Optical and Elastic Constants of Crystalline Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ by Brillouin Light Scattering Spectroscopy
Room-temperature optical and elastic constants of crystalline Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ were determined using data extracted from Brillouin light scattering spectra. Refractive indices at a wavelength of 532 nm obtained from bulk peak linewidth versus frequency shift ratios range from $1.8 \leq n \leq 2.0$ for directions close to the crystallographic $c$-axis, while those determined from an expression relating the refractive index to elastic constants and Rayleigh surface wave velocity in the [010] direction on the (001) plane were found to be higher by $\sim20$%. Extinction coefficients at 532 nm for directions in proximity to the $c$-axis ranged from $0.03 \leq \kappa \leq 0.1$ and are consistent with values determined using optical interference but are several times smaller than those obtained in reflectance experiments. These discrepancies in index and extinction coefficient, also implicit to the optical penetration depth and dielectric function derived from the optical constants, are attributed to a reduction in surface mode frequencies and increased optical absorption due to the presence of surface roughness. A roughness-induced reduction in surface mode frequencies also appears to account for why the value of elastic constant $C_{44}=20\pm2$ GPa obtained in the present work via measurement of bulk phonon modes is 25% larger than that previously reported.