Fig. 1. (a) Experimental data from Schreiber et al.^[57] and the prediction of the nonrelativistic TNSA model. (b) Optimal pulse duration for the specific example of , , for the nonrelativistic consideration (Equation (12), dashed) and the relativistic consideration (Equation (29), solid). The ultra-relativistic limit is given by the dash-dotted line. For larger the curves are globally shifted to respective larger optimal pulse durations.

Fig. 2. Maximum proton and carbon ion energies for varying thicknesses of nm-thin DLC foils reported in Henig et al.^[51]. The solid curve represents the prediction for RPA, Equation (41), using the parameters , , , , . The optimum mass/thickness is indicated by the transition of the solid to a dashed curve.

Fig. 3. Optimum ion energies predicted by the models for TNSA, Equations (29)–(32) (black), and RPA, Equations (41)–(43) (red). The parameters are , , (solid) and (dashed). Some selected experimental results are represented by blue squares (Bin et al.^[112], Henig et al.^[51], Mackinnon et al.^[108], Zeil et al.^[113], Ogura et al.^[68], Jong Kim et al.^[114], Green et al.^[115], Jung et al.^[116]) and theoretical results obtained from PIC simulations are marked by green circles (Pukhov^[71], Wang et al.^[117], Qiao et al.^[40], Sgattoni et al.^[118], Yan et al.^[75], Esirkepov et al.^[35]). For details, see text.