Modelling the effect of laser focal spot size on sheath- accelerated protons in intense laser--foil interactions

We present an approach to modelling the effect of the laser focal spot size on the acceleration of protons from ultra-thin foil targets irradiated by ultra-short laser pulses of intensity 1016–1018 W cm−2. An expression is introduced for the proton acceleration time, which takes account of the time taken for the laser-accelerated electrons, which expand laterally at the rear surface of the target, to escape the region of the sheath. When incorporated into an analytical model of plasma expansion, this approach is found to provide a good fit to measured scaling of the maximum proton energy as a function of intensity at large focal spot sizes.