Since the first observation of odd and even high-order harmonics generated from ZnO crystals in 2011, the dependence of the harmonic yields on the orientation of the laser polarization with respect to the crystal axis has never been properly interpreted. This failure has been traced to the lack of a correct account of the phase of the transition dipole moment between the valence band and the conduction band. Using a simple one-dimensional two-band model, here we demonstrate that the observed odd harmonics is directly related to the orientation dependence of the magnitude of the transition dipole, while even harmonics is directly related to the phase of the transition dipole. Our result points out the essential role of the complex transition dipole moment in understanding harmonic generation from solids that has long been overlooked so far.

We carry out a combined theoretical and experimental investigation on the population distributions in the ground and excited states of tunnel-ionized nitrogen molecules at various driver wavelengths in the near- and midinfrared range. Our results reveal that efficient couplings (i.e., population exchanges) between the ground ${\mathrm{N}}_2^{+}(X^2{\mathrm{\Sigma }}_g^{+})$ state and the excited ${\mathrm{N}}_2^{+}(A^2{\mathrm{\Pi }}_u)$ and ${\mathrm{N}}_2^{+}(B^2{\mathrm{\Sigma }}_u^{+})$ states occur in strong laser fields. The couplings result in a population inversion between the ${\mathrm{N}}_2^{+}(X^2{\mathrm{\Sigma }}_g^{+})$ and ${\mathrm{N}}_2^{+}(B^2{\mathrm{\Sigma }}_u^{+})$ states at wavelengths near 800 nm, which is verified by our experimental observation of the amplification of a seed at $\sim 391\mathrm{nm}$. The result provides insight into the mechanism of free-space nitrogen ion lasers generated in remote air with strong femtosecond laser pulses.

High-order harmonic spectra from solid materials driven by single-color multicycle laser fields sometimes contain even harmonics. In this work we attribute the appearance of even harmonics to the nonzero transition dipole phase (TDP) when the solid system has broken symmetry. By calculating the harmonic efficiency from graphene and gapped graphene by using the semiconductor Bloch equations under the tight-binding approximation, we demonstrate the role of the TDP, which has been ignored for a long time. When the crystal has inversion symmetry, or reflection symmetry with the symmetry plane perpendicular to the laser polarization direction, the TDP can be neglected. Without such symmetry, however, the TDP will lead to the appearance of even harmonics. We further show that the TDP is sensitive to the crystal geometry. To extract the structure information from the harmonic spectra of a solid the TDP cannot be ignored.

Interaction of intense lasers with solid materials offers an alternative way to achieve high-order harmonic generation (HHG). Since the underlying mechanisms of the harmonic emission remain uncertain, a large number of theoretical studies have been proposed to explain the experimental findings in this fast-growing field. Here, we review the primary numerical methods and present a brief perspective of HHG in solids. Graphical Abstract

High-order-harmonic generation in α-quartz $\mathrm{Si}{\mathrm{O}}_2$ is theoretically investigated under a strong laser field by solving the extended semiconductor Bloch equations. The accurate band structures as well as dipole moments between different bands are obtained from state-of-the-art first-principles calculations. We find that the shapes of $k$-space-dependent dipole moments play an important role in harmonic generation. The calculated results show that harmonic conversion efficiency is significantly enhanced and the cutoff energy is distinctly increased when the dipole moments change greatly along a valley in the $k$ direction in the solid. Based on that dependence on the dipole moment, we also show that symmetry groups greatly affect the harmonic spectra from the solid materials. Moreover, a two-color synthesized field is used to achieve a supercontinuum harmonic spectrum near the cutoff region, and isolated attosecond pulses can be obtained directly by filtering out the harmonic radiation. We hope the contribution presented in this work provides a useful reference for future studies on laser-crystal interactions.

We present a theoretical investigation of high-order harmonic generation in spatially inhomogeneous two-color laser fields by solving three dimensional time dependent Schrödinger equation. The cutoff in the harmonic spectra can be significantly extended by means of our proposed method (i.e., from helium interacting with the plasmon-enhanced two-color laser fields), and an ultrabroad supercontinuum up to 1.5 keV is generated by selecting proper carrier-envelope phase of the controlling field. Moreover, classical trajectory extraction, time-dependent ionization and recombination rates, and time-frequency analyses are used to explain the generation of this ultrabroadband supercontinuum. As a result, an isolated 8.8 attosecond pulse can be generated directly by the superposition of the supercontinuum harmonics.