Petawatt laser-driven wakefield accelerator: All-optical electron injection via collision of laser pulses and radiation cooling of accelerated electron bunches.

We explore an electron injection into the laser wakefield accelerator (LWFA) using nearly head-on collision of the petawatt ultrashort ($\sim$30 fs) laser pulse (driver) with a low- amplitude laser (seed) beam of the same duration and polarization. To eliminate the threat to the main laser amplifier we consider two options: (i) a frequency-shifted seed and (ii) a seed pulse propagating at a small angle to the axis. We show that the emission of synchrotron radiation due to betatron oscillations of trapped and accelerated electrons results in significant transverse cooling of quasi- monoenergetic accelerated electrons (with energies above 1 GeV). At the same time, the energy losses due to the synchrotron emission preserve the final energy spread of the electron beam. The ``dark current'' due to the electron trapping in multiple wake buckets and the effect of beam loading (wake destruction at the instant of beams collision) are discussed.