High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses

We investigate the high-order-harmonic generation (HHG) from solids by simulating the dynamics of a single active electron in periodic potentials. The corresponding time-dependent Schrödinger equations (TDSEs) are solved numerically by using $B$-spline basis sets in coordinate space. The energy-band structure and wave vectors can be directly retrieved from the eigenfunctions. The harmonic spectra obtained agree well with the results simulated by TDSE in $k$-space using Bloch states and show a two-plateau structure. Both of the cutoff energies of the two plateaus in the harmonic spectrum scale linearly with the field strength. We also study HHG driven by intense few-cycle laser pulses and find that the cutoff energy of the harmonic spectrum is sensitive to the changes of the carrier envelope phase.

Figure 1

Band structures calculated by our proposed method compared to results obtained by Bloch states expansion. (a) Four bands calculated by using the $B$-spline basis. (b) Four bands calculated by using the Bloch state basis. Interband dynamics describes the transition of electrons between different bands, while intraband transitions involve the dynamics of electrons in the same band. (c) Fast Fourier transform (FFT) of the initial state in our calculations. (d) FFT of the 150th eigenstate.

Figure 2

High-order-harmonic spectra of laser-induced currents. The laser intensity is $I=8.087\times {10}^{11}$ W/${\mathrm{cm}}^2$, and the wavelength is $\lambda =3.2 \mu \mathrm{m}$. The upper red dashed line shows the harmonic spectrum calculated by the Bloch-state basis, while the lower solid black line is the harmonic spectrum calculated by using the $B$-spline basis. For clarity, the latter is down-shifted.

Figure 3

(a) Intensity dependence of the cutoff of the first plateau. (b) Intensity dependence of the cutoff of the second plateau. (c) Scaling coefficient of the cutoff of the first plateau as a function of the wavelengths. (d) Scaling coefficient of the cutoff of the second plateau as a function of the wavelengths.

Figure 4

(a) HHG induced by intraband and interband currents in the $B$-spline method. (b) HHG induced by intraband and interband currents in the Houston basis. The laser intensity is fixed at $I=4.5\times {10}^{11}$ W/${\mathrm{cm}}^2$ and the other laser parameters are the same as those in Fig. 2.

Figure 5

CEP effect on HHG from periodic potentials. The laser intensity is $I=5.068\times {10}^{11}$ W/${\mathrm{cm}}^2$ and the laser wavelength is $\lambda =3.2 \mu \mathrm{m}$. The total pulse duration is two cycles. (a) Laser pulses with different CEPs. (b) HHG obtained with different CEPs.

Figure 6

Interband and intraband current contributions with different CEPs: (a) ${\varphi }_{\text{CEP}}=0$, (b) ${\varphi }_{\mathrm{CEP}}=0.25\pi$, (c) ${\varphi }_{\mathrm{CEP}}=0.5\pi$, and (d) ${\varphi }_{\mathrm{CEP}}=0.75\pi$. Other laser parameters are the same as those in Fig. 5.

Figure 7

Time-frequency analysis of HHG in Figs. 6 and 5 and isolated attosecond pulse generation. (a)–(d) Time-frequency analysis of HHG from interband and intraband contributions with ${\varphi }_{\text{CEP}}=0$ and $0.5\pi$. Also shown is time-frequency analysis of HHG with full contributions with (e) ${\varphi }_{\text{CEP}}=0$ and (f) ${\varphi }_{\text{CEP}}=0.5\pi$. (g) Attosecond pulse by synthesis of harmonics with order 60–80 with ${\varphi }_{\text{CEP}}=0$. (h) Attosecond pulse by synthesis of harmonics with order 40–60 with ${\varphi }_{\text{CEP}}=0.5\pi$.

Figure 8

Laser chirp effect on HHG. (a) Laser fields with different chirps. (b) HHG with different chirps. (c) HHG induced by interband and intraband currents with $\alpha =0$. (d) HHG induced by interband and intraband currents with $\alpha =1.2$.

Figure 9

Time-frequency analysis of inter- and intraband contributions in Fig. 8: (a) $\alpha =0$, time-frequency analysis of the first plateau; (b) $\alpha =1.2$, time-frequency analysis of the first plateau; (c) $\alpha =0$, time-frequency analysis of interband contributions; (d) $\alpha =1.2$, time-frequency analysis of interband contributions; (e) $\alpha =0$, time-frequency analysis of intraband contributions; and (f) $\alpha =1.2$, time-frequency analysis of intraband contributions.

Figure 10

Time-frequency analysis of HHG in Fig. 8 and isolated attosecond pulse generation: (a) $\alpha =0$ and (b) $\alpha =1.2$. (c) Attosecond pulse by synthesis of harmonics with order 85–120 with $\alpha =1.2$.