  A new scheme of proton acceleration from a laser-driven near-critical-density
plasma is proposed. Plasma with a tailored density profile allows a two-stage
acceleration of protons. The protons are pre-accelerated in the laser-driven
wakefields, and are then further accelerated by the collisionless shock,
launched from the rear side of the plasma. The shock has a small transverse
size, and it generates a strong space-charge field, which defocuses protons in
such a way, that only those protons with the highest energies and low energy
spread remains collimated. Theoretical and numerical analysis demonstrates
production of high-energy proton beams with few tens of percents energy spread,
few degrees divergence and charge of few nC. This scheme indicates the
efficient generation of quasi-monoenergetic proton beams with energies up to
several hundreds of MeV with PW-class ultrashort lasers.
