Hofstadter butterfly in optical multilayers

Abstract

This research focuses on the theoretical examination of the propagation of electromagnetic waves in a multilayer structure in which the refractive index in each medium is modulated by a similar expression of Harper’s potential for the electronic 1D model, or, equivalently, to on-site energies in the Aubry–André model. We applied the transfer matrix approach to different multilayers, with a refractive index modulated by a similar expression of Harper’s potential to obtain the optical transmittance spectra for normal incidence. We calculate the transmittance spectrum as a function of frequency and the Harper model phase for two scenarios in order to search for probable edge states that might arise in this system: first, when the quarter wavelength ratio relates to layer thicknesses through njdj=λ0/4, with nj being the refractive index of the jth layer of the 1D photonic crystal and λ0 is the central wavelength; second, we consider the effects when the thicknesses of the layers are related by d2j=2d2j+1. Besides, to obtain the photonic equivalent for Hofstadter butterfly fractal spectra, we calculate the transmittance spectrum as a function of the periodicity control parameter of the Harper model at the critical point.