Intensity scaling limitations of laser-driven proton acceleration in the TNSA-regime

We report on experimental results on laser-driven proton acceleration using high-intensity laser pulses. We present power law scalings of the maximum proton energy with laser pulse energy and show that the scaling exponent ξ strongly depends on the scale length of the preplasma, which is affected by the temporal intensity contrast. At lower laser intensities, a shortening of the scale length leads to a transition from a square root toward a linear scaling. Above a certain threshold, however, a significant deviation from this scaling is observed. Two-dimensional particle-in-cell simulations show that, in this case, the electric field accelerating the ions is generated earlier and has a higher amplitude. However, since the acceleration process starts earlier as well, the fastest protons outrun the region of highest field strength, ultimately rendering the acceleration less effective. Our investigations thus point to a principle limitation of the proton energy in the target normal sheath acceleration regime, which would explain why a significant increase of the maximum proton energy above the limit of 100 MeV has not yet been achieved.