Theory and numerical simulation of optical properties of fractal clusters

A theory of linear optical properties of fractal clusters is developed. The theory is based upon exact properties of the dipole polarizability and the assumption of the existence of scaling for the dipole excitations (eigenstates) of the fractal. This assumption is self-consistently validated by the results of the theory and is also confirmed by numerical simulation in the framework of the Monte Carlo method. With use of exact relations and the scaling requirements, it is shown that the fractal absorption and density of eigenstates scale with the same exponent ${\mathit{d}}_{\mathit{o}}$-1. The index ${\mathit{d}}_{\mathit{o}}$, which is called the optical spectral dimension, is a counterpart of the Alexander-Orbach fracton dimension d¯¯. A dispersion law for the fractal eigenstates is found; it is governed by ${\mathit{d}}_{\mathit{o}}$ and the Hausdorff dimension D. By using this law, the condition of the scaling is determined. According to this, the scaling region occupies the center of the absorption band of the fractal. The spectral dimension ${\mathit{d}}_{\mathit{o}}$ is found for three types of fractals studied by use of numerical simulations. The wings of the fractal absorption contour are described by introducing the model of diluted fractals with the use of the binary approximation. The general conclusion is that large fluctuations of local fields dominate in the determining of fractal optical spectra.