Topology can come in two forms. One is global, describing bulk invariants that span entire energy bands, while the other is local, revealed as striking defects or vortices. In photonics, the former has been linked to quantum-Hall-like phenomena of near field light, while the latter to polarization vortices in the light emitted from a structure in the far field. Until now, global and local topology lived apart, connected only by brute-force numerics with little physical intuition.

In our new work, we build a common language: a symmetry-based Hamiltonian in real space that treats light’s electric field consistently in both near and far fields. This framework bridges global invariants with local defects, and smoothly connects localized (tight binding) and delocalized (photonic) modes. We can also design photonic structures with tailor-made topological properties. Our approach extends well beyond photonics to system such as circuit lattices or mechanical metamaterials that allow for direct design of site-to-site couplings.

Read more on arXiv.