An ultra-short (about 30 fs) petawatt laser pulse focused with a wide focalspot (about 100 microns) in a rarefied plasma (electron density of order10^{17} per cm^3) excites a nonlinear plasma wakefield which can accelerateinjected electrons up to the GeV energy without any pulse channelling. In theseconditions, propagation of the laser pulse with an over-critical power forrelativistic self-focusing is almost the same as in vacuum. The nonlinearquasi-plane wake plasma wave, whose amplitude and phase velocity vary along thelaser path, effectively traps and accelerates injected electrons with a widerange of initial energies. Electrons accelerated along two Rayleigh lengths(about eight centimeters) can gain an energy up to 1 GeV. In particular, theelectrons trapped from quite a long (of order 330 fs) non-resonant electronbeamlet of 1 MeV particles eventually form a low emittance bunch with energiesin the range 900 MeV and energy spread about 10%. All these conclusions followfrom two-dimensional simulations performed in cylindrical geometry by means ofthe fully relativistic time-averaged particle code WAKE.
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