Spectral caustics of high-order harmonics in one-dimensional periodic crystals

We theoretically investigate the spectral caustics of high-order harmonics in solids. We analyze the 1-dimension model of solids HHG and find that, apart from the caustics originated from the van Hove singularities in the energy-band structure, another kind of catastrophe singularities also emerge when the different branches of electron-hole trajectories generating high-order harmonics coalesce into a single branch. We solve time-dependent Schr\"odinger equation in periodic potential and demonstrate the control of this kind of singularities in HHG with the aids of two-color laser fields. The diffraction patterns of the harmonic spectrum near the caustics agree well with the inter-band electron-hole recombination trajectories predicted by the semiconductor semi-classical equation. This work is expected to help to understand the HHG dynamics in solids and manipulate the harmonic spectrum by adjusting driving field parameters.