Interference effects in above-threshold ionization from diatomic molecules: Determining the internuclear separation

We calculate angle-resolved above-threshold ionization spectra for diatomic molecules in linearly polarized laser fields, employing the strong-field approximation. The interference structure resulting from the individual contributions of the different scattering scenarios is discussed in detail, with respect to the dependence on the internuclear distance and molecular orientation. We show that, in general, the contributions from the processes in which the electron is freed at one center and rescatters off the other obscure the interference maxima and minima obtained from single-center processes. However, around the boundary of the energy regions for which rescattering has a classical counterpart, such processes play a negligible role and very clear interference patterns are observed. In such energy regions, one is able to infer the internuclear distance from the energy difference between adjacent interference minima.

Figure 1

(Color online) ATI spectra of the direct electrons in the laser polarization direction in the atomic and molecular case, for a molecule aligned parallel to a laser field of frequency $\omega =0.058\text{a.u.}$ and ponderomotive potential $U_p=2.08\text{a.u.}$ We consider a symmetric combination of atomic orbitals [$c_1=c_2=1∕\sqrt{2}$ in Eq. (5)], with ionization potential $\mid E_0\mid =0.9\text{a.u.}$ In order to facilitate the comparison, the same ionization potential $V_0\left(\mathbf{r}\right)$ was chosen in the atomic and molecular cases. The arrows mark the various destructive interference energies $(n=0,1,2)$ for $R=6\text{a.u.}$

Figure 2

(Color online) Comparison of the complete ATI spectra consisting of direct and rescattered electrons in the atomic and the molecular case for internuclear distances of $R=2\text{a.u.}$ and $R=4\text{a.u.}$ The molecule is aligned parallel to the laser-polarization direction, and the electrons are emitted in the same direction. The arrows mark the destructive interferences $(n=0,1)$ of the molecule for $R=2\text{a.u.}$ The destructive interference for $n=1$ is already in the plateau region, where the role of exchange terms becomes important. The remaining parameters are the same as in Fig. 1.

Figure 3

(Color online) The same as Fig. 2 but with the molecular axis perpendicular to the laser polarization. The electrons are emitted parallel to the laser polarization.

Figure 4

(Color online) Individual contributions of the various rescattering scenarios to the total amplitude (18) for a diatomic molecule with internuclear distance $R=2\text{a.u.}$ aligned parallel to the laser-field polarization, for the same molecular and field parameters as in Fig. 2. The electrons are emitted in the polarization direction. The arrows mark the respective cutoff energies for the various transition amplitude matrix elements. The inset at the lower left-hand side is an enlargement of the region near the cutoff where the direct terms and the exchange terms differ in a characteristic fashion, allowing for the determination of the internuclear separation.

Figure 5

(Color online) Electron yield for the parameters of Fig. 1 and internuclear distance $R=2$ a.u. for different emission angles $\psi$ with respect to the polarization of the laser field. The molecule is aligned parallel (perpendicular) to the laser field in the upper (lower) panel.

Figure 6

(Color online) Angle-resolved ATI spectra on a logarithmic scale for a diatomic molecule with internuclear distances $R=2\text{a.u.}$ and $R=3\text{a.u.}$ (middle and bottom panels, respectively), aligned parallel to the laser-field polarization, compared to the single-atom case (upper panel). The binding energy is $E_0=0.9\text{a.u.}$ in all cases, and the laser frequency and the ponderomotive potential are $\omega =0.058\text{a.u.}$ and $U_p=2.08\text{a.u.}$, respectively. The plotted lines depict the minima (solid lines) and the maxima (dashed lines) of the energy distribution given by Eq. (11).

Figure 7

(Color online) Angle-resolved ATI spectra on a logarithmic scale for a diatomic molecule with ionization potential $E_0=0.9\text{a.u.}$ and internuclear distances $R=4\text{a.u.}$, $R=6\text{a.u.}$, and $R=8\text{a.u.}$ (upper, middle, and bottom panels, respectively), aligned parallel to the laser-field polarization. The field parameters are the same as in Fig. 6. The plotted lines depict the minima (solid lines) and the maxima (dashed lines) of the energy distribution given by Eq. (11).

Figure 8

(Color online) Enlargement of a limited energy-angle region of Fig. 7 for $R=8\text{a.u.}$ (lowest panel) with increased resolution. The indents of Fig. 7 are distinctly visible as valleys deeply cut into the high ridge that precedes the cutoff.