Caustic effects on the high-order harmonic generation in graphene

TL;DR

This paper investigates how caustic effects influence high-order harmonic generation in graphene, revealing broad energy enhancement regions due to trajectory convergence, with implications for other 2D and bulk materials.

Contribution

It introduces a novel analysis of caustic effects in HOHG in graphene, highlighting broad energy enhancement regions and significant trajectory convergence effects.

Findings

Identification of a harmonic enhancement structure (HES) in graphene.

Broad energy region of HOHG enhancement due to caustic effects.

Enhancement magnitude scales as approximately N^{2/3} with harmonic order.

Abstract

We employ the two-band density-matrix equations and time-dependent density functional theory to calculate high-order harmonic generation (HOHG) in graphene under a femtosecond laser irradiation. Our investigation uncovers a striking harmonic enhancement structure (HES) within a specific energy range of the HOHG spectrum. In this regime, we find the convergence of multiple interband electron-hole recombination trajectories, leading to the zero determinant of the Hessian matrix of the semiclassical action. This trajectory convergence exhibits the characteristics akin to the focusing behavior of light rays, commonly known as caustic effects. In contrast to atom situation, where caustic effects are confined to a narrow energy regime around the HOHG cut-off energy and the enhancement due to caustic trajectory convergence is less apparent, the two-dimensional nature of graphene results in a broad energy region for HOHG enhancement that the caustic trajectories can even dominate the entire interband harmonic generation regime. The magnitude of enhancement is significant and can be estimated to be on the order of $\sim N^{2/3}$, with $N$ representing the harmonic order, according to the catastrophe theory. These mechanisms have broad applicability and hold significant implications for other two-dimensional materials, as well as bulk materials, providing crucial insights into the understanding of HOHG phenomena in diverse material systems.