Angular distribution parameters in two-photon double ionization of helium beyond the dipole approximation

We study nondipole effects in the two-photon double ionization of helium by solving the full-dimensional time-dependent Schrödinger equation (TDSE). The angular distributions of the photoelectrons in TDSE calculations are parametrized by several angular distribution parameters. The electron correlation effects are revealed and discussed by analyzing angular distribution parameters of the photoelectrons. Compared with the linearly polarized laser, the values of dipole angular parameters for circularly polarized laser pulses in the nonsequential double-ionization regime are closer to the pure dipole emission of the one-photon single ionization, which indicates a weaker electron correlation for circular pulses. For the nondipole angular parameters, in the sequential double-ionization regime, we find that the parameters show peak structures from the sequential ionization channel over the energy sharing between two electrons and these peak positions are slightly shifted by the electron correlation. In general, due to the nondipole effects, the angular distributions of photoelectrons are shifted towards the direction of laser propagation, due to the transfer of the photon linear momentum.

Figure 1

Dipole angular parameters for a linearly (left panel) and circularly (right panel) polarized pulse for a duration of 2 fs at different photon energies. The two horizonal gray dashed lines at 2 and 0 in each panel represent the cases of an uncorrelated emission of electrons. The vertical gray solid lines are respectively related to the sequential process of the ground state of ${\mathrm{He}}^{+}$ (the inner two) and the first excited state of ${\mathrm{He}}^{+}$ (the outer two), for the case of 70 eV.

Figure 2

Dipole angular distribution parameters for 42 eV (upper panel) and 68 eV (lower panel) under different ellipticity of the laser field. $\epsilon =0$ (1) represents linear (circular) polarization laser pulse. Besides, the pulse duration is 2 fs for 42 eV and 20 cycles for 68 eV. The two horizonal gray dashed lines at 2 and 0 in each panel represent the cases of an uncorrelated emission of electrons. The vertical gray solid lines are respectively related to the sequential process of the ground state of ${\mathrm{He}}^{+}$ (the inner two) and the first excited state of ${\mathrm{He}}^{+}$ (the outer two), for the case of 68 eV.

Figure 3

Nondipole angular parameters $\gamma ,\delta$, and ${\tilde{\beta }}_5$ for a 2 fs pulse of circular polarization at different photon energies. The horizonal gray dashed line at zero represents the results if nondipole effects are not considered.

Figure 4

Relatively asymmetric parameter $A$ as a function of the energy sharing $\eta$ for circularly polarized laser pulses. The horizonal gray dashed line at zero in each panel represents the results in the case where nondipole effects are not considered. (a) The duration of laser pulses is 2 fs, but for different photon energies. (b) The photon energy is fixed at 68 eV, but for different pulse durations. The vertical gray solid lines are related to the sequential process of the ground state of ${\mathrm{He}}^{+}$ (the inner two) and the first excited state of ${\mathrm{He}}^{+}$ (the outer two).

Figure 5

Reconstructed one-electron angular distribution in the NSDI regime (the first row) and the SDI regime (the second row). The angle $0^{\circ }$ represents the direction of the laser propagation. The left column represents distributions reconstructed by the dipole parameters only (dipole) and by all the parameters (total). The right column shows the angular distributions reconstructed only by the nondipole parameters.

Figure 6

Relative scale of nondipole angular distribution $P_{\mathrm{ND}}\left({\theta }_1\right)$ and the dipole angular distribution $P_{\mathrm{D}}\left({\theta }_1\right)$ with photon energy 42 eV (left panel) and 68 eV (right panel). The horizonal dashed gray line represents the value zero. The direction perpendicular to the propagation direction laser pulse is shown by the vertical dashed gray line. The positive (negative) value represents the enhancement (reduction) of the angular distribution because of nondipole corrections.