Ionization of atomic hydrogen by an intense x-ray laser pulse: An ab initio study of the breakdown of the dipole approximation

Solving the time-dependent Schrödinger equation numerically within the framework of an ab initio model, the breakdown of the dipole approximation in modeling the ionization and excitation dynamics of a hydrogen atom exposed to an intense 1.36-keV x-ray laser pulse is investigated in some detail. The relative importance of the ${\mathit{A}}^2$ diamagnetic term in comparison with the $\mathit{A}·\mathit{p}$ contribution to the resulting beyond-dipole (nondipole) light-matter interaction is studied for laser pulse intensities ranging from the weak perturbative to the strong-field regime. It is found that the diamagnetic interaction represents by far the most important correction to the dipole approximation at higher field strengths, while nondipole corrections induced by the $\mathit{A}·\mathit{p}$ operator are generally small and largely independent of the laser intensity. The most profound finding of the present study was the discovery of a forward-backward asymmetry in the underlying electron ejection dynamics: Depending on the electron's kinetic energy in the final state, the photoelectron tends to be emitted in the laser propagation (forward) and/or counterpropagation (backward) directions, for energies corresponding to the low-energy and/or high-energy side of the multiphoton resonances, respectively.

Figure 1

(a) Total ionization probability vs electric-field strength for a 15-cycle laser pulse with $\omega =50$ a.u. The solid black line shows the (full) nondipole result obtained by solving the TDSE with the Hamiltonian in Eq. (5), the solid blue line the nondipole result obtained by solving the TDSE with only the $\mathit{A}·\mathit{p}$ term, i.e., the Hamiltonian in Eq. (6), and the dashed red line the dipole approximation result obtained by solving the TDSE with the Hamiltonian in Eq. (7). (b) Corresponding survival probability. (c) Corresponding probability of excitation.

Figure 2

Kinetic energy spectrum of the emitted photoelectron for a 15-cycle laser pulse with $\omega =50$ a.u. and for two different electric-field strengths (a) $E_0=10$ a.u. and (b) $E_0=400$ a.u. The solid black line shows the (full) nondipole result obtained by solving the TDSE with the Hamiltonian in Eq. (5), the solid blue line the nondipole result obtained by solving the TDSE with only the $\mathit{A}·\mathit{p}$ term, i.e., the Hamiltonian in Eq. (6), and the dashed red line the dipole result obtained by solving the TDSE with the Hamiltonian in Eq. (7).

Figure 3

Total probability for the net absorption of one photon from the laser field (i.e., differential probability integrated over the electron energies $25<E<75$ a.u.) vs electric-field strength for a 15-cycle laser pulse with $\omega =50$ a.u. The solid black line shows the (full) nondipole result obtained by solving the TDSE with the Hamiltonian in Eq. (5) and the dashed red line the dipole approximation result obtained by solving the TDSE with the Hamiltonian in Eq. (7).

Figure 4

Projections corresponding to the net absorption of one photon from the laser field as a function of the electric-field strength (see the main text for details). The solid black line shows Eq. (19), the solid blue line Eq. (20), and the dashed red line Eq. (21).

Figure 5

Electron angular distributions, integrated over the electron energy interval $25<E<75$ a.u. (corresponding to the net absorption of one photon from the field), for $\omega =50$ a.u., a 15-cycle laser pulse linearly polarized in the horizontal direction and propagating in the upward direction (indicated with an arrow), and two different electric-field strengths: $E_0=10$ a.u. (left column) and $E_0=400$ a.u. (right column). The top panels show the (full) nondipole result obtained by solving the TDSE with the Hamiltonian in Eq. (5), the middle panels the nondipole result obtained by solving the TDSE with only the $\mathit{A}·\mathit{p}$ term, i.e., the Hamiltonian in Eq. (6), and the bottom panels the corresponding dipole result obtained by solving the TDSE with the Hamiltonian in Eq. (7).

Figure 6

Full (nondipole) electron angular distributions for a 15-cycle laser pulse linearly polarized in the horizontal direction and propagating in the upward direction (indicated with an arrow) and for three different electric-field strengths: $E_0=400$ a.u. (top panels), $E_0=500$ a.u. (middle panels), and $E_0=600$ a.u. (bottom panels). The left column shows the angular distribution as obtained by integrating over the energies corresponding to the low-energy side of the one-photon resonance, i.e., $25<E<49.1$ a.u., and the right column the angular distribution as obtained by integrating over the energies corresponding to the high-energy side of the one-photon resonance, i.e., $49.1<E<75$ a.u. The total (integrated) one-photon ionization yields are 0.0013, 0.0017 and 0.0019, in the top, middle, and bottom panels, respectively.

Figure 7

Same as Fig. 6, but for the corresponding results obtained with the simple model (26). The total (integrated) one-photon ionization yields are 0.0018, 0.0025 and 0.0030, in the top, middle, and bottom panels, respectively.

Figure 8

Double differential angular and energy probability ${\partial }^{2}P/\partial \mathrm{\Omega }\partial E$ extracted at the final continuum electron energies as indicated in the figure, i.e., for energies ranging from 45 to 53 a.u. (from top to bottom) in steps of 1 a.u. The laser pulse is the same as before with $E_0=400$ a.u. The left column shows the ab initio result and the right column the simple model (SFA) result calculated by means of Eq. (26).