Macroscopic effect of plasmon-driven high-order-harmonic generation

We present a numerical method to calculate the macroscopic harmonic spectrum generated from the gas-exposed nanostructure. This method includes the propagation of plasmonic and harmonic fields in the macroscopic medium as well as the response of the single atom exposed to plasmonic field. Based on the simulation, we demonstrate that the macroscopic harmonic yields drop dramatically in the high-energy region. This result well interprets the disagreement in the cutoff between the single-atom prediction and the experimental detection. Moreover, we also show that the harmonic cutoff difference induced by a $\pi$ shift in carrier-envelope phase (CEP) of laser pulses depends sensitively on the spatial position. However, when the collective effect of plasmon-driven high-order-harmonic generation is considered, this cutoff difference is eliminated.

Figure 1

Geometry of a single bow-tie nanostructure. The incident pulse linearly polarizes along the $x$ direction.

Figure 2

(a) Enhancement factor of field intensity as a function of the bow-tie height and angle. (b)–(d) Spatial profile of plasmonic field in the plane of $z=15$ nm, $y=0$ nm, and $x=0$ nm at $t=17.88$ fs. (e) $E_x\left(t\right)$ at the point (0, 0, 15). Here, the geometrical parameters are $h=180$ nm, $\theta ={80}^{\circ }, t=30$ nm, and $d=20$ nm.

Figure 3

(a) Single-atom harmonic spectrum in the spatially homogeneous field. (b) Single-atom harmonic spectrum in the spatial inhomogeneity field. (c) Plasmon-driven macroscopic harmonic spectrum. In Figs. 3 and 3, the spatial point is $x_0=4$ nm, $y_0=0$ nm, and $z_0=15$ nm.

Figure 4

(a) Plasmon-driven single-atom harmonic spectra at the spatial points (0, 0, 15), (4, 0, 15), and (8, 0, 15). (b) and (c) Same as (a), but at the spatial points (0, 0, 15), (0, 4, 15) (0, 8, 15) and (0, 0, 0), (0, 0, 15), (0, 0, 30). For easier comparison, the harmonic intensity is multiplied by a factor ${10}^{(10n-10)}$ with $n$ changes from 1 to 3 from bottom to top in Figs. 4–4.

Figure 5

(a)–(c) Plasmon-driven single-atom harmonic spectrum, time-frequency image and classical electron trajectories (black lines) of HHG at the spatial point (5, 0, 15) with the CEP of 0. (d)–(f). Same as (a)–(c), but with the CEP of $\pi$. The background color in Figs. 5 and 5 represents the distribution of the the plasmonic field amplitude $E_x$ as a function of $x$ and $t$.

Figure 6

Same as Fig. 5, but at the spatial point ($-5$, 0, 15).

Figure 7

Same as Fig. 5, but at the spatial point (0, 0, 15).

Figure 8

Macroscopic harmonic spectra with the CEP of 0 and $\pi$. The harmonic intensity with the CEP of $\pi$ is multiplied by a factor ${10}^3$ for easier comparison.