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.

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.

Band structure and transition dipole moment play important roles in high-order harmonic generation from solid materials. In this work we provide a new all-optical technique to reconstruct the momentum-dependent transition dipole moment using the harmonic spectrum from MgO crystal driven by an ultrashort mid-infrared laser pulse. Under the influence of the ultrashort laser pulse, the emitted photon energy and the crystal momentum form a one-to-one match, in the same way between the intensity of the harmonic above the minimum bandgap and the square of the amplitude of the transition dipole moment, resulting in a realization of directly probing the transition dipole moment. Our all-optical method paves a way to image the two-dimensional transition dipole moment of crystals with the inversion symmetry.

Attosecond pulses in the soft x-ray (SXR) to water-window energy region offer the tools for creating and studying target-specific localized inner-shell electrons or holes in materials, enabling monitoring or controlling charge and energy flows in a dynamic system on attosecond timescales. Recently, a number of laboratories have reported generation of continuum harmonics in the hundred-electron-volt to kilovolt region with few-cycle long-wavelength mid-infrared lasers. These harmonics have the bandwidth to support pulses with durations of a few to a few-ten attoseconds. But harmonics generated in a gas medium have attochirps that cannot be fully compensated by materials over a broad spectral range; thus, realistically what are the typical shortest attosecond pulses that one can generate? To answer this question, it is essential that the temporal attosecond pulses be accurately characterized. By reanalyzing the soft x-ray attosecond metrology reported in three recent experiments [Li et al. 53-attosecond x-ray pulses reach the carbon K-edge, Nat. Commun. 8, 186 (2017); Gaumnitz et al. Streaking of 43-attosecond soft x-ray pulses generated by a passively CEP-stable mid-infrared driver, Opt. Exp. 25, 27506 (2017); Cousin et al. Attosecond Streaking in the Water Window: A New Regime of Attosecond Pulse Characterization, Phys. Rev. X 7, 041030 (2017)] using a newly developed broadband phase retrieval algorithm, we demonstrate that the generated attosecond pulses in the first two papers have durations of about 60 as, longer than what they have reported. Similarly, the duration from the third experiment is retrieved to be about half of the 322-as upper limit cited in that work. We also introduce the autocorrelation (AC) of the streaking spectrogram. By comparing the ACs from the experiments and from the retrieved SXR pulses, the accuracy of the retrieved results can be directly visualized. Our retrieval method is fast and accurate, and it shall provide a powerful tool for the metrology of the emerging few-ten-attosecond pulses.

Cooper minimum structure of high-order harmonic spectra from atoms or molecules has been extensively studied. In this paper, we demonstrate that the crystal harmonic spectra from an ultrashort midinfrared laser pulse also exhibit the Cooper minimum characteristic. Based on the accurate band dispersion and $k$-dependent transition dipole moment (TDM) from the first-principles calculations, it can be found that the harmonic spectra from the MgO crystal along the $\mathrm{\Gamma }\text{−}X$ direction present a dip structure in the plateau, which is originated from the valley of TDM by examining the distribution of the harmonic intensity at the $k$ space. The Cooper minimum feature in crystal high-order-harmonic generation (HHG) will pave a new way to retrieve the band information of solid materials by using HHG from the ultrashort midinfrared laser pulse.