Piezo-optic effect of high-harmonic generation in semiconductors

We theoretically investigate the piezo-optic effect of high-harmonic generation (HHG) in shear-strained semiconductors. By focusing on a typical semiconductor, GaAs, we show that there is optical activity, meaning different responses to right-handed and left-handed elliptically polarized electric fields. We also show that this optical activity is more pronounced for higher harmonics whose perturbative order exceeds the band-gap energy. These findings point to a useful pathway for strain engineering of nonlinear optics to control the reciprocity of HHG.

Figure 1

Schematic diagram of the current $\mathit{J}\left(t\right)$ generated by a linearly polarized electric field. The left figure indicates the generated current in the absence of strain, while the right figure indicates the generated current when an external shear strain is applied. In the absence of strain, current is generated only in the $z$ direction parallel to the electric field of the incident light. When shear strain is applied to the system, the direction of the current is distorted and $\mathit{J}\left(t\right)$ has both $z$ and $x$ components.

Figure 2

(a) Schematic diagram of HHG in shear-strained GaAs caused by a linearly polarized electric field ($\eta =0$). The polarization axis of the light rotates as it propagates through the shear-strained material. As a result, the electric field of the emitted light includes components along the major axis (the $z$ direction) and the minor axis (the $x$ direction). (b) Numerical result of HHG spectra in shear-strained GaAs caused by linearly polarized electric field ($\eta =0$) in the case of ${\delta }_{zx}=0.01$. The blue and red lines show the HHG spectra emitted along the major axis ($z$ axis) and minor axis ($x$ axis), respectively.

Figure 3

(a) Schematic diagram of HHG for shear-strained GaAs caused by elliptically polarized electric fields ($\eta \ne 0$). (b) Numerical result for the calculated ellipticity dependence of HHG in the absence of strain. The blue and red lines show the ellipticity dependencies of HHG emitted along to the major axis ($z$ axis) and minor axis ($x$ axis), respectively. The finite shear strain changes these dependencies and is expected to cause optical activity, i.e., different responses to the right-handed ($\eta >0$) and left-handed ($\eta <0$) elliptically polarized electric fields.

Figure 4

Calculated ellipticity dependencies of high-order harmonic intensities focusing on the third [(a1) and (b1)], fifth [(a2) and (b2)], seventh [(a3) and (b3)], and ninth harmonics [(a4) and (b4)]. Figures (a1)–(a4) show the harmonic intensities emitted along the major axis ($z$ axis), while figures (b1)– (b4) show those along the minor axis ($x$ axis). The red, green, and blue lines indicate the ellipticity dependencies for ${\delta }_{zx}=0$, 0.005, and 0.01, respectively. These figures indicate that the right-handed ($\eta >0$) and left-handed ($\eta <0$) elliptically polarized electric fields yield the different HHG spectra. This breakdown of reciprocity becomes significant for the high-order harmonics above the band-gap energy (n $\ge E_g/\hslash \omega =7$).

Figure 5

Ellipticity of emitted fifth- (a) and seventh-order harmonics (b) as a function of the ellipticity of the incident electric field. The red, blue, and green curves denote the ellipticity of the harmonics for the cases of ${\delta }_{zx}=0$, 0.005, and 0.01, respectively.