Orientation dependence of caustic effect in high-order harmonic generation of graphene

The propagation of light can exhibit an interesting behavior termed as caustic phenomenon, in which multiple light rays converge and give rise to the bright focusing features. In analogous to light propagation, the generalized caustic effects occur when multiple particle trajectories converge, resulting in a caustic singularity and the subsequent enhancement of physical phenomena. The caustic phenomena have been observed in diverse fields, including acoustics, radio propagation, as well as high-order harmonic generation (HHG). When an atom is irradiated by a femtosecond laser, caustic effects occur at the cut-off energy of HHG. At this critical point, two branches of electron trajectories, commonly referred to as "short" and "long" trajectories, converge and contribute to the same harmonic energy, resulting in an enhancement in the spectrum magnitude. Recently, the discussions of caustic effects on HHG are extended to solid material, in which the the authors indicate that the Van Hove singularities in energy band structure may lead to the caustic effect. The caustic effects in a one-dimensional periodic potential model have been discussed theoretically. Similar to atomic scenarios, it is found that the caustic enhancement emerges only at a cut-off regime determined by the maximum electron-hole recombination energy. In this work, we employ the two-band density-matrix equations to calculate total high-order harmonic generation in graphene under a infrared laser irradiation. In contrast to atom, in which the caustic effect occur in the cutoff frequency, the simulation results indicate that in graphene, the enhanced harmonic induced by the caustic effect are significantly different from the cutoff frequency and have significant orientation dependence. By solving the saddle-point equations, we uncover that the orientation dependence of the caustic effect is due to the asymmetry of two Dirac points in the first Brillouin region when the laser polarization direction changes. We also investigate the dependence of the caustic effect on the laser intensity and find that the dependence on the laser intensity gradually decreases when the laser polarization direction gradually deviated from the armchair direction of graphene. The investigation of orientation dependence of caustic effect in this paper may have potential application value in determining the orientation of graphene lattice.