Constructing helical edge states with two-dimensional C6 and C6 -like symmetric photonic crystals

In recent years, helical edge states (HESs) in two-dimensional (2D) topological pseudospin photonic crystals (PhCs) have attracted more and more attention due to their pseudospin-locked unidirectional propagation of electromagnetic (EM) waves. Most of the studies are based on a kind of triangular lattice PhCs with the primitive cell containing six identical scatterers with C₆ symmetry. By shrinking or expanding the six-scatterer structure in the primitive cell, topological pseudospin PhCs are formed and HESs can be constructed along the zigzag interface of two PhCs with distinct topological pseudospins. In this paper, based on several 2D C₆ symmetric dielectric PhCs with the primitive cell containing different shapes of regular polygon prisms, we design C₆-like symmetric PhCs by partially replacing regular polygon prisms in the primitive cell of C₆ symmetric PhCs. Numerical calculations show all these different PhCs can possess similar band structures by tuning sizes of the prisms even in the C₆-like symmetric PhCs in which the C₆ symmetry is lacking. Then, by splicing these C₆ and C₆-like symmetric PhCs along their zigzag interfaces, we construct the similar HESs. Through calculating the energy propagation of these HESs, we demonstrate their same unidirectional propagation properties and the similar immunity to corners and defects along the interface. Our results indicate that the formation and the property of HESs in 2D topological pseudospin PhCs are only determined by the topological band structures and the topological pseudospins of PhCs on both sides of the interface, rather than the shapes of the scatterers in PhCs in real space. Our study provides further understanding on the formation of HESs and paves the way for control of topological edge states in practice.