Photonic Equivalent to Hofstadter Butterfly
optical transmission, edge states, butterfly Hofstadter
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 w/w0. In order to identify possible edge states or topological states, we calculate the transmittance spectrum as a function of w/w0 and the parameter f for three cases: first, when layer thicknesses are given by the optical-length ratio Ijdj=l0/4, 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 l=0.5, we reproduce the photon equivalent of the Hofstadter butterfly, which corresponds to a critical state in the metal-insulating transition, that is, for l >0.5, the system is equivalent to a state and when l >0.5, the system is equivalent to an insulating state.