Phase-stable limited relativistic acceleration or unlimited relativistic   acceleration in the laser-thin-foil interactions
  To clarify the relationship between phase-stable acceleration (PSA) and unlimited relativistic acceleration (URA) (Phys. Rev. Lett. 104, 135003 (2010)), an analytical relativistic model is proposed in the interactions of the ultra-intense laser and nanometer foils, based on hydrodynamic equations. The dependence of the ion momentum on time is consistent with the previous results and checked by PIC simulations. Depending on the initial ion momentum, relativistic RPA contains two acceleration processes: phase-stable limited relativistic acceleration (PS-LRA) and URA. In PS-LRA, the potential is a deep well trapping the ions. The ion front, i.e., the bottom, separates it into two parts: the left half region is PSA region; the right half region is PSD region, where the ions climb up and are decelerated to return back. In PS-LRA, the maximum ion energy is limited. If the initial ion momentum large enough, ions will experience a potential downhill and drop into a bottomless abyss, which is called phase-lock-like position. URA is not phase-stable any more. At the phase-lock-like position the ions can obtain unlimited energy gain and the ion density is non-zero. You cannot have both PSA and URA. 
Huang Yongsheng and Wang Naiyan and Tang Xiuzhang and Yan Xueqing
physics.plasm-ph physics.acc-ph