Fig. 3

Experimental maximum proton energy vs. on-target peak intensity (\({{I}}_{{\rm L}}{{\lambda }}_{{\rm L}}^2\)). For the LULI experiments (black or blue open symbols), the targets were Al and Au foils of thickness between 0.5 and 2 μm and of transverse dimensions > 1 × 1 mm². For the SNL experiment (red open symbols), the targets were 1.1 μm thick Au foils with transverse dimensions between 50 × 50 μm² and 300 × 300 μm² (note that no trend on the size was observed). Each point corresponds to a single laser shot. Lines plot predictions of the 1D plasma expansion model considered in Figs 1 and 2, and which neglects magnetic field effect (see Methods). Filled symbols represent 2D PICLS simulation results with 2 μm-thick Al targets: filled circles and triangles are for 50 ×50 μm² wide targets and filled red squares for 20 × 20 μm² wide targets. The error bars on the proton energy quantify the energy range of the radiochromic films used to diagnose the TNSA protons. The error bars in the laser intensity arise from the uncertainty in estimating the encircled laser energy within the FWHM spot. (inset) Experimental setup employing a refocusing ellipsoidal plasma mirror (EPM) to reduce the laser focal spot and increase the laser intensity³¹