Equivalente fotônico a borboleta de Hofstadter

Abstract

In the present work, we do the theoretical study of the propagation of electromagnetic waves in a multilayer system, whose refractive index of each layer is modulated by the function that describes the potential of the one-dimensional Harper model. We apply the transfer matrix method to obtain the transmittance spectra as a function of the reduced frequency ω/ω0. The application of the topology in the photonics is arousing great interest in the physics of condensed matter, together with the discovery of new phases of the matter called topological insulators, which are characterized by the emergence of phenomena of topologically protected edges. In order to identify possible edge states or topological states, we calculate the transmittance spectrum as a function of reduced frequency ω/ω0 and phase φ for three cases: first, when layer thicknesses are given by the optical-length ratio Ijdj = λ0/4, where Ij is the refractive index of the layer j; the second, when all thicknesses of the layers are equal, dj = d; and finally when the thicknesses of the layers are related by d2j = 2d2j+1. In addition, we obtained the transmittance spectrum as a function of the periodicity control parameter b. At the critical point λ = 0.5, we reproduce the photonic equivalent of the Hofstadter butterfly, which corresponds to a critical state in the metal-isolante transition, that is, for λ < 0.5, the system is equivalent to a state and when λ > 0.5, the system is equivalent to an insulating state.