All-optical reconstruction of k-dependent transition dipole moment by solid harmonic spectra from ultrashort laser pulses

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.