Figure 1
(A) Schematic of 2D simulation set-up used to study QED cascade saturation. (B) The number of e−e+ and γ-photons as a function of laser intensity (I0) at t = 13T0 (T0 ≈ 3.3 fs is the laser cycle) in 2D simulations. Only the γ-photons with energy higher than 1.022 MeV are counted. The magenta dashed line shows the estimates of \({N}_{e0}\) at different laser intensities. The blue and red meshed bands indicate the analytical calculations from Eqs (4) and (5) after substituting the scaling function of \(\overline{{\Gamma }_{+}}\) in (C). The band width is attributed to the variation of \({N}_{e0}\) caused by varying laser intensities from 1023 to 3.2 × 1024 W/cm2. (C) Average cascade growth rate \(\overline{{\Gamma }_{+}}\) (normalized to T0) for two different initial plasma densities as a function of laser intensity. The dashed line corresponds to the fitting curve at plasma density of 280nc (n c = \({m}_{e}{\omega }_{l}^{2}/4\pi {e}^{2}\) is the critical plasma density) and \({I}_{24}\) = 1024 W/cm2. (D) Temporally and spatially averaged quantum parameter \({\bar{\chi }}_{e}\) of electrons as a function of laser intensity. The average \({\chi }_{e}\) is obtained by supposing that the produced electrons are located at the antinodes of the electric field. In the absence of two QED processes, one can obtain the maximum \({\chi }_{emax} \sim 2{a}_{0}^{2}/{a}_{s}\)33, which is shown by the green solid curve for comparison. Here a0 = \(eE/{m}_{e}c{\omega }_{l}\) is the normalized laser field amplitude and as = \({m}_{e}{c}^{2}/\hslash {\omega }_{l}\) the normalized critical field amplitude35.
