Enhancement of high-order harmonic generation in two-dimensional materials by plasmonic fields

High-order harmonic generation (HHG) is responsible for high-energy photons and isolated attosecond pulses. The plasmonic fields excited by nanostructures exhibit both field enhancement and spatial dependence. And two-dimensional (2D) materials exhibit attractive features for coherent HHG. Here, by using ab initio simulations, we explore the possibility of combining 2D materials and plasmonic fields generated by nanoscale dimers to produce efficient HHG. In our scheme, we find the spatial inhomogeneity of plasmonic fields plays a key role for HHG enhancement in a wide range of the high-energy region, which enables the production of isolated attosecond pulses. By the classical trajectory simulation based on the single-electron approximation, we further analyze and confirm the role of the spatial inhomogeneity. We also investigate the HHG enhancement in bilayer 2D materials and analyze the influences of the stacking forms between two layers. According to our simulation results, different stacking forms will induce distinctive interlayer plasmons which cause significant differences in low-energy harmonics. Our exploration supplies a scheme to generate high-energy photons and ultrashort isolated pulses.

Figure 1

(a) Schematic illustration of the composite system composed of 2D materials and nanoscale Au dimers for HHG. The incident laser pulse is polarized along the $z$ axis and the monolayer or bilayer graphene spread in the $x\text{−}y$ plane is placed at the middle of the gap of Au dimers. The inset on the bottom right (the magenta box) shows the structure of graphene and its unit cell used in our simulation (the blue diamond). $E_0$ is the maximal incident electric-field strength. See the text for the detailed parameters. (b) Plasmonic field enhancement in the $x\text{−}z$ plane obtained from FDTD simulation. The red line is the position profile of plasmonic field strength along the axial line of the dimers. (c) The time profiles for the incident pulse (black dashed line) and the induced plasmonic field (red solid line) at the crossing point of the axial line of the dimers and the plane of the graphene ($z=0$). (d) Plasmonic field enhancement in the plane of $z=0$ (the graphene layer) obtained from FDTD simulation.

Figure 2

(a) The HHG spectra from the monolayer graphene ignoring the spatial dependence of the plasmonic field (black line) and considering the spatial inhomogeneity (blue line). The green arrow shows the atomiclike HHG cutoff position. (b) The time profile of the generated ultrashort pulse obtained by performing inverse Fourier transformation of the region between red arrows in (a). (c) Total electric-field distribution along the out-of-plane direction (the $z$ axis) near the gap's center. The monolayer graphene is placed at $z=0$.

Figure 3

(a, b) The time-frequency analyses of HHG spectra from monolayer graphene (a) ignoring the spatial dependence of the plasmonic field [corresponding to the black line in Fig. 2] and (b) considering the spatial inhomogeneity [corresponding to the blue line in Fig. 2]. (c, d) The evolution of the induced electronic density, integrated in the plane of 2D material, (c) ignoring the spatial dependence of the plasmonic field [corresponding to (a)] and (d) considering the spatial inhomogeneity [corresponding to (b)]. The magenta curves in (c) and (d) represent the time profiles of the driving plasmonic field.

Figure 4

The classical trajectory simulation based on the single-electron approximation. (a) Collection of classical trajectories for the homogeneous field case. (b) Collection of classical trajectories for the inhomogeneous field case. The black curves in (a) and (b) represent the electron trajectories emitted at the same moment, but, respectively, for the homogeneous field case and the inhomogeneous field case. The black arrow in (b) shows the recollision energy of the black trajectory.

Figure 5

The HHG spectra from the bilayer graphene considering the spatial inhomogeneity of the plasmonic field. The black line in (a) is the HHG spectrum from the AB-stacking form of bilayer graphene and the blue line in (b) is the HHG spectrum from the AA-stacking form. The insets show the corresponding stacking forms between two layers.

Figure 6

The polarization of induced electronic density in bilayer graphene, averaged in the plane of graphene layers, (a) for AB stacking and (b) for AA stacking. The red arrows in (a) and (b) represent the positions of graphene layers. The black curves in (a) and (b) represent the minus time profiles of the plasmonic field $[-e{E}_{\mathrm{plas}}(0,t)]$. (c) The dipole moment of interlayer plasmons for AB stacking (black line) and AA stacking (blue line), respectively. (d) The HHG spectra from interlayer plasmons for AB-stacking (black line) and AA-stacking (blue line) forms.