Alignment and pulse-duration effects in two-photon double ionization of ${\mathrm{H}}_2$ by femtosecond XUV laser pulses

We present calculations for the dependence of the two-photon double ionization (DI) of ${\mathrm{H}}_2$ on the relative orientation of the linear laser polarization to the internuclear axis and the length of the pulse. We use the fixed-nuclei approximation at the equilibrium distance of 1.4 $a_0$, where $a_0=0.529\times {10}^{-10}\mathrm{m}$ is the Bohr radius. Central photon energies cover the entire direct DI domain from 26.5 to 34.0 eV. In contrast to the parallel geometry studied earlier [X. Guan, K. Bartschat, B. I. Schneider, and L. Koesterke, Phys. Rev. A 83, 043403 (2011)], the effect of the pulse duration is almost negligible for the case when the two axes are perpendicular to each other. This is a consequence of the symmetry rules for dipole excitation in the two cases. In the parallel geometry, doubly excited states of ${}^1{\Sigma }_u^{+}$ symmetry affect the cross section, while in the perpendicular geometry only much longer-lived ${}^1{\Pi }_u$ states are present. This accounts for the different convergence patterns observed in the calculated cross sections as a function of the pulse length. When the photon energy approaches the threshold of sequential DI, a sharp increase of the generalized total cross section (GTCS) with increasing pulse duration is also observed in the perpendicular geometry, very similar to the case of the molecular axis being oriented along the laser polarization direction. Our results differ from those of Colgan et al. [J. Colgan, M. S. Pindzola, and F. Robicheaux, J. Phys. B 41, 121002 (2008)] and Morales et al. [F. Morales, F. Martín, D. A. Horner, T. N. Rescigno, and C. W. McCurdy, J. Phys. B 42, 134013 (2009)], but are in excellent agreement with the GTCSs of Simonsen et al. [A. S. Simonsen, S. A. Sørngård, R. Nepstad, and M. Førre, Phys. Rev. A 85, 063404 (2012)] over the entire domain of direct DI.

Figure 1

GTCS for two-photon DI of the ${\mathrm{H}}_2$ molecule. In panel (a), the molecular axis is oriented perpendicular to the linear polarization vector (represented by a double-headed arrow) of the laser pulse. Our present results were obtained for pulse lengths of 10, 20, and 30 optical cycles. Also shown are the ab initio and the model results of Simonsen et al. [11], as well as the cross sections of Colgan et al. [9] and Morales et al. [10] at 30 eV. Panel (b) shows a comparison between our ab initio results obtained in the perpendicular ($\perp$) and parallel ($\parallel$) geometries, with the corresponding predictions from a simple model suggested by Simonsen et al. [11].

Figure 2

Angular distributions of the electrons ejected from the ${\mathrm{H}}_2$ molecule at equal energy sharing ($E_1=E_2=4.3$ eV) for the central photon energy of 30.0 eV in the FNA at the equilibrium distance of 1.4 $a_0$. The laser polarization axis is represented by a double-headed arrow. (a) 10-cycle pulse in the parallel geometry. (b) 40-cycle pulse in the parallel geometry. (c) 10-cycle pulse in the perpendicular geometry. (d) 30-cycle pulse in the perpendicular geometry. One electron (${\mathit{k}}_1$) is observed, respectively, at fixed angles (${\theta }_1$) of $0^{\circ }$, ${30}^{\circ }$, ${60}^{\circ }$, and ${90}^{\circ }$ with respect to the laser polarization axis. Each panel has been individually scaled.

Figure 3

Angular distributions of the electrons ejected from the ${\mathrm{H}}_2$ molecule at equal energy sharing ($E_1=E_2=2.5$ eV) for the central photon energy of 26.5 eV in the FNA at the equilibrium distance of 1.4 $a_0$. (a) 10-cycle pulse in the perpendicular geometry. (b) 30-cycle pulse in the perpendicular geometry.

Figure 4

Angular distributions of the electrons ejected from the ${\mathrm{H}}_2$ molecule at equal energy sharing ($E_1=E_2=6.3$ eV) for the central photon energy of 34.0 eV in the FNA at the equilibrium distance of 1.4 $a_0$. (a) 10-cycle pulse in the perpendicular geometry. (b) 30-cycle pulse in the perpendicular geometry.

Figure 5

GTDCS in the coplanar geometry for electrons ejected from the ${\mathrm{H}}_2$ molecule at equal energy sharing ($E_1=E_2=4.3$ eV) in the parallel case for the central photon energy of 30.0 eV. The detection angle ${\theta }_1$ of one electron is held fixed at $0^{\circ }$(a), ${30}^{\circ }$(b), ${60}^{\circ }$(c), or ${90}^{\circ }$(d), while the detection angle ${\theta }_2$ is varied. The results from the present calculation for 10 and 40 o.c. are compared with predictions from earlier TDCC [9] and ECS [10] calculations. Note the scale factors in the legend of panel (a), which apply to all panels of this figure.

Figure 6

GTDCS in the coplanar geometry for electrons ejected from the ${\mathrm{H}}_2$ molecule at equal energy sharing ($E_1=E_2=4.3$ eV) in the perpendicular case for the central photon energy of 30.0 eV. The results from the present calculation for 10 and 30 o.c. are compared with predictions from earlier TDCC [9] and ECS [10] calculations. No scale factors were applied here.

Figure 7

Same as Fig. 6 for central photon energies of 26.5 eV (a) and 34.0 eV (b), respectively. The results are from the present calculation for 30 o.c. No TDCC or ECS results are available for these cases. Note the scale factors for ${\theta }_1={60}^{\circ }$ and ${90}^{\circ }$.

Figure 8

Coplanar GTDCS for central photon energies of 30.0 eV [left column, panels (a)–(d)] and 34.0 eV [right column, panels (e)–(h)]. The results are from the present calculation for 30 o.c. in the perpendicular case, for fixed angles ${\theta }_1=0^{\circ }$, ${30}^{\circ }$, ${60}^{\circ }$, and ${90}^{\circ }$ for different portions of the excess energy shared between the two outgoing electrons. The radius of the outer circle corresponds to $3\times {10}^{-54}{\mathrm{cm}}^4\mathrm{s}/\left({\mathrm{sr}}^2\mathrm{eV}\right)$. Note the scale factors in the panels for ${\theta }_1\ne 0^{\circ }$.