Laser Ion Acceleration Scaling Laws Seen in Muti-Parametric PIC   Simulations
  The ion acceleration driven by a laser pulse at intensity $I = 10^{20} - 10^{22} $W/cm$^2\times(\mu$m$/\lambda)^2$ from a double layer target is investigated with multi-parametric Particle-in-Cell (PIC) simulations. For targets with a wide range of thickness $l$ and density $n_e$, at given intensity the highest ion energy gain occurs at certain {\it critical depth} of the target $\sigma = n_e l$, which is proportional to the square root of intensity. In the case of thin targets and optimal laser pulse duration, the ion maximum energy scales as the square root of the laser pulse power. When the radiation pressure of the laser field becomes dominant, the ion maximum energy becomes proportional to the laser pulse energy. 
T. Esirkepov, M. Yamagiwa, T. Tajima