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Bound states in the continuum (BICs) in a periodic structure sandwichedbetween two homogeneous media have interesting properties and usefulapplications in photonics. The topological nature of BICs was previouslyrevealed based on a topological charge related to the far-field polarizationvector of the surrounding resonant states. Recently, it was established thatwhen a symmetry-protected BIC (with a nonzero topological charge) is destroyedby a generic symmetry-breaking perturbation, a pair of circularly polarizedresonant states (CPSs) emerge and the net topological charge is conserved. Aperiodic structure can also support propagating BICs with a nonzero wavevector.These BICs are not protected by symmetry in the sense of symmetry mismatch, butthey need symmetry for their robust existence. Based on a highly accuratecomputational method for a periodic array of slightly noncircular cylinders, weshow that a propagating BIC is typically destroyed by a structural perturbationthat breaks only the in-plane inversion symmetry, and when this happens, a pairof CPSs of opposite handedness emerge so that the net topological charge isconserved. We also study the generation and annihilation of CPSs when astructural parameter is varied. It is shown that two CPSs with oppositetopological charge and same handedness, connected to two BICs or in acontinuous branch from one BIC, may collapse and become a CPS with a zerocharge. Our study clarifies the important connection between symmetry andtopological charge conservation.

Circularly polarized states and propagating bound states in the continuum in a periodic array of cylinders