Origins and conservation of topological polarization defects in resonant photonic-crystal diffraction.

We present a continuative definition of topological charge to depict the polarization defects on any resonant diffraction orders in photonic crystal slab regardless they are radiative or evanescent. By using such a generalized definition, we investigate the origins and conservation of polarization defects across the whole Brillouin zone. We found that the mode crossings due to Brillouin zone folding contribute to the emergence of polarization defects in the entire Brillouin zone. These polarization defects eventually originate from the spontaneous symmetry breaking of line degeneracies fixed at Brillouin zone center or edges, or inter-band coupling caused by accidental Bloch band crossings. Unlike Bloch states, the polarization defects live and evolve in an unbound momentum space, obeying a local conservation law as a direct consequence of Stokes' theorem, but the total charge number is countless.