  Bound states in the continuum (BICs) realized in two-dimensional (2D)
photonic crystal slabs (PhCSs) have attracted considerable attentions, owing to
the advantages in the fabrications and on-chip applications. The polarization
vortex centered at BICs in the momentum space displayed by the linear
polarization vectors of far-field radiation is a striking property of BICs.
Here, by constructing BICs at K point in photonic graphenes, we theoretically
demonstrate that the far-field polarization states close to BICs can exhibit
remarkable polarization diversity. Interestingly, C-points (circular
polarizations) close to BIC at K point are observed. Thus, we propose that,
along a closed loop enclosing BICs in the momentum space, the trajectory of the
far-field polarization states on the shell of the Poincare sphere could provide
a general and straightforward means to characterize the topological natures of
BICs. The topological charge carried by the BIC can be defined by the half of
the winding number of the trajectory around the S3 axis, which is an integer
and half integer for the non-degenerate and double-degenerate BICs at K point,
respectively. Our findings could open a gateway towards the applications of
BICs in lasers with full polarizations and the generation and manipulation of
vector beams of light.
