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To clarify the relationship between phase-stable acceleration (PSA) andunlimited relativistic acceleration (URA) (Phys. Rev. Lett. 104, 135003(2010)), an analytical relativistic model is proposed in the interactions ofthe ultra-intense laser and nanometer foils, based on hydrodynamic equations.The dependence of the ion momentum on time is consistent with the previousresults and checked by PIC simulations. Depending on the initial ion momentum,relativistic RPA contains two acceleration processes: phase-stable limitedrelativistic acceleration (PS-LRA) and URA. In PS-LRA, the potential is a deepwell trapping the ions. The ion front, i.e., the bottom, separates it into twoparts: 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, themaximum ion energy is limited. If the initial ion momentum large enough, ionswill experience a potential downhill and drop into a bottomless abyss, which iscalled phase-lock-like position. URA is not phase-stable any more. At thephase-lock-like position the ions can obtain unlimited energy gain and the iondensity is non-zero. You cannot have both PSA and URA.

Phase-stable limited relativistic acceleration or unlimited relativistic acceleration in the laser-thin-foil interactions