Wyckoff positions and the expression of polarization singularities in photonic crystals

We reveal the fundamental relation between linear photonic crystal symmetries and the local polarization states of its Bloch modes, in particular the location and nature of polarization singularities as established by rigorous group theoretic analysis, encompassing the full system symmetry. This is illustrated with the fundamental transverse electric mode of a two-dimensional hexagonal photonic crystal, in the vanishing contrast limit and at the K point. For general Wyckoff positions within the fundamental domain, the transformation of a local polarization state is determined by the nature of the symmetry operations that map to members of its crystallographic orbit. In particular, the site symmetries that correspond to specific Wyckoff positions constrain the local polarization state to singular character--circular, linear or disclination. Moreover, through the application of a local symmetry transformation relation, and the group's character table, the precise natures of the singularities may be determined from self-consistency arguments.