Optical Properties of One-Dimensional Inflation Lattice. I. Fractal Structure of Optical Spectrum

Optical properties are studied of an inflation lattice generated by the inflation rule α→{α 1 α 2 …α n } and β→{β 1 β 2 …β m }. A matrix recurrence equation is derived for the optical structure factors which is useful in the numerical calculation. Based on the multi-product representation, an analysis is presented of the fractal structure of the spectrum. The assumption of successive enhancement yields an analytic expression of the peak frequencies. The new labeling scheme of the peak frequencies enables us to describe the fractal structure of the spectrum in terms of the self-similarity factor σ.