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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.

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., Comment: 4 pages, 3 Postscript figures

Recently, self-sustained oscillations in complex networks consisting of nonoscillatory nodes (network oscillators) have attracted great interest in diverse natural and social fields. Due to complexity of network behaviors, little is known so far about the basic structures and fundamental rules underlying the oscillations, not to mention the principles of how to control it. In this article we propose a common design principle for oscillations; predict novel and universal Branched Circle (BC) structures of oscillatory networks based on this principle; and suggest an operable Complexity Reduction Method to reveal the BC structures. These ideas are applied to excitable cell networks (including neural cell networks), and genomic regulatory networks. Universal BC structures are identified clearly in these two considerably different systems. These BC structures reveal for the first time both oscillation sources and wave propagation pathways of complex networks, and guide us to control the oscillations with surprisingly high efficiency., Comment: 37 pages, 12 figures