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Hopf insulators are topological insulators whose topological behavior arisesfrom the nontrivial mapping from a 3D sphere to a 2D sphere, known as the Hopfmap. The Hopf map, typically encountered in the study of spinor and skyrmionsystems, is classified topologically by an integer invariant called the Hopfindex. Here we show that due to the periodic circulation of light inside eachmicroring, a 2D lattice of microring resonators can emulate an $N$-bandphotonic Hopf insulator with nontrivial Hopf index. In particular, we show bynumerical computation and direct analytical proof that the $N$-band Hopf indexof the microring lattice is identical to its winding number. The result showsthat the Hopf index is an alternative topological invariant for classifying 2Dmicroring photonic lattices and establishes a correspondence between the Hopfinsulator phase and the anomalous Floquet insulator phase of the lattice. Moregenerally, our work shows that 2D microring lattices can provide a versatilenanophotonic platform for studying non-Abelian topological photonic systems.

N-Band Photonic Hopf Insulators Based on 2D Microring Lattices