Figure 4. WAKE simulation results.

Initial conditions are: pre-ionized plasma of density n_e=5.0 × 10¹⁷ cm⁻³; incident laser pulse of duration τ=160 fs, energy=100 J (peak laser strength parameter a₀=0.58, normalized power P/P_cr=20). (a) Laser strength parameter a(r=0,z) and peak intensity I(r=0,z) along propagation axis inside plasma for initially Gaussian (G, dashed orange curve) and third-order super-Gaussian (SG-3, solid blue) pulses of radius w₀=275 μm, the latter showing two rapid self-focus/defocus cycles near the cell exit. Inset: image of incident laser profile at approximately the cell entrance, ~2 cm from its focal plane, for shot ‘a’ in Fig. 2, compared with white dashed circle of radius w₀=275 μm and white solid circle of radius w₀=80 μm. Colour scale indicates intensity as a multiple of 10¹⁷ W cm⁻². (b) Transverse profiles at z=30 mm of initially Gaussian (G, short-dashed orange), second- (SG-2, short-dashed black), third- (SG-3, solid blue) and fourth- (SG-4, long-dashed green) order super-Gaussian pulses, showing Airy-ring-like structure for non-Gaussian profiles that is evidently critical for self injection. Each profile is plotted for the longitudinal position of maximum ∫₀^∞drrI(r) within the pump profile; this position differs slightly for each plot, and from the position of maximum peak intensity shown in a. For comparison, curve SG-3(φ) (red squares) shows the transverse profile at z=30 mm, calculated with the NLSE, of an initially third-order super-Gaussian multiplied by the azimuthally varying function 1+0.25(r/w₀)⁹ cos (9φ) to mimic an evolving array of hotspots. (c) Simulated electron energy spectrum at z=6.8 cm for initially third-order super-Gaussian pulse evolving as in a, showing 2 GeV peak and low-energy tail. Similar spectra were obtained for other super-Gaussian orders and for ±10% variations in a₀. Inset: electron density profile, showing background plasma electrons (black, top) and test electrons (red, bottom). Injected electrons are the red feature inside evacuated bubble. ξ denotes longitudinal distance along the bubble’s axis. Black dashed curve: 8 × 10¹⁸ W cm⁻² iso-intensity contour of laser pulse at z=6.8 cm.