The selection rules of high harmonic generation (HHG) are investigated using three-dimensional time-dependent density functional theory (TDDFT). From the harmonic spectra obtained with various real molecules and different forms of laser fields, several factors that contribute to selection rules are revealed. Extending the targets to stereoscopic molecules, it is shown that the allowed harmonics are dependent on the symmetries of the projections of molecules. For laser fields, the symmetries contributing to the selection rules are discussed according to Lissajous figures and their dynamical directivities. All the phenomena are explained by the symmetry of the full time-dependent Hamiltonian under a combined transformation. We present a systematic study on the selection rules and propose an intuitive method for the judgment of allowed harmonic orders, which can be extended to more complex molecules and various forms of laser pulses.

We propose an intuitive method, called the time-dependent population imaging (TDPI) method, to map the dynamical processes of high-order-harmonic generation (HHG) in solids by solving the time-dependent Schrödinger equation. It is shown that the real-time dynamical characteristics of HHG in solids, such as the instantaneous photon energies of emitted harmonics, can be read directly from the energy-resolved population oscillations of electrons in the TDPI picture. Meanwhile, the short and long trajectories of solid HHG are illustrated clearly from the TDPI picture. By using TDPI, we investigate the effects of the carrier-envelope phase in few-cycle pulses and demonstrate intuitively the HHG dynamics driven by two-color fields. Our results show that TDPI provides a powerful tool to study the ultrafast dynamics in strong fields for various laser-solid configurations and to gain insight into HHG processes in solids.

The wavelength scaling of the cutoff energy in solid high harmonic generation (HHG) is investigated theoretically by separating the contributions into the intraband and interband components. We find that, the cutoff energies of intraband and interband HHG exhibit the essentially different wavelength dependences. Specifically, the cutoff energy of the intraband HHG is wavelength independent, whereas the cutoff energy of the interband HHG depends linearly on the laser wavelength. Our results can uniformly interpret two current different wavelength scalings of the cutoff energy reported in previous researches. The fundamentally different wavelength scalings of the cutoff energy for intraband and interband HHG can be used to distinguish the two mechanisms.

We theoretically demonstrate a scheme to produce elliptically polarized isolated attosecond pulses, based on the high-order harmonic generation from the current-carrying state of an atom exposed in linearly polarized laser fields. It is shown that the nonzero ellipticity of the attosecond pulse generated at the single-atom level is attributed to the nonvanishing angular momentum of the state, which is further elucidated by the strong-field approximation model. With specific carrier envelope phases (CEPs), isolated attosecond pulses with large ellipticity can be obtained. Moreover, the ellipticity of high-order harmonics remains almost invariable versus the CEP of the few-cycle driving field. Hence, the temporal profile and ellipticity of the attosecond-pulse radiation can be manipulated separately, which is beneficial to shape the attosecond-pulse radiation without compromising the ellipticity. The obtained elliptical attosecond pulse may serve as a potential tool to explore and manipulate the ultrafast dynamics in magnetic materials and chiral media.

We show that the processes of $\pi -{\pi }^{*}$ transitions imprint themselves on the high-harmonic spectra when conjugated molecules interact with intense femtosecond laser pulses. It is found that a noninteger order peak appears in the harmonic spectrum with the photon energy equaling the excitation energy of the $\pi -{\pi }^{*}$ excitation. Further studies prove that this radiation is caused by the $\pi -{\pi }^{*}$ transition. The transition signals are prominent and can be easily measured as the corresponding radiation intensities are comparable to those of integer order harmonics. Our results pave the way for the study of excited-state electron-ion dynamics using high-harmonic spectroscopy. In comparison to the traditional absorption spectroscopy method relying on the synchrotron radiation source, the present approach is easily accessible for the use of a tabletop laser-based source. Furthermore, our study also provides a potential tool to probe the $\pi -{\pi }^{*}$ transition processes in femtosecond resolution.

We theoretically investigate the wavelength dependence of high-order harmonic yields in solids driven by a mid-infrared laser field. By solving the three-dimensional two-band density matrix equations in the wavelength range of $2.0–7.5\mu \mathrm{m}$, it is shown that, in the limit of slow dephasing (dephasing time $T_2\to \infty$), the high-order harmonic yield from a crystal follows a scaling of ${\lambda }^{-4}$ for a fixed energy interval. The ${\lambda }^{-4}$ scaling is attributed to the wave-packet spreading (${\lambda }^{-3}$) for the overall yield and the energy distribution effect (${\lambda }^{-1}$) due to the increase of the cutoff. For the crystal with a finite dephasing time $T_2$, we find that the exponential factor $x$ of ${\lambda }^{-x}$ increases with a decay of $T_2$. An apparent and rapid fluctuation on a fine wavelength mesh is also observed in the harmonic yields. The fine-scale oscillation originates from the quantum interference effect, and the corresponding modulation period $\delta \lambda$ scales as ${\lambda }^{-1}$.