Multiple topological states in photonic crystals with generalized Kekulé modulation

We theoretically proposed a feasible way to design multi-topological states in one two-dimensional photonic crystal (PC). By sliding and rotating the meta-atoms of its unit cell, the topological phase of the PC can be continuously tuned, but without a closing of bandgap; it is similar to the Kekulé modulation in honeycomb lattices. We first show the existence of helical topological edge states in conventional topological insulator, which intersect together and form a Dirac cone in the two-dimensional synthetic space of the PC. Then, in full synthetic space, the PC behaves high-order topological insulator, and the corner state is then constructed in a heterostructure that consists of three different PCs. Furthermore, we demonstrate the photonic crystal with non-uniform modulation can behaves as a splitter and a rainbow trapper. Our proposal may provide interesting insight for designing of multiple topological states, which have potential in photonic on-chip devices.