Angularly resolved two-photon above-threshold ionization of helium

Angularly resolved two-photon single ionization yields of helium resulting after the interaction with an ultrashort XUV pulse are obtained by numerically solving the full dimension time-dependent Schrödinger equation. The angular distributions reveal the underlying dominant mechanism, which depends on the effective photon energy absorbed and the pulse parameters. We specifically explore the contributions of radial and angular electron correlation terms. A single active electron picture is a qualitatively valid approach for the lowest photon energies, even in the above-threshold ionization region. Nonetheless, angular correlation plays a detectable role in the low-energy region and a major role at higher energies when autoionizing states are populated. As the photon energy increases, sequential ionization-excitation dominates; therefore, the resulting probability distributions are explained as the result of two active uncorrelated electrons. This uncorrelated picture fails again for photon energies above ionization potential of the ion.

Figure 1

Schematic representation of the relevant transitions for two-photon single ionization of helium. We indicate four regions discussed in the text leading to two-photon single ionization with photon energies: (a) below the single ionization threshold, 24.6 eV, (b) above-threshold ionization where the two-photon absorption reaches the doubly excited states ($\omega >28.95$ eV), (c) two-photon single ionization where core-excited resonances dominate, and (d) region where the simultaneous excitation-ionization is energetically open already for one-photon absorption.

Figure 2

Total cross sections for two-photon single ionization of helium using a 2 fs pulse as a function of the photon energy absorbed. Black thick full line: total ionization yield. Blue with dots: $D$ contribution. Red with dots: $S$ contribution. The photoelectron angular distributions corresponding to a given energy of the emitted electron are plotted as insets. Distributions are plotted for effective photon energies of 15, 18, 22, 26, 34, 40.8, 45, 48.38, 57, 61, and 65 eV.

Figure 3

Two-photon single ionization cross section of helium as a function of the photon energy absorbed. Comparison with previous works using time-independent perturbation theory approaches. Full red line: Ref. [9]. Blue filled dots: Ref. [31]. Turquoise dashed line: Ref. [32]. Purple circles: Ref. [33]. Full lines with symbols: apparent cross sections resulting after solving the TDSE for pulses with a laser intensity of ${10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and durations of 2 fs (stars), 3 fs (squares), and 4.2 fs (triangles), as indicated in the legend.

Figure 4

(a) Asymmetry parameters, ${\beta }_2$ and ${\beta }_4$, as a function of photon energy. (b) Corresponding total and partial ($S$ and $D$) cross sections for the two-photon single ionization using a 2 fs pulse. Dotted vertical lines indicate the energy of the ionization potential for helium ($I_p=24.6$ eV) and the excitation energies for the ion from ${\mathrm{He}}^{+}\left(1s\right)$ to ${\mathrm{He}}^{+}(n=2)$, ${\mathrm{He}}^{+}(n=3)$, and ${\mathrm{He}}^{+}(n=4)$. (c) Contribution of the different ionization channels with one electron in the remaining ion, ${\mathrm{He}}^{+}(ns$, $np$, $nd$, etc.), and the emitted electron with a particular angular momentum ($ks,kp,kd$,...) for a total final $S$ symmetry. (d) Same as (c); contributions from different ionization channels for a total final symmetry $D$.

Figure 5

Asymmetry parameters, ${\beta }_2$ (upper panel) and ${\beta }_4$ (bottom panel), as a function of the photon energy. Comparison with the $\beta$ parameters resulting from the ab initio simulation (TDSE 2 fs) and a truncated calculation removing the angular correlation terms (TDSE 2 fs WoAC), both employing a 2 fs pulse. We also include the results of a single-active electron model based on TDPT (see main text) using a 2 fs (model TDPT 2 fs) and a 4.2 fs pulse (model TDPT 4.2 fs), and the results by Ishikawa and Ueda [18] reported for a 7 fs pulse duration at FWHM, equivalent to $T=4.2$ fs.

Figure 6

As in Figs. 4 and 4, $\beta$ parameters and cross sections zoomed in the energy region where the doubly excited states are populated upon two-photon absorption. The vertical dotted lines indicate the energies at which the photoelectron angular distributions plotted in the bottom of the figure are computed, as accordingly labeled. The first row of PADs corresponds to the first ${}^{1}S^e$ doubly excited state [${}_2{(1,0)}_2^{+}$ or $2s2s$]. The second row corresponds to the first ${}^{1}D^e$ doubly excited state [${}_2{(1,0)}_2^{+}$ or $2p2p$] and the third row corresponds to the second ${}^{1}S^e$ doubly excited state [${}_2{(-1,0)}_2^{+}$ or $2p2p$].

Figure 7

$\beta$ parameters (upper panel) and total cross sections for two-photon single ionization into ${}^{1}S^e$ (middle panel) and ${}^{1}D^e$ (bottom panel) final symmetry using the $S$-wave “single-active electron” approximation described in the text. Contributions from each ionization channel are also included. The ground state of helium is retrieved within the $S$-wave approximation using an $l_{\max }=0$, and the subsequent electronic transitions are only allowed within a single active electron approach. Note nevertheless that all electron correlation terms within the $S$-wave SAE approximation are accounted for.

Figure 8

Same as Fig. 7, but for a truncated simulation without angular correlation (WoAC), i.e., solving the TDSE in full dimensionality where the angular correlation terms are neglected.

Figure 9

Photoelectron angular distributions for two-photon single ionization of helium. Red full line corresponds to cuts of the three-dimensional distributions in Fig. 2 taken along the plane that contains the light polarization vector. These are the angular differential probabilities including all ionization channels ($S$, $P$, and $D$ contributions), i.e., one- and two-photon absorption probabilities are coherently added. Blue dashed line correspond to only the $S$ and $D$ contributions, i.e., only resulting upon a two-photon absorption. Purple bars in each subplot represent the absolute value of the ionization amplitudes associated to the $S$, $P$, and $D$ contributions.