Fig. 1. (a) Proton energy spectra for cases I (blue), II (red), and III (black) at t = 100T₀ (T₀ is the laser period). (b) Laser-to-proton energy conversion efficiency as a function of simulation time.

Fig. 2. (a)–(c) Proton angular distributions at t = 100T₀ for cases I–III, respectively. (d)–(f) Spatial distributions of electron density n_e at t = 75T₀ for cases I–III, respectively. The corresponding spatial distributions of the averaged transverse electric field 〈E_y〉 and self-generated azimuthal B-field 〈B_z〉 are shown in (g)–(i) and (j)–(l), respectively. Here 〈 〉 denotes the average over one laser period T₀, e.g., 〈Ey〉=(1/T0)∫t−T0/2t+T0/2Ey(t)dt. The line y = 0 corresponds to the laser axis.

Fig. 3. Time-integrated spectra of (a) the forward and (b) the backward electrons as they pass by the boundaries placed at the target rear side (x = 26 µm) at t = 75T₀.

Fig. 4. (a) and (b) Evolution of an on-axis proton initially located at (25.02, 60 µm): (a) transverse position y; (b) averaged longitudinal sheath field 〈E_x〉. (c) and (d) Evolution of an off-axis proton initially located at (25.02, 45 µm): (c) transverse position y; (d) averaged transverse sheath field 〈E_y〉. 〈 〉 denotes the average over one laser period. The line y = 0 corresponds to the laser axis.

Fig. 5. Maximum proton energy Eimax and maximum sheath field strength Esmax as functions of the peak laser intensity. The circles and diamonds are the results from the 2D PIC simulations. The fitted curve segments for the proton energy shown by the dashed and solid black lines satisfy Ei∝(Iλ2)0.71 and Ei∝(Iλ2)0.34, respectively. The other two fitted curve segments shown by the dashed and solid blue lines satisfy Es∝(Iλ2)0.4 and Es∝(Iλ2)0.66, respectively.