Time-dependent energetic proton acceleration and scaling laws in   ultra-intense laser pulses interactions with thin foils
  A two-phase model, where the plasma expansion is an isothermal one when laser irradiates and a following adiabatic one after laser ends, has been proposed to predict the maximum energy of the proton beams induced in the ultra-intense laser-foil interactions. The hot-electron recirculation in the ultra-intense laser-solid interactions has been accounted in and described by the time-dependent hot-electron density continuously in this model. The dilution effect of electron density as electrons recirculate and spread laterally has been considered. With our model, the scaling laws of maximum ion energy have been achieved and the dependence of the scaling coefficients on laser intensity, pulse duration and target thickness have been obtained. Some interesting results have been predicted: the adiabatic expansion is an important process of the ion acceleration and cannot be neglected; the whole acceleration time is about 10-20 times of laser pulse duration; the larger the laser intensity, the more sensitive the maximum ion energy to the change of focus radius, and so on. 
Yongsheng Huang, Yuanjie Bi, Yijin Shi, Naiyan Wang, Xiuzhang Tang,   Zhe Gao