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