-breaking field [9]. However, as the laser pulse propagated to \(x\) = 5.84 mm (Figure 4e), \(E_{max}/E_{0}\) increased up to 0.033, which implies that the driving source of the wakefield had gradually changed from the laser pulse to the \(e\)-beam. Figure 3: (**a**) Evolution of the transverse beam intensity along the laser propagation. (**b**) Evolution of the on-axis normalized vector potential. The dark dotted line represents the distance of laser pump depletion. (**c**) Evolution of the \(e\)-beams’ peak energy and FWHM energy spread. The red line represents the energy and the blue line represents the FWHM energy spread. The star corresponds to the first \(e\) beam and the dot corresponds to the second \(e\)-beam. (d–f) Snapshots of the \(e\)-beams’ density distribution and wakefield structure for (**d**) \(x\) = 2.4 mm, (**e**) \(x\) = 4.8 mm, and (**f**) \(x\) = 7.04 mm. Finally, we analyzed the energy spread variation of both \(e\)-beams when the acceleration mode changed from LWFA to PWFA. The energy spread of an \(e\)-beam consists of two components: the transverse energy spread \(\Delta\varepsilon_{T}\), and the longitudinal energy spread \(\Delta\varepsilon_{x}\). Because \(\Delta\varepsilon_{T}\) remains unchanged in the LWFA stage [28] and is compressed in the PWFA stage [15], the analysis of only \(\Delta\varepsilon_{x}\) is considered in the following discussion for simplicity. For an initial injected \(e\)-beam in an LWFA, the energy chirp is usually negative, with higher-energy electrons in the front and lower-energy electrons at the tail, so the initial longitudinal energy spread \(\Delta\varepsilon_{x0}<0\). The evolution of the e-beam longitudinal energy spread can be described as \(\Delta\varepsilon_{x}=\Delta\varepsilon_{x0}+\text{e}E^{\prime}l_{x}l_{a}\), where \(E^{\prime}\)=\((E_{front}-E_{back})/l_{x}\) is the slope of the wakefield, corresponding to its dispersion, \(l_{x}\) is the e-beam length, and \(l_{a}\) is the acceleration length [25]. When the \(e\)-beam entered the acceleration region in the LWFA, \(|\Delta\varepsilon_{x}|\). decreased because the dispersion of wakefield in this region was positive (\(E^{\prime}>0\)) and therefore, compensated for the negative energy chirp of the \(e\)-beam.