Generation of Hofstadter's butterfly spectrum using circular arrays of microring resonators.

Hofstadter's butterfly spectrum, which characterizes the energy bands of electrons in a 2D lattice under a perpendicular magnetic field, has been emulated and experimentally characterized in periodic bandgap structures at microwave and acoustic frequencies. However, measurement of the complete spectrum at optical frequencies has yet to be demonstrated. Here, we propose a simple topological photonic structure based on a circular array of microrings with periodic resonant frequency detunings that can be implemented on an integrated optics platform. We show that this ring-of-rings structure exactly emulates the Harper equation and propose an experimental approach for measuring Hofstadter's butterfly spectrum at optical frequencies.