Optical Properties of One-Dimensional Quasi-Periodic Crystals. III. Optical Reflectivity Spectrum and Structure Factor of a Generalized Fibonacci Lattice

Optical properties are studied of a generalized semiconductor Fibonacci superlattice generated by the rule S_n+1=S^p _nS^q _n-1 with a pair of positive integers p and q. The initial generations S₁ and S₂ are arbitrary. Based on the multi-product representation of the reflectance function or the structure factor an analysis is presented of the fractal structure of the spectrum. An analytic expression is derived for the peak positions with a new labeling scheme. This scheme enables us to generalize the description of the self-similarity and the nested structure of the spectrum. The self-similarity factor σ(p,q) is obtained as a function of p and q which is rational for p=q-1.