Geometrical dependence in photoionization of ${{\mathrm{H}}_2}^{+}$ in high-intensity, high-frequency, ultrashort laser pulses

The ionization dynamics of ${{\mathrm{H}}_2}^{+}$ exposed to high-intensity, high-frequency, ultrashort laser pulses is investigated with two theoretical approaches. The time-dependent Schrödinger equation is solved by a direct numerical method, and a simple two-center interference-diffraction model is studied. The energy and angular distributions of the photoelectron for various internuclear distances and relative orientations between the internuclear axis of the molecule and the polarization of the field are calculated. The main features of the photoelectron spectrum pattern are described well by the interference-diffraction model, and excellent quantitative agreement between the two methods is found. The effect of quantal vibration on the photoelectron spectrum is also calculated. We find that vibrational average produces some broadening of the main features, but that the patterns remain clearly distinguishable.

Figure 1

(Color online) Coordinates (molecular frame); The protons are at a fixed distance $R$ with the molecular axis defining the polar axis ($z$ direction). The $xz$-plane is chosen so that it contains the polarization vector which defines an angle $\theta$ with respect to the $z$ axis. The electron coordinate, with its origin at the midpoint between the nuclei, is denoted by $\mathbf{r}$.

Figure 2

(Color online) Outgoing waves from each of the scattering centers at three different instants of time (from left to right: $t=$4, 7, and $10\mathrm{a}.\mathrm{u}.$). The central energy of the wave packets is $1.1\mathrm{a}.\mathrm{u}.$, and the internuclear separation is $3\mathrm{a}.\mathrm{u}.$ The upper panels correspond to ionization in the direction parallel to the internuclear axis, and the lower ones correspond to perpendicular ionization direction. The scattering centers are shown as circular dots. In the upper case, one of the two outgoing waves is distorted by the upper scattering center, whereas no such interaction takes place in the lower case.

Figure 3

(Color online) The momentum wave function in the scattering plane, $\mid {\tilde{\Psi }}_{\mathrm{out}}(k_x,k_y=0,k_z){\mid }^2$, for various internuclear distances $R$ and relative orientations between the linearly polarized field and the vertical internuclear axis. $R$ increases from top to bottom. To the left, the field and the molecule are parallel $(\theta =0^{\circ })$ to each other, to the right they are perpendicular to each other $(\theta =90°)$, and in the middle we have $\theta =45°$. The internuclear axis points in up/down direction, and the momenta $k_x$ and $k_z$ span from $-3$ to $3\mathrm{a}.\mathrm{u}.$

Figure 4

(Color online) The energy distribution of the photoelectron as a function of kinetic energy of the outgoing electron and the angle $\theta$ between the internuclear axis and the field. The distributions correspond to $R=$1, 2, 3, and $4\mathrm{a}.\mathrm{u}.$ Peaks are seen at $E\approx \omega -I_P$ and $E\approx 2\omega -I_P$ [$\omega =2\mathrm{a}.\mathrm{u}.$, and $I_P\left(R\right)=$1.45, 1.1, 0.91, and $0.80\mathrm{a}.\mathrm{u}.$, respectively]. The one-photon ionization signal dominates the spectrum, and therefore it also governs the dependence of the ionization probability on the relative orientation $\theta$.

Figure 5

(Color online) The differential ionization probability at various polarization directions and internuclear separations. The figures are displayed in the same manner as in Fig. 3. The line indicates the direction of the field. The overall scaling varies from figure to figure.

Figure 6

(Color online) The differential ionization probability for one-photon ionization plotted in the scattering plane. The ordinate is the angle of the outgoing electron relative to the direction of the field. The full curves are the results obtained from solving the Schrödinger equation, the dotted curves are the prediction of the simple interference model, Eq. (13), while the dashed curves are the predictions of the model including diffraction, Eq. (16). The panels are displayed according to geometry in the same manner as Figs. 3, 5.

Figure 7

(Color online) The total differential ionization probability with the initial vibrational state taken into consideration. The upper panels correspond to the vibrational ground state, and the lower ones correspond to a Gaussian distribution centered at $R=2.5\mathrm{a}.\mathrm{u}.$ with a standard deviation of $0.7\mathrm{a}.\mathrm{u}.$ These vibrational wave functions are shown to the left. As in Fig. 4, the direction of the polarization with respect to the vertical internuclear axis is given by $\theta =0°$, $45°$, and $90°$, respectively.