Higher-Order Fractal Geometry; Application to Multiple Light Scattering

A hierarchy of fractal geometrical exponents $D\left(l\right)\mathrm{}$, based upon $l$-rank orientational fluctuations, is proposed; $D\left(0\right)=D$ is the usual fractal dimension. The first three $D\left(l\right)\mathrm{}$ are calculated via computer simulation for a growth model with a tunable fractal dimension for several values in the range $3>D>\mathrm{}1\mathrm{}$, and for bond percolation. The new exponents are used to discuss fractal structure. The second-order light-scattering intensity is evaluated for the growing fractal clusters, and is shown to be sensitive to the higher-order $D\left(l\right)\mathrm{}$.