Surface scattering amplitude for a spatially dispersive model dielectric

Recently, it was shown [H.-Y. Deng and E. A. Muljarov, Phys. Rev. B 106, 195301 (2022)] that the so-called additional boundary conditions (ABCs) had significant issues when describing the surface properties of spatially dispersive media and an alternative ABC-free theory was developed introducing the surface scattering amplitude (SSA), which determines how polarization waves are scattered by a surface. Here we analytically calculate the SSA for a spatially dispersive model dielectric using wave-scattering theory and discuss some emerging generic properties independent of the model. The model consists of a lattice of interacting harmonic oscillators which had historically been used to mimic local excitons. It was shown that for short-range interactions the SSA is a constant independent of where the polarization waves were originated, which is not the case for long-range interaction. The calculated optical properties of a slab verify the general ABC-free theory