Many works showed that controlling the instabilities during the foil acceleration is the key to maintaining Radiation Pressure Acceleration（RPA）for a long time and suggested possible solutions. Researchers at State Key Laboratory of High Field Laser Physics——Shanghai Institute of Optics and Fines Mechanics (SIOM/China) explore in detail the parameter regime where the front of the laser pulse has only few laser cycles.[ PHYSICS OF PLASMAS 18, 013103 (2011)]

They find that by controlling the ions initially in the middle of the target to enter the compressed electron layer just at the end of the hole-boring stage, the instabilities can be to a certain extent suppressed for RPA in the SFL interaction. To reach that, a suitable laser-front steepness a0/ tup has been obtained. For both sharper and flatter laser fronts, the foil is more easily destroyed and the ions can be accelerated only for short distance. Based on our theoretical model, we also find that the efficiency of the ion acceleration can be improved by decreasing the areal density n0l0 for the same laser parameters. All our theoretical results are based on the assumption that the incident laser pulse is fully reflected in the SFL interaction. Therefore, the transparency caused by the ultrathin foil, instabilities, or other reasons may jeopardize our theoretical expectation.