Caustic effects on high-order harmonic generation in graphene

We employ two-band density-matrix equations and time-dependent density-functional theory to calculate high-order harmonic generation (HHG) in graphene under femtosecond laser irradiation. Our investigation uncovers a striking harmonic enhancement structure (HES) in the HHG spectrum. We find that the HES is associated with the bunching of multiple interband electron-hole recombination trajectories, in analogy to the focusing behavior of the light rays known as caustics. In contrast to the atom situation, where the caustic effects are confined to a narrow energy regime around the HHG cutoff energy, for graphene, the caustic effects can 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 catastrophe theory.

Figure 1

The harmonic spectra calculated by (a) TBDMEs and (b) TDDFT with a laser intensity of $8\times {10}^{11}$ W/${\mathrm{cm}}^2$ and a wavelength of 5500 nm. In (a) and (b), the obvious HES are marked by blue dashed rectangles, and the HES peak corresponding to the local harmonic maximum intensity is indicated by red arrows at ${\omega }^{*}=0.34$ in (a) and ${\omega }^{*}=0.31$ in (b). The upper and lower boundaries of the blue dashed rectangle marked by $I_{\text{peak}}$ and $I_0$ correspond to the harmonic energy of the HES peak and the background harmonic energy away from the HES, respectively. The vertical dotted lines label the energy difference between the $c$ and $v$ bands at the Van Hove singularities ($M$ and $\mathrm{\Gamma }$ points of graphene).

Figure 2

For the harmonic frequency of $\omega =0.22$ a.u. [(a) and (c)] and $\omega =0.33$ a.u. [(b) and (d)], the red dots in (a) and (b) mark the saddle-point momenta ${\mathbf{K}}_0^{st}$ calculated using Eqs. (2). The harmonic intensities $H({\mathbf{K}}_0,\omega )$ calculated by TBDMEs are shown in (c) and (d). The different subpatterns marked by A, B, and C correspond to recombination times in the different half cycles of the short pulse as shown in Figs. 3 and 3. (For details, see Sec. IV of the Supplemental Material [37].)

Figure 3

(a), (b) The time-frequency distributions corresponding to the harmonic spectra shown in Fig. 1. In (a) and (b), the dark cyan points represent the recombination trajectories calculated by Eqs. (2). The subpatterns marked by A, B, and C correspond to the recombination times in the different half cycles of the short laser pulse. (c) The recombination energy $\omega$ as a function of the saddle-point momenta ${\mathbf{K}}_0^{st}=({\text{K}}_{0x}^{st},{\text{K}}_{0y}^{st})$ for branch A. The black curves show the relationship between the recombination energy $\omega$ and ${\text{K}}_{0x}^{st}$ for specific ${\text{K}}_{0y}^{st}$. In (a)–(c), the blue dots indicate 1D caustic trajectories with $\partial \omega /\partial {\text{K}}_{0x}^{st}=0$, and the red points ${\omega }^{*}$ indicate 2D caustic trajectories with $\partial \omega /\partial {\text{K}}_{0x}^{st}=\partial \omega /\partial {\text{K}}_{0y}^{st}=0$ corresponding to branch A.

Figure 4

(a) The gray area corresponds to the region where the interband currents dominate the harmonic generation according to TBDMEs (for details, see Sec. V of the Supplemental Material [37]). The HES regions are indicated by bars, which represent the energy intervals associated with the harmonic enhancement structure, corresponding to the energy regions of the blue dashed rectangles shown in Fig. 1. The pink area indicates the energy regions of 1D caustic singularities where $\partial \omega /\partial {\text{K}}_{0x}^{st}=0$. (b) The black squares and blue stars show the energy of HES peaks as a function of $A_0$, calculated by TBDMEs and TDDFT. At specific $A_0=0.58$ a.u., they correspond to local harmonic peaks denoted by the red arrows in Fig. 1. The red lines are the predictions of caustic equation (5). (c) The black square and blue star show the amplification of harmonic intensity at the HES peaks, calculated by $I_{\text{peak}}/I_0$. The red line is the prediction of the catastrophe theory.