Bulk-radiation correspondence in square photonic lattices

We develop an effective model to describe the guided and leaky modes of photonic crystal slabs with ultra long-range couplings and non-local responses. Within this framework, we identify the bulk Berry curvature and construct the corresponding far-field radiation Berry curvature. We apply the model to a square lattice and numerically investigate the bulk-radiation correspondence in the vicinity of the Γ-point. The results show that the comparability between the bulk topology and the far-field topology is not universal; the validity depends on the states. Notably, the correspondence completely breaks down around the far-field singularities, while it can hold in smooth regions under special symmetry conditions.