Figure 2. Extended electron trajectories.

For He gas interacting with a 30-fs-long laser pulse with a central wavelength of 800 nm, extended electron trajectories were calculated for two intensities: (a), (b) 5 × 10¹⁵ W/cm² and (c), (d) 8 × 10¹⁵ W/cm². (a)–(d) The distance of the electron from the ion was calculated throughout several re-scattering events and plotted as a function of time, assuming that ionisation occurred a few optical half-cycles (OHC) before the intensity maximum. The scale of the colour bars is proportional to the probability of finding the electron at a certain distance from the atom. (e)–(h) At every re-scattering, the moment of re-scattering (re-collision) and the kinetic energy of the electron were also calculated and exhibit strong correlation. In every OHC, two re-collisions occur, and two soft X-ray pulses are generated. The scale of the colour bars is proportional to the probability of the electron having a certain kinetic energy at the time of collision. (i), (j), (l), (m) For both intensities, two corresponding electron trajectories with a re-collision time of approximately 2 optical cycle are presented; they fulfil the condition of X-ray parametric amplification in the corresponding X-ray energy range because they collide with suitably small impact parameters. (k), (n) The temporal evolution of the distance of the electron from the ion is also plotted separately for both chosen trajectories.