Fig. 1. (a) Simulation setup of HC target configurations. (b) Self-discharged current generated by the emitted particles. (c) Spectrum of the current pulse from fast Fourier transform (FFT).

Fig. 2. (a) The spatial-temporal distribution of the current on the straight aluminum wire, where the black dashed line refers to the speed of light. (b) Dependence of the surface current velocities on the ratio of the wavelength to the coil diameter (), with different radii and pitches in the HC and straight wire. The β and d are both set to 1 in the case of straight wire. (c) Distribution of current in the HC, with the velocity mark of βc (black dashed line) and the fitting longitudinal velocity 1.6βc of the main positive peak (green dashed line). (d) Snapshot of the current distributions in the HC at 60, 120 and 180 ps.

Fig. 3. (a) Snapshot of the longitudinal electric field on the central axis of the HC at 60, 120 and 180 ps. The red balls represent the positions of protons with cut-off energy at different times. (b) Spatial-temporal distribution of the longitudinal electric field in an HC, with the mark of the extreme points of positive fields (green dashed lines).

Fig. 4. (a) Snapshots of proton distributions in phase space (x, p_x) and the longitudinal electric field at 60, 240 and 360 ps, in a single-stage HC. (b) Longitudinal electric field (red curve) in the coordinate frame of the traveling highest-energy protons, and the evolution of the cut-off energy (blue curve) in a single-stage HC, where the three groups of green circles mark the three statuses in (a).

Fig. 5. (a) Energy gains of traveling protons in the HC with different input energies. (b) The spatial-temporal distribution of E_x and proton trajectories with input energies of 20 and 30 MeV.

Fig. 6. (a) The structure of a single-stage HC and a two-stage HC. (b) The spatial-temporal distribution of the current in the two-stage HC. (c), (d) The temporal profiles of the current pulses at 4 and 8 mm in the case of single- and two-stage HCs, respectively.

Fig. 7. (a) Longitudinal electric field in the two-stage HC, where the black dashed line indicates the velocity mark of 1.2βc and the green dashed line indicates the extreme points of positive fields. (b) The E_x distribution and the position of protons with initial energy of 25 MeV in single-stage (top) and two-stage (bottom) HCs at 60, 240 and 360 ps, respectively. The red balls represent the protons’ positions at the cut-off energy, and the vertical black lines in (a) and (b) indicate the position of the drift section.

Fig. 8. (a) Snapshots of the proton distributions in phase space (x, p_x) and E_x at 60, 240 and 360 ps, in the two-stage HC. (b) Longitudinal electric field (red curve) in the coordinate frames of the traveling protons at the cut-off energy, and the evolution of the cut-off energy (blue curve) in the two-stage HC. The three groups of green circles mark the three snapshots in (a).

Fig. 9. (a) Snapshots of the current distributions and the positions of protons at cut-off energy in the single-stage HC at 90, 96 and 102 ps. (b) Energy gain by varying the helical length of the single-stage HC and the drift length of the two-stage HC. (c) Spectrum of the input protons (black line); spectrum after a single-stage HC of 8 mm (green dashed line) and 20 mm (blue dashed line); spectrum after a two-stage HC (red line).

Fig. 10. (a) Expected target charge of escaped electrons in the logarithmic scale calculated from the model as a function of laser intensity (blue solid line). The green dashed line shows the cut-off energy of laser-driven protons against laser intensity. The requirements of the hundreds-of-terawatts laser and the petawatt laser in the simulations are marked with red circles and rhombuses, respectively. (b) Spectrum of the input protons in the simulations with the petawatt laser (black line); spectrum after a single-stage HC (blue line); spectrum after a two-stage HC (red line). The lengths of the single-stage HC and two-stage HC are 10 and 40 mm, with and , respectively, and the drift length is 6.6 mm.

Fig. 11. (a) Scheme of the reflection ringing of a three-stage HC structure. (b) Spatial-temporal distribution of the current in a three-stage HC. (c) Spatial-temporal distribution of E_x in a three-stage HC, where the black dashed lines in (c) mark the velocity of 1.2βc and the vertical black lines in (b) and (c) represent drift sections. (d) Energy gain (red curve) and maximum intensity of E_x (blue curve) at different stages of the HC (from one to four stages in different yellow regions).