Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields

We theoretically study high-order harmonic generation (HHG) from solid-phase systems in spatially inhomogeneous strong laser fields originated by resonant plasmons within a metallic nanostructure. The intensity of the second plateau in HHG may be enhanced by two to three orders and be comparable with the intensity of the first plateau. This is due to bigger transition probabilities to higher conduction bands. It provides us with a practical way to increase the conversion efficiency of HHG with laser intensity below the damage threshold. It presents a promising way to triple the range of HHG spectra in experimental measurements. It also allows us to generate intense isolated attosecond pulses from solids driven by few-cycle laser fields.

Figure 1

The nanostructure of bow-tie element. Schematic illustration of laser field enhancement using a nanostructure of bow-tie elements growing on the substrate of semiconductor crystals. The incident direction of the laser pulses is vertical to the substrate.

Figure 2

Comparison of HHG spectra between homogeneous and inhomogeneous fields. HHG spectra in the case of the homogeneous field (black solid curve), the inhomogeneous fields with $\epsilon$ = 0.0002 (violet dashed curve), and $\epsilon$ = 0.0005 (red dotted curve). (a) HHG with full range. (b) The first HHG plateau with $\epsilon$ = 0.0005. (c) The second HHG plateau with $\epsilon$ = 0.0005. The intensity, wavelength, and duration of the driving laser pulses are $8.77\times {10}^{11}$ W/${\mathrm{cm}}^2$, 3200 nm, and six optical cycles (o.c.), respectively.

Figure 3

The distinction of contributions within the conduction bands. HHG induced by full (black solid curve), conduction bands C1 (red dotted curve), and C2 plus C3 (blue dashed curve) contributions in an inhomogeneous field with $\epsilon$ = 0.0005. The inset shows the HHG induced by full conduction bands C1 and C2 plus C3 contributions in the homogeneous field. The laser pulse parameters are the same as those in Fig. 2.

Figure 4

Time-frequency analysis of HHG. Time-frequency analysis of HHG in a homogeneous field in panels (a) and (c) and an inhomogeneous field in panels (b) and (d). Panels (a) and (b) are time-frequency analyses with full contribution. Because the two HHG plateaus are mainly determined by interband contribution, we show time-frequency analysis of interband contribution in homogeneous field in panel (c) and inhomogeneous field in panel (d). The color is plotted on the logarithmic scale.

Figure 5

Energy band structures and populations of conduction bands. Band structures calculated by Bloch states expansion are compared with the result obtained by the B-spline basis in panels (a) and (b), respectively. The electron populations as a function of time for the C1 band and C2 plus C3 band in homogeneous field (red curve) are illustrated in panels (c) and (d), respectively. The olive curves present results in the inhomogeneous field.

Figure 6

Transitional probabilities of conduction bands. Transitional probabilities calculated by $|\langle {\varphi }_{n_0}|\widehat{x}|{\varphi }_n{\rangle |}^2$ (black solid curve) for homogeneous field and an additional term $|\langle {\varphi }_{n_0}|x^2|{\varphi }_n{\rangle |}^2$ (orange dashed curve) for spatial inhomogeneous field.

Figure 7

Synthesis of attosecond pulses. (a) HHG in few-cycle laser fields. The inhomogeneity $\left(\epsilon \right)$ of laser fields is 0.0007. We choose two different CEP values, 0 and $0.5\pi$. The violet dotted curve in panel (a) shows the HHG in the case of homogeneous field with CEP = $0.5\pi$. The field parameters are the same as those in Fig. 2 except for the pulse duration. Attosecond pulses generated by synthesis of harmonics with order 50–100 in the case of the inhomogeneous field with $\epsilon$ = 0.0007, CEP = $0.5\pi$ and in the case of the homogeneous field with CEP = $0.5\pi$ are presented in panels (b) and (c), respectively. The intensity has been normalized.