Percolation effects and sum rules in the optical properties of composities

The optical properties of binary composites may be affected by the presence or absence of infinite connected paths of either component. In a model composite of Drude metal and insulator, we find not only that the Drude peak in the real conductivity, $\mathrm{Re}{\sigma }_{\mathrm{eff}}\left(\omega \right)$, disappears below the metal percolation threshold, but also that the metal plasmon peak in the energy-loss function $-\mathrm{Im}{\epsilon }_{\mathrm{eff}}^{-1\mathrm{}}\left(\omega \right)$ vanishes at the insulator threshold. The integrated strength of the percolation modes is found to vary near the percolation threshold in the limit of small damping, according to the conductivity exponents $t$ and $s$ as defined by Straley. These effects are illustrated by elementary calculations based on the effective medium approximation. Similar phenomena are found in other kinds of composites, and the possibility that these effects may have been observed in polarized transmission experiments is discussed. New sum rules, analogous to those of Bergman, are derived within the quasistatic approximation for $\mathrm{Re}{\sigma }_{\mathrm{eff}}\left(\omega \right)$ and $-\mathrm{Im}{\epsilon }_{\mathrm{eff}}^{-1\mathrm{}}\left(\omega \right)$. These are used to make statements about the center of gravity of the impurity band in these quantities, and the way in which this is affected by percolation phenomena.