Time-resolved dynamics of odd and even harmonic emission from oriented asymmetric molecules

We study the time-resolved dynamics of high-order harmonic generation (HHG) from oriented asymmetric molecules in intense laser fields theoretically. Previous studies have shown that the odd-even HHG spectra of asymmetric molecules don't show the striking two-center-interference-induced minimum, as the symmetric molecules do, due to the symmetry breaking. Surprisingly, with considering only the short-trajectory contribution, an apparent groove with small amplitudes is observed in the HHG time-frequency distribution, which implies that the harmonic emission is strongly suppressed in a specific time-frequency region. The position of this groove is sensitive to the molecular parameters and the orientation. Our analyses on this origin of the groove reveal different time-frequency properties of odd versus even signals, where the interplay of intramolecular interference and the permanent-dipole effect plays an important role. We show that the even-odd ratio often used in high-harmonic spectroscopy can be influenced significantly by the interference effect.

Figure 1

Time-frequency distribution of the HHG for the 3D model $\mathrm{HeH}^{2+}$, at (a) $\theta =0^{\circ }$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={40}^{\circ }$, and (d) $\theta ={50}^{\circ }$. The molecular parameters are shown at the top of the panels.

Figure 2

Time-frequency distribution of the HHG for the 3D model $\mathrm{HeH}^{2+}$ restricted to a short trajectory, at (a) $\theta =0^{\circ }$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={40}^{\circ }$, and (d) $\theta ={50}^{\circ }$. The molecular parameters are the same as those in Fig. 1. The approximate location of the center of the groove is marked in each panel.

Figure 3

Odd-even HHG spectra restricted to a short trajectory (left-hand panels) and the corresponding odd dipoles of Eq. (3) and even dipoles of Eq. (4) (right-hand panels) at different orientation angles $\theta$. The molecular parameters are the same as those in Fig. 2. The angle $\theta$ is as shown. Here, the odd spectra and dipoles are plotted in bold black curves, while the even ones are plotted in red thin curves. The approximate locations of the minima in dipoles are marked.

Figure 4

Same as Fig. 2, but for different molecular parameters.

Figure 5

Same as Fig. 3, but for different molecular parameters.

Figure 6

Odd and even HHG spectra restricted to a short trajectory and relevant dipoles for the 1D model $\mathrm{HeH}^{2+}$. The molecular parameters are shown at the top of the panels. In panel (a), we show the odd (solid black curve) and even (dashed-dotted red curve) spectra obtained using Eq. (1). In panel (b), we show the odd dipole $|\langle 0|{\vec{\mathbf{e}}}_x·\vec{\nabla }V|{\mathbf{p}}_u{\rangle |}^2/{\omega }^4$ (solid black curve) and the even one $|\langle 0|{\vec{\mathbf{e}}}_x·\vec{\nabla }V|{\mathbf{p}}_g{\rangle |}^2/{\omega }^4$ (dashed-dotted red curve). In panels (c) and (d) for odd and even harmonics, respectively, we show the spectra obtained using Eq. (7) of the ground-state-$|{\mathbf{p}}_u\rangle$ channel and Eq. (8) of the ground-state-$|{\mathbf{p}}_g\rangle$ channel (solid black curves), Eq. (5) of the ground-state-all-continuum-states channel (dotted blue curves), and Eq. (1) of the exact expression (dashed-dotted red curves). The solid arrow indicates the minimum in the odd spectrum of Eq. (7). The location of the minimum in the odd dipole is marked.

Figure 7

Time-frequency distribution of the HHG (color coding) for the 1D model $\mathrm{HeH}^{2+}$ restricted to a short trajectory, obtained using Eq. (2) of the exact expression (a), Eq. (6) of the ground-state-all-continuum-states channel (b), Eq. (9) of the ground-state-$|{\mathbf{p}}_u\rangle$ channel for odd harmonics (c), and Eq. (10) of the ground-state-$|{\mathbf{p}}_g\rangle$ channel for even harmonics (d). The molecular parameters are shown at the top of the panels.

Figure 8

Same as Fig. 7, but for different molecular parameters.