Acta Photonica Sinica, Volume. 51, Issue 3, 0302001(2022)

Internal Relationship between Symmetrical and Asymmetric Molecular Harmonic Radiation and Its Structure

For symmetric molecules, only odd harmonics are emitted. In particular, the high-order harmonic spectrum shows a significant minimum which corresponds to the minimum in the dipole moment of the bound-continuous state transition. For asymmetric molecules, both odd and even harmonics are emitted. However, in many cases, the striking minimum disappears in the odd or even harmonic spectrum. Fortunately, when the minimum cannot be read from the harmonic spectra directly, it can be probed through the polarization measurement of the odd-even high-order harmonic generation. Specifically, the position of the minimum in the odd or even dipole corresponds to the harmonic order for the maximal ellipticity of the odd or even harmonics. However, for symmetric molecules ${\mathrm{H}}_{\mathrm{2}}^{+}$, the minimum in the transition dipole is not completely consistent with the minimum of the harmonic spectra. For asymmetric molecules $\mathrm{H}\mathrm{e}{\mathrm{H}}^{\mathrm{2}+}$, the prediction of the dipole minimum by the polarization measurement does not always agree well with the theoretical evaluation. In some cases, a remarkable difference is also observed. This remarkable difference may arise from other mechanisms beyond the description of the simple model, or the inaccurate calculation of dipole moment may be caused only by some rough approximate in relevant theoretical treatments. In this paper, this question is explored by improving the calculation of the dipole moment. The intrinsic relationship between harmonic radiation and the structure of symmetric and asymmetric molecules is studied by a combination of numerical and analytical methods. First, the numerical expressions of the ground state wave functions of symmetric and asymmetric molecules are obtained using the virtual time evolution method. Starting from the accurate ground state wave functions of symmetric and asymmetric molecules, the bound-continuous state transition dipole moments are calculated. The term proportional to the nuclear separation is further subtracted from the transition dipole moment. For symmetric molecules ${\mathrm{H}}_{\mathrm{2}}^{+}$, the calculated odd dipole moment is compared with the harmonic spectrum and the transition dipole moments obtained by the pure analytical method. For asymmetric molecules $\mathrm{H}\mathrm{e}{\mathrm{H}}^{\mathrm{2}+}$, the calculated odd dipole moment is compared with the harmonic spectrum, the ellipticity of the harmonics and the transition dipole moments obtained by the pure analytical method. Simulation results show that the minimum in the improved odd dipole moments agree more well with that predicted by odd harmonics compared with the transition dipole moments obtained by the pure analytical method for symmetric molecules${\mathrm{H}}_{\mathrm{2}}^{+}$. For asymmetric molecules $\mathrm{H}\mathrm{e}{\mathrm{H}}^{\mathrm{2}+}$, the calculated dipole moment shows a clear minimum, which arises from the effect of two-center interference. However, there is usually no minimum value appearing in the high-order harmonic spectrum of asymmetric molecules. A further comparison between the ellipticity of the harmonics and the corresponding dipole moments shows that the harmonic order at which the ellipticity is maximal corresponds to the order at which the dipole has a minimum. The polarization measurement of harmonics can be used as a tool to detect the position of the minimum value of dipole moment. The obtained ground state wave function significantly improves the consistency between the minimum odd-even dipole moment and the maximum odd-even harmonic polarization at different molecular parameters. These phenomena reveal that the recombination process plays a key role in the harmonic radiation of symmetric and asymmetric molecules and verifies the one-to-one matching between the high-order harmonic spectra and the corresponding dipoles. And molecular orbitals can be reconstructed by transition dipole elements. The research results provide deep insights into the relation between odd high-order harmonic generation and symmetric molecular orbital and the relation between odd-even high-order harmonic generation and asymmetric molecular orbital. The research results have some significance for the role of odd-even harmonic radiation in the ultrafast detection of asymmetric molecules.