Ellipticity of the harmonic emission from graphene irradiated by a linearly polarized laser

We investigate the high-order harmonic generation in graphene irradiated by a linearly polarized intense laser, addressing the ellipticity or polarization properties of the harmonics. We exploit time-dependent density functional theory to calculate the harmonic spectra for a laser wavelength of 4770 nm and an intensity of 1.7 $\mathrm{TW}/{\mathrm{cm}}^2$, and our numerical results can qualitatively reproduce recent experimental data. Our simulations also reveal that the harmonic ellipticity depends on both the harmonic order and the orientation angle between the graphene symmetric axis and the laser polarization direction. It can reach 0.68 for the ninth-order harmonic at the orientation angle of ${20}^{\circ }$. To understand the mechanism of the high ellipticity, we develop a two-band model based on the tight-binding approximation. We may explain the ellipticity of high-order harmonic generation by investigating the transition dipole moments in the two-band model. Our theory further predicts a sensitive dependence of the harmonic ellipticity on the laser intensity for various laser wavelengths.

Figure 1

(a) Hexagonal lattice structure of two-dimensional graphene. Each primitive cell contains two atoms labeled A and B, and ${\mathbf{d}}_i$, $i=1,2,3$, are the nearest-neighbor vectors. $\theta$ is the orientation angle between the laser polarization direction $\vec{e}$ and the $x$ axis. (b) Sketch of a harmonic with ellipticity. $|F_{\parallel }|$, $|F_{\perp }|$, and $|F_{\mathrm{total}}|$ are the amplitudes of the parallel, perpendicular, and total harmonics, respectively

Figure 2

(a) Normalized coefficients $C_i$ used to fit the spectrum of the laser in Ref. [29]. (b) Waveforms of the vector potentials ${\mathbf{A}}_G\left(t\right)$ (black line) and $\mathbf{A}\left(t\right)$ (red line). (c) Currents with (red line) and without (black line) filters. (d–f) We compare our simulation results with the experimental results in Ref. [29].

Figure 3

(a) Total harmonic spectra calculated by Eq. (6). For comparison, the gray dashed line presents the intensity dependence of the harmonics from the perturbation theory. (b) Ellipticity of emitted harmonics for typical orientation angles. (c) Ellipticity of emitted harmonics as a function of the ratio $r=|F_{\perp }|/|F_{\parallel }|$ and phase difference $\delta$.

Figure 4

(a) The comparison between simulated spectra (TDDFT and the tight-binding (TB) approximation) and experimental results of Ref. [29] at $\theta =0^{\circ }$. (b) The ellipticity of the ninth-order harmonic calculated by TDDFT and the tight-binding approximation with respect to the orientation angles.

Figure 5

Imaginary part of (a) parallel and (b) perpendicular transition dipole moments ($p_{\mathbf{k},\parallel }^{cv}$ and $p_{\mathbf{k},\perp }^{cv}$) for $\theta =0^{\circ }$. In (a) and (b), the dark yellow arcs indicate the resonance regions where the energy difference between the valence and conduction bands is equal to the ninth-order harmonic. The typical $\mathbf{k}$ points that correspond to large parallel transition dipole moments are indicated by black dots and labeled from 1 to 8. (c) Parallel harmonic spectrum calculated by Eq. (22) with $\mathbf{k}$ points of the resonance regions. The perpendicular spectrum, whose yield is close to zero, is not displayed. (d) Fourier transform coefficients calculated by Eq. (23) for eight typical $\mathbf{k}$ points. The black [coordinates ($-0.8,-1.8$)] and red [coordinates (0,0)] dots are average parallel and perpendicular currents, respectively.

Figure 6

Same as Fig. 5, but for $\theta ={30}^{\circ }$.

Figure 7

Same as Fig. 5, but for $\theta ={20}^{\circ }$.

Figure 8

Ellipticity calculated by the two-band model as a function of the laser intensity for (a) the 3rd-order harmonic of wavelength $\lambda =1064$ nm, (b) the 9th-order harmonic of wavelength $\lambda =4770$ nm, and (c) the 19th-order harmonic of wavelength $\lambda =10600$ nm. The colored lines are the ellipticity as a function of ratio $r$ for several typical phase differences $\delta$.

Figure 9

Parallel transition dipole moments of (a) valance and (b) conduction bands