  When a circularly polarized laser pulse interacts with a foil target, there
are three stages: pre-hole-boring, hole-boring and the light sail acceleration.
We study the electron and ion dynamics in the first stage and find the minimum
foil thickness requirement for a given laser intensity. Based on this analysis,
we propose to use a shaped foil for ion acceleration, whose thickness varies
transversely to match the laser intensity. Then, the target evolves into three
regions: the acceleration, transparency and deformation regions. In the
acceleration region, the target can be uniformly accelerated producing a
mono-energetic and spatially collimated ion beam. Detailed numerical
simulations are performed to check the feasibility and robustness of this
scheme, such as the influence of shape factors and surface roughness. A GeV
mono-energetic proton beam is observed in the three dimensional
particle-in-cell simulations when a laser pulse with the focus intensity of
1022W=cm2 is used. The energy conversion efficiency of laser pulse to
accelerated proton beam is more than 23%. Synchrotron radiation and damping
effects are also checked in the interaction.
