Dependence of high-order-harmonic generation on dipole moment in $\mathrm{Si}{\mathrm{O}}_2$ crystals

High-order-harmonic generation in α-quartz $\mathrm{Si}{\mathrm{O}}_2$ is theoretically investigated under a strong laser field by solving the extended semiconductor Bloch equations. The accurate band structures as well as dipole moments between different bands are obtained from state-of-the-art first-principles calculations. We find that the shapes of $k$-space-dependent dipole moments play an important role in harmonic generation. The calculated results show that harmonic conversion efficiency is significantly enhanced and the cutoff energy is distinctly increased when the dipole moments change greatly along a valley in the $k$ direction in the solid. Based on that dependence on the dipole moment, we also show that symmetry groups greatly affect the harmonic spectra from the solid materials. Moreover, a two-color synthesized field is used to achieve a supercontinuum harmonic spectrum near the cutoff region, and isolated attosecond pulses can be obtained directly by filtering out the harmonic radiation. We hope the contribution presented in this work provides a useful reference for future studies on laser-crystal interactions.

Figure 1

(a) The lowest conduction band and highest valence band of $\mathrm{Si}{\mathrm{O}}_2$ with the $p3121$ symmetry group along the Γ-$M$ direction. (b) The $k$-dependent dipole moments calculated by first-principles theory (red solid line) and first-order $k·p$ theory (blue dashed line).

Figure 2

(a) High-harmonic spectra of $\mathrm{Si}{\mathrm{O}}_2$ with the $p3121$ symmetry group from dipole elements calculated by first-principles theory (red solid) and first-order $k·p$ theory (blue dashed). Time-frequency analyses of HHG based on dipole moments from (b) first-order $k·p$ theory and (c) real first-principles calculations.

Figure 3

(a) Electric field profile of a 6 fs/800 nm laser. Populations of conduction band for two cases based on dipole moments from (b) first-order $k·p$ theory and (c) real first-principles calculations.

Figure 4

The crystal structures of (a) $p3121$ and (b) $p3221$ symmetry groups by using real first-principles dipole element in the unit cell. (c) The band structures of conduction band and valence band along the Γ-$M$ direction with $p3121$ (red solid) and $p3221$ (blue) symmetry groups. (d) The $k$-dependent dipole moments of $p3121$ (red) and $p3221$ (blue dashed) $\mathrm{Si}{\mathrm{O}}_2$.

Figure 5

(a) and (b) The experimental laser profiles and corresponding harmonic spectra from Figs. 3(a) and 3(d) in Ref. [15], respectively. The harmonic spectra with symmetry group $p3121$ using first-principles dipole moments from a few-cycle laser field (c) and a half-cycle laser field (d). (e) and (f) are the same as (c) and (d) respectively but for symmetry group $p3221$. Here, the intensity of the harmonic spectrum $I\left(\omega \right)={\omega }^{2}S\left(\omega \right)$ is used for better comparison with experimental data, and $T_2=0.5\mathrm{fs}$.

Figure 6

HHG spectra generated from $p3121$ (red solid) and $p3221$ (blue dashed) $\mathrm{Si}{\mathrm{O}}_2$ in laser fields with peak intensities of (a) $5.0\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, (b) $8.0\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, and (c) $1.0\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^3$.

Figure 7

(a) HHG spectra of $\mathrm{Si}{\mathrm{O}}_2$ with $p3121$ group in two-color synthesized laser field. The laser peak intensities corresponding to the fundamental field and controlling field are $I_1=5.0\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and $I_2=5.0\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, respectively. The wavelength of the fundamental field is 800 nm, and the wavelength of the controlling field changes from 200 to 3200 nm. (b) Attosecond pulses generated by directly filtering out harmonics below 70 eV as shown in (a).

Figure 8

(a) The wavelength-dependent attosecond emission spectrum for $\mathrm{Si}{\mathrm{O}}_2$ with $p3121$ group in a two-color synthesized laser field. The laser parameters are identical to those in Fig. 7 but using a longer 15 fs controlling laser. (b) ${\phi }_2$-dependent harmonic spectra with the wavelength of the controlling laser at 2000 nm. Other laser parameters are identical to those in (a).

Figure 9

Band structure of $p3121$ $\mathrm{Si}{\mathrm{O}}_2$ calculated by high-level HSE06 density functional calculations.