Fig. 4: Enhanced proton acceleration in conduction-restricted foils.

a, b Distributions of the electron energy density D_en at the simulation time of 520 fs, as well as the trajectories of two representative electrons evolved with time. c, d Distributions of the longitudinal acceleration field E_x and transverse current density J_y at the simulation time of 480 fs. J_y is zoomed in by four times along the x axis to indicate the details. The dark dots represent the positions of the most energetic protons. a and c are the results of a D = 30 μm foil, while b and d are those of a planar target, all with an initial foil position of x = 40 μm. e, f Temporal evolution of the net charge and of the maximum proton energy, respectively. The total charge in the region of 38 μm < x < 43 μm and ∣y∣ < 15 μm is considered. g Isosurfaces (1.5n_c) of the carbon and proton densities, as well as the density distribution of electrons at the simulation time of 100 fs. Here, the critical density n_c = 1.74 × 10²⁷ m⁻³ for the 800-nm laser. h Dependence of the maximum proton energy on the target size. The planar targets are treated as D = 500 μm and the error bars indicate the standard deviation. The dashed line shows the calculated proton energy in the Coulomb repulsion model. i Comparison of the measured and simulated proton spectra using a D = 30 μm foil. a–f are obtained from 2D PIC simulations, while g–i are based on a 3D simulation.