Application of the gauge-invariant formalism to the description of ionization by ultrastrong few-cycle pulses

We analyze energy spectra of photoelectrons ejected by strong laser pulses, using the manifestly gauge-invariant approach. Gauge-invariant expressions for the ionization amplitude can be obtained by grouping together all terms of the same order in atomic potential in the standard strong-field approximation (SFA) expansion. We calculate energy spectra of photoelectrons for the hydrogen atom, and compare them with results based on numerical solutions of the time-dependent Schrödinger equation. Agreement, at least qualitative, is observed for an electric-field amplitude higher than 1 a.u., and for larger energies of ionized electrons, basically beyond the region, where the standard SFA is usually applied.

Figure 1

Energy distribution of photoelectrons for two-cycle laser pulses with $\omega =0.57$ a.u., corresponding to a wavelength of 80 nm, and with (a) $E_0=3$ a.u. and (b) $E_0=10$ a.u. Solid black lines corresponds to TDSE results, dashed red lines corresponds to present calculations. The results were scaled to fit TDSE for photoelectron energy equal to 8 a.u.

Figure 2

Same as Fig. 1, but for ten-cycle laser pulses with $\omega =0.5$ a.u., corresponding to a wavelength of 91 nm, and with (a) $E_0=1$ a.u. and (b) $E_0=3$ a.u.