Atomic-orbital-dependent photoelectron momentum distributions for ${\mathrm{F}}^{-}$ ions by orthogonal two-color laser fields

We theoretically investigate the two-dimensional photoelectron momentum distributions (PMDs) of ${\mathrm{F}}^{-}$ ions in an orthogonal two-color laser field with equal intensities. The PMDs for different atomic orbitals are simulated by an exact solution to the three-dimensional time-dependent Schrödinger equation and the strong-field approximation method, respectively. Through the comparison of the calculations of these methods, we confirm that the asymptotic behavior of initial bound-state wave function plays a crucial role in forming the main shape of PMDs at large momenta. Based on the saddle-point method and the imaginary time theory, we show that the PMDs of ${\mathrm{F}}^{-}$ ions can be decoded to reveal the definite imprint of the photoelectron sub-barrier phase from the subcycle interference structures. We demonstrate the sub-barrier phases from different atomic orbitals have different impacts on the subcycle interference structures.

Figure 1

The two-dimensional photoelectron momentum distributions of ${\mathrm{F}}^{-}$ ions in the polarization plane of a few-cycle OTC laser field with $\mathrm{\Delta }\phi =0$, for the $2p_x$ and $2p_y$ orbitals, respectively. The values in the first column are calculated by the TDSE method. The values in the second column are calculated by the SFA method with the asymptotic wave function, and are shown as the AS orbital on the top. The values in the third column are calculated by the SFA method with the Hatree-Fock–type wave function, and are shown as the HF orbital on the top. The data are normalized to the maximum probability and the color is plotted on the logarithmic scale.

Figure 2

The radial probability density distributions of the initial-state wave functions used in this work. The solid red line corresponds to the initial-state wave function of the TDSE, which is obtained by solving the stationary Schrödinger equation with the double Yukawa potential in Ref. [21]. The dot-dashed olive line corresponds to the asymptotic wave function of Eq. (5). The dotted blue line corresponds to the Hartree-Fock–type wave function of Eq. (6).

Figure 3

The temporal sketch of the three-cycle OTC field with $\mathrm{\Delta }\phi =0$ and the corresponding saddle-point distributions. The dotted olive line, dashed blue line, and solid orange line represent the magnitude of the fundamental field, the second harmonic field, and the synthesized electric field, respectively. Each group of points represents the positions of saddle points for a range of photoelectron energies from 0 to 1.0 a.u. and ejection angles $\theta ={90}^{\circ }$ and $\theta ={270}^{\circ }$. The red (dark gray) points and green (light gray) points indicate the SPs for $\theta ={90}^{\circ }$ and $\theta ={270}^{\circ }$, respectively. The numbers of several dominant SPs are also marked in the complex-time plane.

Figure 4

The PMDs obtained from the SP method by considering five dominant SPs (SPs 5–9), with and without sub-barrier phase, respectively. The values in the first column are the results for the $2p_x$ orbital, and the values in the second column are those for the $2p_y$ orbital. The values in the first row are calculated with the sub-barrier phase. The values in the second row are calculated without the sub-barrier phase. The white rectangle in (b) marks one of the four minima on the edge of the two-dimensional PMDs for the $2p_y$ orbital. The white rectangle in (c) marks one of the four minima in the PMDs for the $2p_x$ orbital when the sub-barrier phase is absent.

Figure 5

The PMDs for the $2p_x$ orbital obtained from the SP method with and without sub-barrier phase, by coherently adding different saddle points, respectively. The values in the first row are the results obtained by considering SPs 5–7, and the values in the the second row are obtained by considering SPs 5 and 6. The values in the first column are calculated with the sub-barrier phase. The values in the second column are calculated without the sub-barrier phase. The white rectangle in (d) marks the minima corresponding to those in Fig. 4.

Figure 6

The same as Fig. 5 but for the $2p_y$ orbital. The white rectangle in (c) marks the minima corresponding to those in Fig. 4.