Frequency dependence of the photoelastic coefficients of silicon

We have measured the frequency dependence of the individual photoelastic coefficients of silicon using a simplified acousto-optic technique. We define the photoelastic tensor $q$ by $\Delta {\epsilon }_{\mathrm{ij}}=q_{\mathrm{ijkl}}{e}_{\mathrm{kl}}$, where $e$ is the particle displacement gradient tensor and $\Delta \epsilon$ is the strain-induced change in the optical dielectric tensor. We find $q_{1111}\left(3.39\mathrm{} \mu m\right)=+13.0\mathrm{}\pm 0.6$, $q_{1111}\left(1.15\mathrm{} \mu m\right)=+15.7\mathrm{}\pm 1.1$, $q_{1122}\left(3.39\mathrm{} \mu m\right)=-2.41\mathrm{}\pm 0.1$, and $q_{1122}\left(1.15\mathrm{} \mu m\right)=-1.45\mathrm{}\pm 0.1$. The extrapolated long-wavelength limit of the average photoelastic coefficient agrees well with our previous estimate for the frequency-independent Phillips—Van Vechten model. However, within a proper frequency-dependent Penn model we have shown that the "oscillator strength" does not vary as $r^{-3\mathrm{}}$. Further we have demonstrated that the dispersion energy $E_d$ in the Wemple-DiDomenico model is proportional to $r^{-1.9\mathrm{}\pm 0.8}$ and not volume independent as would be expected from the model. It is concluded that whereas a simple single-gap model works well to describe the low-frequency dispersion in the dielectric constant of silicon, it is incapable of describing the dispersion in the photoelastic tensor.