Evolution of topological polarization singularities in honeycomb photonic crystal slabs

We study the evolution of polarization singularities in dielectric photonic crystal slabs, consisting of six rectangular holes arranged as a ring-shaped cluster with C₆ symmetry in the unit cell of a honeycomb lattice. Symmetry-protected (SP) bound states in the continuum (BICs) with extremely large quality factors occur at the center of the Brillouin zone (Γ point) for the third TE-like and the first TM-like modes, which are recognized as high-order polarization singularities with topological charge q = -2. In particular, the SP BICs are recognized as vortex singularities (V points) in the far-field polarization states and identified as hexapole modes in the near-field electromagnetic distribution. By rotating three non-adjacent rectangular holes about each of their centroids to break C₂ symmetry, while preserving C₃ symmetry, the high-order SP BIC is transformed into a low-order BIC with q = +1, accompanied by a group of six circularly polarized states (C points) with q = -1/2, symmetrically located around the Γ point. The evolution of polarization singularities is further manifested in the translation and rotation of the C points in the momentum space when the degree of symmetry breaking is increased.