Dirac bound states in the continuum in honeycomb photonic crystal slabs

We investigate bound states in the continuum (BICs) in dielectric photonic crystal slabs, which occur at the double Dirac point with four-fold degeneracy. The underlying structure consists of a hexagonal cluster of equilateral triangular holes with [Formula: see text] symmetry in the unit cell of a honeycomb lattice. The lowest four TE bands intersect at the center of Brillouin zone, exhibiting linear dispersions to form double Dirac cones when the ratio of cluster radius R (the distance from unit cell center to the centroid of each hole) to lattice constant a satisfies the 'degenerate' condition, where the honeycomb lattice can also be described as a triangular lattice due to symmetry-enforced geometric self-duality. In particular, two resonant states at the double Dirac point are identified as symmetry-protected (SP) BICs with extremely high quality factors. They are high-order vortex polarization singularities (V points) characterized by two irreducible representations ([Formula: see text] and [Formula: see text]) in the [Formula: see text] symmetry group, each carrying a topological charge [Formula: see text]. As the ratio R/a deviates from the degenerate condition, the four-fold degeneracy is lifted and a gap is opened between two pairs of doubly degenerate bands. In this situation, Dirac BICs no longer exist and a pair of SP BICs appear on either the upper or lower two bands, with similar polarization patterns and topological charges. By reducing the size of three non-adjacent triangular holes to break [Formula: see text] symmetry, while maintaining [Formula: see text] symmetry, Dirac BICs are transformed to low-order V points with [Formula: see text], accompanied by a group of six circularly polarized states (C points) with [Formula: see text] that surround the V point, which preserve the total topological charge.