Numerical modeling of laser tunneling ionization in Particle in Cell Codes with a laser envelope model

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 block 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.