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 time-dependent population imaging (TDPI), to map the dynamical processes of high harmonic generation (HHG) in solids by solving the time-dependent Schrödinger equation (TDSE). 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 TDPIs. Meanwhile, the short and long trajectories of solid HHG are illustrated clearly from TDPI. By using the TDPI, we also investigate the effects of carrier-envelope phase (CEP) in few-cycle pulses and intuitively demonstrate the HHG dynamics driven by two-color fields. Our results show that the TDPI provides a powerful tool to study the ultrafast dynamics in strong fields for various laser-solid configurations and gain an insight into HHG processes in solids.

High-order-harmonic generation (HHG) from small linear molecules driven by a circularly polarized laser pulse (CPLP) is investigated. It is found that the obtained high-order harmonics are more pronounced than those from reference atoms with equal ionization potential driven by the same CPLP. By analyzing the dependence of the cutoff position on laser parameters and calculating the recollision trajectories, it is shown that this molecular HHG originates from the recollision mechanism, instead of the bound-bound transition mechanism found to be responsible for molecular HHG by CPLP in earlier works. A semiclassical model is used to analyze the HHG process and discuss the origin of the higher efficiency of molecular HHG. It is found that the higher HHG efficiency for molecules is mainly contributed in the recombination step and at least partly due to the higher recollision probability of continuum electrons.