  The resources needed for Particle in Cell simulations of Laser Wakefield
Acceleration can be greatly reduced in many cases of interest using an envelope
model. However, the inclusion of tunneling ionization in this time averaged
treatment of laser-plasma acceleration is not straightforward, since the
statistical features of the electron beams obtained through ionization should
ideally be reproduced without resolving the high frequency laser oscillations.
In this context, an extension of an already known envelope ionization procedure
is proposed, valid also for laser pulses with higher intensities, which
consists in adding the initial longitudinal drift to the newly created
electrons within the laser pulse ionizing the medium. The accuracy of the
proposed procedure is shown with both linear and circular polarization in a
simple benchmark where a nitrogen slab is ionized by a laser pulse, and in a
more complex benchmark of laser plasma acceleration with ionization injection
in the nonlinear regime. With this addition to the envelope ionization
algorithm, the main phase space properties of the bunches injected in a plasma
wakefield with ionization by a laser (charge, average energy, energy spread,
rms sizes, normalized emittance) can be estimated with accuracy comparable to a
non-envelope simulation with significantly reduced resources, even in
cylindrical geometry. Through this extended algorithm, preliminary studies of
ionization injection in Laser Wakefield Acceleration can be easily carried out
even on a laptop.
