Ionization dynamics beyond the dipole approximation induced by the pulse envelope

When atoms and molecules are ionized by laser pulses of finite duration and increasingly high intensities, the validity of the much-used dipole approximation, in which the spatial dependence and magnetic component of the external field are neglected, eventually breaks down. We report that, when going beyond the dipole approximation for the description of atoms exposed to ultraviolet light, the spatial dependence of the pulse shape, the envelope, provides the dominant correction, while the spatial dependence of the carrier is negligible. We present a first-order beyond-dipole correction to the Hamiltonian which accounts exclusively for nondipole effects stemming from the carrier envelope of the pulse. We demonstrate by ab initio calculations for hydrogen that this approximation, which we refer to as the envelope approximation, reproduces the full interaction beyond the dipole approximation for absolute and differential observables and proves to be valid for a broad range of high-frequency fields. This is done both for the Schrödinger and the Dirac equation. Moreover, it is demonstrated that the envelope approximation provides an interaction-term which gives rise to faster numerical convergence in terms of partial waves compared to its exact counterpart.

Figure 1

The total ionization probability as a function of the laser peak intensity (given in electric-field strength). The hydrogen atom was exposed to a 15-cycle sine-square-shaped XUV laser pulse of angular frequency $\omega =3.5$ a.u. The results are obtained with four different theoretical frameworks. The time-dependent Schrödinger equation (TDSE) is solved within the dipole approximation (8) (solid blue) and for a Hamiltonian including the first-order expansion of the full nondipole interaction (7) (thin solid black). The envelope approximation is shown both when applied to the TDSE (15) (dashed red) and the time-dependent Dirac equation (TDDE) (19) (green diamonds).

Figure 2

Differential probability distribution for the photoelectron as a function of kinetic energy. Here, the laser field parameters are the same as in Fig. 1 with the field strength $E_0=45$ a.u. The energy distribution is shown for three different interactions used to model the laser-atom dynamics in the time-dependent Schrödinger equation. First, the dipole-approximation results are shown in a solid blue line whereas the corresponding predictions by a beyond-dipole interaction to first order is depicted in a thinner solid black line. Lastly, the results obtained by applying the envelope approximation, i.e., Eq. (15), are drawn in a dashed red line. The energy distributions are shown on absolute (upper) and logarithmic (lower) scale. The inset in the upper panel is a closeup of the low-energy region.

Figure 3

Angular-resolved probability distributions for the final continuum state obtained after the conclusion of the laser pulse used in Fig. 1 for the field strength $E_0=45$ a.u. Here, the axis of polarization is oriented horizontally whereas the propagation direction is denoted by the vertical black arrows. The panels correspond to, from the top to bottom, the dipole approximation (8), beyond the dipole approximation to first order, Eq. (7), and the envelope approximation (15).

Figure 4

The total ionization probability as a function of the laser angular frequency. The field intensity is adjusted so that the quiver velocity of the free (classical) electron is kept fixed at 10% of the speed of light for all the calculations. The pulse is described by a sine-squared envelope function lasting 10 optical cycles. Three different interactions within the time-dependent Schrödinger equation (TDSE) are shown and compared to the results obtained with the time-dependent Dirac equation (green diamonds). The TDSE Hamiltonians include the dipole approximation (8) (solid blue), the dipole interaction and the full first-order nondipole correction (thin solid black) (7), and the envelope approximation applied to the latter (dashed red), Eq. (15). The probabilities are shown on absolute (upper) and logarithmic (lower) scale.

Figure 5

Population in all $m\ne 0$ states as a function of time. Here, the photon energy is $3.5$ a.u., and the laser intensity corresponds to $E_0=45$ a.u. at maximum. The plots show the results obtained with both the full first-order nondipole interaction Hamiltonian (7) (thin black lines) and the envelope approximation (15) (thick red lines). The dotted black lines depict the shape of the laser pulse, $f\left(t\right)$. Three different envelope functions described by Eq. (21) with $\sigma =0.4$ (top panel), $\sigma =0.8$ (center panel), and $\sigma =1.6$ (bottom panel) have been used to model the laser. In all three cases the pulse duration parameter $T$ corresponds to 7.5 optical cycles.

Figure 6

Energy distribution of the ejected photoelectron after a 40-cycle laser-atom interaction. The photon energy and the field intensity are set to $\hslash \omega =3.5$ a.u. and $E_0=45$ a.u., respectively. The distributions are drawn around the one-photon absorption resonance $E\sim \hslash \omega -I_p$, and the different curves pertain to different values of $l_{\max }$. The upper panel shows comparison between fully converged dipole, beyond-dipole (1st) and envelope approximation (Env) results. The distribution obtained within the dipole approximation features oscillations which vanish when the spatial dependence of the external field is included. The second and third panels show the results as obtained with the first-order nondipole expansion (1st) and with the envelope approximation (Env) at different values of $l_{\max }$, respectively.