Abstract
Exotic electronic states have been studied with different boundary conditions and on various geometries, especially topological states. However, geometries, including the boundary conditions in photonic crystals, have not drawn sufficient attention and have rarely been investigated. Here, we propose photonic crystals on a Möbius band, which is topologically equivalent to the lattice on cylinder with reflection symmetries. Through this mapping, we can obtain some exotic phenomena in magnetic and all-dielectric photonic systems, such as two photonic carrier channels, one-way edge modes and interface states. Along this line, the photonic states on more complex geometries can be raised and studied based on the topological equivalence principle for exploring the richer edge and interface physics, even topological physics.
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