Fig. 5: Simulation results of the density scan.
a Summary of the conducted simulation runs using a normalized laser amplitude of a0 = 33. The plasma density profile is represented by the shadow diameter d (lower x-axis) and the peak density at the target center (upper x-axis). Maximum proton energies \({E}_{\max }\) calculated at 240 fs are displayed in the upper part. 0 fs refers to the time when the peak intensity reaches the target center. The green line shows the energy for each target configuration averaged over 9 simulation runs and the shaded green area represents the standard deviation (refer to Supplementary Fig. 5). For simulation runs with average \({E}_{\max }\), the maximum on-axis density \({n}_{\max }\) (blue triangle, retrieved at 0 fs) are shown below. The blue dashed line indicates the theoretical threshold for relativistic transparency γnc,800nm. The bottom part also shows the laser pulse energy transmitted through the target (red plus). For comparison to the experiment, proton energies of the best-performing shots (orange circles as in Fig. 3a) are included as well. Error bars indicate the uncertainties of the shadow diameter measurement. b–d display the calculated (at 240 fs) proton energy spectra in the xy-plane as a function of the emission angle ϕ for representative simulation runs indicated by arrows in a with d = 6.2 μm (b), d = 18.5 μm (c) and d = 33.5 μm (d). ϕ = 0∘ indicates the laser propagation direction.
