Attosecond photoelectron microscopy of $\text{H}_2{}^{+}$

We present a numerical study of the ultrafast ionization dynamics of $\text{H}_2{}^{+}$ exposed to attosecond extreme ultraviolet (xuv) pulses that goes beyond the Born-Openheimer approximation. The four-dimensional, time-dependent Schrödinger equation was solved using a generalization of the finite-element discrete-variable-representation/real-space–product technique used in our previous calculations to include the dynamical motion of the nuclei. This has enabled us to expose the target to any polarized light at arbitrary angles with respect to the molecular axis. Calculations have been performed at different angles and photon energies ($\hslash \omega =50\text{eV}$ up to 630 eV) to investigate the energy and orientation dependence of the photoionization probability. A strong orientation dependence of the photoionization probability of $\text{H}_2{}^{+}$ was found at a photon energy of $\hslash \omega =50\text{eV}$. At this energy, we found that the ionization probability is three times larger in the perpendicular polarization than in the parallel case. These observations are explained by the different geometric “cross sections” seen by the photoejected electron as it leaves the molecule. This ionization anisotropy vanishes at the higher-photon energy of $\hslash \omega \ge 170\text{eV}$. When these higher-energy xuv pulses are polarized perpendicular to the internuclear axis, a “double-slit-like” interference pattern is observed. However, we find that the diffraction angle only approaches the classical formula ${\varphi }_n={\text{sin}}^{-1}\left(n{\lambda }_{\text{e}}/R_0\right)$, where $n$ is the diffraction order, ${\lambda }_{\text{e}}$ is the released electron wavelength, and $R_0$ is the internuclear distance, when $n{\lambda }_{\text{e}}$ becomes less than 65% of $R_0$. These results illustrate the possibility of employing attosecond pulses to perform photoelectron microscopy of molecules.

Figure 1

(Color online) The schematic diagram of a hydrogen molecular ion $\text{H}_2{}^{+}$.

Figure 2

(Color online) The ground state of 4D $\text{H}_2{}^{+}$ obtained from imaginary-time propagation: (a) the system energy as a function of propagation time, (b) the expectation value of internuclear distance $⟨R⟩$ versus propagation time, (c) the probability density profile on the $x\text{−}y$ plane, and (d) the probability density profile on the $x\text{−}R$ plane.

Figure 3

(Color online) The concerned xuv pulse has a photon energy of $\hslash \omega =50\text{eV}$, a peak field strength of $E_0\approx 0.53\text{a}\text{.u}\text{.}$, and a pulse length of $\tau =500\text{asec}$ with a ${\text{sin}}^2$ envelope.

Figure 4

(Color online) The time-dependent ground-state population of $\text{H}_2{}^{+}$ driven by attosecond xuv pulses $\left(\hslash \omega =50\text{eV}\right)$ polarized in parallel (red dashed-dotted line), in perpendicular (blue dashed line), and “tilted” $\theta =45°$) relative to the molecular axis in the $x$, $y$ plane (green solid line), respectively. A strong orientation dependence of the ground-state depletion is observed.

Figure 5

(Color online) The molecular system’s energy change $\left[⟨H_0⟩-E_0\right]$ during the attosecond pulse interaction $\left(\hslash \omega =50\text{eV}\right)$, for parallel polarization (red dashed-dotted line), perpendicular polarization (blue dashed line), and $45°$-tilted polarization relative to the molecular axis in the $x\text{−}y$ plane (green solid line), respectively.

Figure 6

(Color online) The electron probability densities of $\text{H}_2{}^{+}$ at the end of the attosecond xuv pulses $\left(\hslash \omega =50\text{eV}\right)$ for (a) parallel polarization and (c) perpendicular polarization. The corresponding momentum distributions of the ejected photoelectrons are shown in (b) and (d), respectively. The ionization probability is three times higher for the perpendicular polarization than that of the parallel polarization.

Figure 7

(Color online) (a) The pulse-ending snapshot of electron probability density of $\text{H}_2{}^{+}$ for the case of tilted polarization $\left(\theta =45°\right)$ with respect to the molecular axis ($x$ axis); (b) the corresponding photoelectron momentum distribution in the $x\text{−}y$ plane. It is noted that the ejected photoelectron peaks off the polarization axis, which is a result of the orientation dependence of the attosecond photoionization of $\text{H}_2{}^{+}$.

Figure 8

(Color online) The xuv pulse with a photon energy of $\hslash \omega =170\text{eV}$, a peak field strength of $\sim 3.76\text{a}\text{.u}\text{.}$, and a pulse length of $\tau =250\text{asec}$.

Figure 9

(Color online) The time-dependent ground-state population of $\text{H}_2{}^{+}$ driven by high photon-energy attosecond xuv pulses $\left(\hslash \omega =170\text{eV}\right)$, whose polarizations are either parallel (red dashed-dotted line) or perpendicular (blue dashed line) to the molecular axis.

Figure 10

(Color online) The molecular system’s energy change $\left[⟨H_0⟩-E_0\right]$ versus the interaction time $\left(\hslash \omega =170\text{eV}\right)$, for both parallel polarization (red dashed-dotted line) and perpendicular polarization (blue dashed line).

Figure 11

(Color online) The electron probability densities of $\text{H}_2{}^{+}$ at the end of the attosecond xuv pulses $\left(\hslash \omega =170\text{eV}\right)$ for (a) parallel polarization and (c) perpendicular polarization. The corresponding momentum distributions of the ejected photoelectrons are shown in (b) and (d), respectively. “Double-slit-like” interference patterns are observed for the perpendicular polarization case. The orientation dependence of attosecond photoionization, observed at $\hslash \omega =50\text{eV}$, disappears in this case.

Figure 12

(Color online) The electron probability densities of $\text{H}_2{}^{+}$ at the end of the attosecond xuv pulses at perpendicular polarization, for (a) $\hslash \omega =350\text{eV}$ and (b) $\hslash \omega =630\text{eV}$. Different field strengths are applied to obtain noticeable ionization. Classical double-slit interference angles are recovered when $n{\lambda }_{\text{e}}\le 65\%R_0$ is satisfied (see more discussions in text).

Figure 13

(Color online) The nuclear momentum spectrum for the parallel polarization cases at $\hslash \omega =50\text{eV}$.