Non-local optical response of nanostructured surfaces from scattering theory

Nanostructured materials are commonly described by local effective permittivities inferred from far-field propagation. However, near-field interactions and large parallel wave vectors probe spatial dispersion and limit the validity of purely local response models. Here, we develop a non-local effective description formulated in terms of the electromagnetic dyadic Green function and wave-vector-dependent reflection operators at planar interfaces. The approach provides a unified description of propagating and evanescent modes and directly connects microscopic scattering processes to observable optical response. As a concrete realisation, we derive the reflection properties of a periodic array of dipolar scatterers on a substrate within a self-consistent scattering formalism. The resulting composite reflection coefficients naturally exhibit collective lattice resonances and surface-modified particle resonances. In the long-wavelength limit, the theory reduces to conventional effective-medium descriptions, while beyond this limit it captures the interplay of particle, lattice, and near-field effects in a controlled manner.