  Bound states in the continuum (BICs) in a periodic structure sandwiched
between two homogeneous media have interesting properties and useful
applications in photonics. The topological nature of BICs was previously
revealed based on a topological charge related to the far-field polarization
vector of the surrounding resonant states. Recently, it was established that
when a symmetry-protected BIC (with a nonzero topological charge) is destroyed
by a generic symmetry-breaking perturbation, a pair of circularly polarized
resonant states (CPSs) emerge and the net topological charge is conserved. A
periodic structure can also support propagating BICs with a nonzero wavevector.
These BICs are not protected by symmetry in the sense of symmetry mismatch, but
they need symmetry for their robust existence. Based on a highly accurate
computational method for a periodic array of slightly noncircular cylinders, we
show that a propagating BIC is typically destroyed by a structural perturbation
that breaks only the in-plane inversion symmetry, and when this happens, a pair
of CPSs of opposite handedness emerge so that the net topological charge is
conserved. We also study the generation and annihilation of CPSs when a
structural parameter is varied. It is shown that two CPSs with opposite
topological charge and same handedness, connected to two BICs or in a
continuous branch from one BIC, may collapse and become a CPS with a zero
charge. Our study clarifies the important connection between symmetry and
topological charge conservation.
