Fig. 1. At t = 862 fs, (a) side view and (b) frontal view of the LS pulse (orange), high-density protons (green) and electrons (blue/purple) from different views. At this time, the field of the LS pulse is considerably weakened. The color bar represents electron density, which is normalized to n_c. (c) At t = 92 fs, the electric field distribution of the LS pulse E_y in the x–y plane at z = 0. (d) Acceleration field E_x in the x–y plane at z = 0 and t = 862 fs. The electric field is normalized to m_eω₀c/e.

Fig. 2. (a) Electron density and (b) proton density in the x–y plane at z = 0 and t = 862 fs.

Fig. 3. (a) At t = 1.02 ps, longitudinal momentum distributions of protons, with the momentum normalized to m_pc and m_p denoting the proton mass. (b) Energy spectra of protons and (c) angular distribution of proton energy, where the color bar represents the number of protons normalized to the maximum number of protons.

Fig. 4. Dependence of proton momentum on the bubble velocity, where v_p is normalized to c, and P_x is normalized to m_pc. The black square indicates the theoretical result, and the red square indicates the simulation result.

Fig. 5. At t = 92 fs, electric field distribution of (a) the LS pulse, (c) the LG pulse and (e) the GS pulse in the x–y plane at z = 0. The black line represents the selection of the line for the one-dimensional plot of the electric field. At t = 862 fs, acceleration field of (b) the LS pulse, (d) the LG pulse and (f) the GS pulse in the x–y plane at z =0. The black line represents the selection of the line for the one-dimensional plot of the acceleration field.

Fig. 6. At t = 862 fs, the acceleration field of (a) the LG pulse and (d) the GS pulse. Electron density in the cases of (b) the LG pulse and (e) the GS pulse in the x–y plane at z = 0. At t = 1.02 ps, the energy spectra of protons driven by (c) the LG pulse and (f) the GS pulse.

Fig. 7. (a) At t = 1.17 ps, 3D longitudinal momentum distributions of witness protons for the LS pulse as the driving pulse. (b) Energy spectra of witness protons driven by LS, LG and GS pulses, respectively.

Fig. 8. Intensity patterns for imperfect LS pulses combined by sub-LG beams with random initial phases in 12 simulation runs.

Fig. 9. (a) Proton density, (b) electron density and (c) acceleration field E_x in the x–y plane at z = 0 and t = 862 fs when the imperfect LS pulse in Figure 8(a12) (combined by sub-LG beams with random initial phases) is used. The proton and electron density has been normalized to n_c. The electric field is normalized to m_eω₀c/e. (d) At t = 1.05 ps, longitudinal momentum distributions of protons in the x–y plane at z = 0, with the momentum normalized to m_pc and m_p denoting the proton mass.

Fig. 10. At t = 862 fs, the energy spectra of protons accelerated by (a) the imperfect LS laser pulse in Figure 8(a12) and (b) the LG laser pulse at the same power.

Table 1. Topological charge of the nth LG sub-beam constituting the synthesized LS pulse, and the corresponding beam-waist radius and angular frequency ω_n.