Topological description for gaps of one-dimensional symmetric all-dielectric photonic crystals

We propose a topological description for gaps of one-dimensional symmetric all-dielectric photonic crystals (PCs). It is shown that, in the propagating direction, the effective electromagnetic parameters of PCs can be derived from one unit cell with mirror symmetry. Besides, at the frequencies of gaps, these symmetric PCs can be described as photonic insulators with effective negative permittivity or negative permeability. Moreover, based on the mapping of Maxwell's equations to the Dirac equation and the band inversion achieved by tuning the material and structural parameters, we demonstrate that the gaps of PCs with effective negative permittivity or negative permeability possess different topological orders. Lastly, we show that a bound state is robust against the disorder under a zero-average-effective-mass condition in a heterostructure made of two PCs with different topological orders.