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 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, and normalized emittance) can be estimated with accuracy comparable to a nonenvelope 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.

Figure 1

Initial state of the ${\mathrm{N}}^{5+}$ ionization benchmark in the envelope simulations along the propagation axis $x$. The envelope module $|\tilde{A}|$ and the charge density of the ${\mathrm{N}}^{5+}$ slab are normalized to 1.

Figure 2

Comparison of (a) the momentum distributions and (b) the position distributions of the electrons created by ionization in the ${\mathrm{N}}^{5+}$ benchmark at time $T=1277.2{\lambda }_0/2\pi c$, with linear polarization along the $y$ axis.

Figure 3

Comparison of (a) the momentum distributions and (b) the position distributions of the electrons created by ionization in the ${\mathrm{N}}^{5+}$ benchmark at time $T=1277.2{\lambda }_0/2\pi c$, with circular polarization.

Figure 4

Adapted from Figs. 2 and 5 of Ref. [54] with the almost the same laser and plasma parameters: $a_0=2, n_0=0.44\%n_c, L_{\mathrm{FWHM}}=44{\lambda }_0/2\pi$. (a) Phase space trajectories of test electrons, including a fluid orbit, the separatrix, and the trajectory of an electron stripped from its atom or ion through ionization; (b) laser vector potential $A$, electrostatic potential $\mathrm{\Psi }$, and longitudinal electric field $E_x$.

Figure 5

(a) Comparison of the evolution of the laser peak transverse electric field in the LWFA benchmark simulations. For the other two panels, all the electrons created by ionization, present in the moving window and with longitudinal momentum $p_x>50m_{e}c$, are considered. The electrons injected in the plasma wave bucket behind the laser are a subset of these electrons. (b) Comparison of the electron total charge evolution; (c) comparison of the average electron energy.

Figure 6

Comparison of (a) the longitudinal electric field and (b) the electron normalized electron density on the propagation axis after $800\mu \mathrm{m}$, computed with the standard laser and envelope simulations in the LWFA benchmark. The maximum value shown for the charge density is 0.01, to highlight the zone near the injected electron beam.

Figure 7

Comparison of the electron normalized density on the $x-y$ plane after $800\mu \mathrm{m}$ of propagation computed with (a) the envelope simulation and (b) the standard laser simulation in the LWFA benchmark.

Figure 8

Comparison of the energy spectrum of all the electrons created by ionization in the moving window after $800\mu \mathrm{m}$ of propagation computed with the standard laser and envelope simulations in the LWFA benchmark.

Figure 9

(a) Comparison of the evolution of the laser peak transverse electric field in the LWFA benchmark simulations. For the other two panels, all the electrons created by ionization, present in the moving window and with longitudinal momentum $p_x>50m_{e}c$, are considered. The electrons injected in the plasma wave bucket behind the laser are a subset of these electrons. (b) Comparison of the electron total charge evolution; (c) comparison of the average electron energy. The results of the standard laser simulation and of the envelope simulations with with $[N_x,N_r,N_{\theta }]=[4,2,1],[1,1,1]$ are reported.

Figure 10

Comparison of (a) the longitudinal electric field and (b) the electron normalized density on the propagation axis after $800\mu \mathrm{m}$, computed with the standard laser and envelope simulations in the LWFA benchmark. The maximum value shown for the charge density is 0.01, to highlight the zone near the injected electron beam.

Figure 11

Comparison of the energy spectrum of all the electrons created by ionization in the moving window after $800\mu \mathrm{m}$ of propagation computed with the standard laser and envelope simulations with $[N_x,N_r,N_{\theta }]=[4,2,1],[1,1,1]$ in the LWFA benchmark.