Fig. 1. Schematic showing the drive laser irradiating a gas jet of polarized HCl. A plasma channel is formed behind the laser bubble. Some background protons are pushed longitudinally by E_z, and protons with v⃗ch are accelerated radially by the transverse field in the plasma channel. Meanwhile, the spin of protons precesses in the magnetic vortex according to Eq. (3). Transverse expansion of the magnetic vortex enhances the acceleration field at the rear boundary of the target. The forward-moving protons are trapped and accelerated to several hundred MeV.

Fig. 2. (a) Snapshot in z–x space of the plasma charge density ρ (comprising the charge densities of electrons, protons, and chlorine nuclei) at t = 250τ. (b) Snapshot in z–x space of the longitudinal electric field E_z at t = 250τ. The solid green curve depicts the longitudinal electric field along the laser propagation axis at x = y = 0. (c) Density plot in z–p_z space of on-axis protons (with |x| < 2λ). (d)–(f) Density plots in z–x space, z–E space, and z–p_z space, respectively, of energetic protons (with energy larger than Emin=9.4 MeV) at t = 250τ.

Fig. 3. (a) Snapshot in z–x space of the azimuthal magnetic field at t = 250λ/c. Here, the y component of the magnetic field B_y represents the azimuthal magnetic field B_φ in the plane y = 0. (b) and (c) Density plots in z–s_z space and z–s_x space, respectively, of energetic protons (with energy E>9.4 MeV) at t = 250λ/c.

Fig. 4. (a)–(c) Snapshots in x–z space of the plasma charge density ρ, the longitudinal electric field E_z, and the azimuthal magnetic field B_y respectively, at t = 270τ. The light green and magenta lines in (a) show the trajectories of two tracer protons, with diamond and square symbols indicating their positions along the respective trajectory. The solid green diamond and solid magenta square in (a)–(c) indicate the positions of each proton at t = 270τ. (d) Comparison of the electric field increment in one laser cycle, ΔE_z = E_x|_t=270.5τ − E_x|_t=269.5τ (solid green curve), with the radial gradient of the azimuthal magnetic field, ∂_rB_φ|_y=0 = ∂_xB_y (dashed red curve), at z = 270λ.

Fig. 5. (a) and (b) Density plots in z–p_z space and θ_x–p_z space, respectively, of energetic protons at t = 270τ. (c) and (d) Corresponding density plots at t = 280τ.

Fig. 6. (a) Snapshot of the phase-space distribution of energetic protons at t = 320τ. (b)–(d) Snapshots in x–z space of the plasma charge density ρ, the longitudinal electric field E_z, and the azimuthal magnetic field B_y respectively, at t = 320τ. The dashed black lines indicate the position z = 284λ at which the maximum energy of protons appears. The corresponding longitudinal electric field and azimuthal magnetic field at z = 284λ are shown by solid dark green curves in (c) and (d), respectively. As in Fig. 4, the light green and magenta lines and the diamond and square symbols in (b) show the trajectories of two tracer protons, with the solid diamond and square symbols in (b)–(d) indicating their positions at t = 320τ.

Fig. 7. (a) Energy spectra of protons at different times during 260τ ≤ t ≤ 400τ. (b) Density plot in s_z–p_z space of protons at t = 400τ, together with the longitudinal momentum spectra of all protons (solid red curve) and beam protons (dashed black curve). (c) Longitudinal momentum spectra at t = 260τ (black curve) and 280τ (blue curve). The shaded areas indicate the protons corresponding to those in the final accelerated beam. (d) Longitudinal spin spectrum of beam protons, dN/ds_z, as a function of time. The solid blue curve describes the dependence of the beam polarization P=|s⃗| on time. The cutoff energy of the accelerated protons is plotted by the solid red curve. The dashed black curve in (a) corresponds to the accelerated beam protons shown by the black dashed curve in (b). The magenta square and the green diamond in (b) indicated the locations of the two tracer particles, as in Fig. 4. The magenta and green curves in (d) show the variations of their respective values of s_z with time.

Fig. 8. (a) Density plot in E–s_z space of protons at t = 500τ, together with the energy spectra of all energetic protons (solid red curve) and the accelerated beam protons (dashed black curve). The magenta square and the green diamond indicate the locations of the two tracer particles, as in Fig. 4. (b) Longitudinal momentum spectra of accelerated protons at t = 500τ for density ramp lengths d = 20λ (solid curve) and 50λ (dashed curve). (c) Evolution of maximum energy (red curves) and beam polarization (blue curves) over time for d = 20λ (solid curves) and 50λ (dashed curves).

Fig. 9. (a) Density plot in E–s_z space of protons from the three-dimensional simulation at t = 500τ, together with the energy spectrum of all energetic protons (solid red curve). (b) Density plot in s_z–θ space of accelerated protons within the FWHM of the energy peak in (a). The solid red curve shows the corresponding longitudinal spin spectrum and the dashed red curve the spectrum of protons within the FWHM of the energy peak for θ < 10°.