Role of the Transition Dipole Amplitude and Phase on the Generation of Odd and Even High-Order Harmonics in Crystals

Since the first observation of odd and even high-order harmonics generated from ZnO crystals in 2011, the dependence of the harmonic yields on the orientation of the laser polarization with respect to the crystal axis has never been properly interpreted. This failure has been traced to the lack of a correct account of the phase of the transition dipole moment between the valence band and the conduction band. Using a simple one-dimensional two-band model, here we demonstrate that the observed odd harmonics is directly related to the orientation dependence of the magnitude of the transition dipole, while even harmonics is directly related to the phase of the transition dipole. Our result points out the essential role of the complex transition dipole moment in understanding harmonic generation from solids that has long been overlooked so far.

Figure 1

(a) The crystal structure of wurtzite ZnO and the definition of the $XYZ$ axes. The (11-20) plane is indicated by light blue color. (b) The arrangement of atoms in real space on the (11-20) plane, where O and Zn atoms are shown as red and gray balls, respectively. The unit cell and laser polarization are indicated. (c) The corresponding $k_y=0$ plane in the reciprocal space where high symmetry $\mathbf{k}$ points are $\mathrm{\Gamma }=(0.0,0.0,0.0)$, $A=(0.0,0.0,0.5)$, and $M=(0.5,0.0,0.0)$. The crystal axis is defined to be the $Z$ axis, and the orientation angle $\theta$ is between the crystal axis and the axis of the laser polarization which lies on the (11-20) plane. (d) Band structure of ZnO. The valence band and the two lowest conduction bands are highlighted. (e) The magnitude of the transition dipole between each pair of bands considered.

Figure 2

(a) Orientation-dependent HHG spectra, redrawn from the experimental data in Ref. [2]. (b) Spectra calculated from the present theory. (c) Spectra calculated from the present theory but artificially removing the phase of the transition dipole between the valence and conduction bands. The efficiency of the harmonic spectra has been normalized to the 11th-order harmonic. Parameters of the lasers used are given in the text.

Figure 3

(a),(b) Comparison of theoretical and experimental orientation-dependent harmonic strength of the 11th-order and the 12th-order harmonics, respectively. (c) The square of the absolute value of the transition dipole moments $\overline{D}$ versus the orientation angle of the laser polarization direction. (d) The same quantity as in (c) but multiplied by the square of the phase of the transition dipole at $k=0.2$. See text for details.

Figure 4

(a) The dependence of the calculated harmonics on the dephasing time and the comparison with experimental data. The colored lines on the top indicate the different dephasing time and the color for the experimental data. (b) Comparison of the strength of the harmonics versus the dephasing time.