Order-disorder effects in structure and color relation of photonic-crystal-type nanostructures in butterfly wing scales.

In order to study local and global order in butterfly wing scales possessing structural colors, we have developed a direct space algorithm, based on averaging the local environment of the repetitive units building up the structure. The method provides the statistical distribution of the local environments, including the histogram of the nearest-neighbor distance and the number of nearest neighbors. We have analyzed how the different kinds of randomness present in the direct space structure influence the reciprocal space structure. It was found that the Fourier method is useful in the case of a structure randomly deviating from an ordered lattice. The direct space averaging method remains applicable even for structures lacking long-range order. Based on the first Born approximation, a link is established between the reciprocal space image and the optical reflectance spectrum. Results calculated within this framework agree well with measured reflectance spectra because of the small width and moderate refractive index contrast of butterfly scales. By the analysis of the wing scales of Cyanophrys remus and Albulina metallica butterflies, we tested the methods for structures having long-range order, medium-range order, and short-range order.