Fibonacci sequences constructed from high-index-contrast GaAs and
Al2O3 quarter-wavelength layers are used as unit cells in a novel multilayer system. Quasi-periodic heterostructures, formed by concatenating repeated Fibonacci sequences of different order, have properties markedly different from classic Fabry-Perot bilayer systems. We employ the transfer matrix method, including imaginary components of the refractive index, to extract transmission and reflection spectra, and consider their sensitivity to material and geometrical variation. We find that these quasi-periodic heterostructures may have a very high quality factor and deep extinction in reflection. By contrast, bilayer structures of similar dimension are so strongly evanescently damped that the coupling to the cavity is negligible. Due to the coupled geometric resonances in the Fibonacci-based heterostructure, the spectral properties are easily tuned by altering the imaginary component of the refractive index in a single layer. We discuss the viability of possible technological applications.