Nonlocal Optical Response and Surface Susceptibilities: A Systematic Derivation via Spatial Moment Expansion

TL;DR

This paper develops a systematic theory linking nonlocal optical response kernels to effective surface susceptibilities, incorporating curvature effects and generalizing classical boundary conditions for various interface geometries.

Contribution

It introduces a comprehensive framework connecting nonlocal response kernels to surface susceptibilities, including curvature corrections, for arbitrary interface shapes.

Findings

Surface susceptibility $^s$ generalizes Feibelman $d$-parameters to curved interfaces.

Explicit curvature corrections proportional to mean and Gaussian curvatures are derived.

The formalism is validated on analytical models for various geometries and kernel choices.

Abstract

We present a systematic theory connecting the nonlocal response kernel of a homogeneous medium to its effective surface susceptibilities for an arbitrary curved interface. Starting from the most general tensorial nonlocal constitutive relation and combining a spatial moment expansion with a distributional thin-layer limit, we show that the full complexity of the interfacial response condenses, at leading order, into a single scalar: the surface susceptibility $\chi^s$, equal for the tangential and normal components of the electric field. These quantities provide a constructive generalization of the Feibelman $d$-parameters to interfaces of arbitrary curvature, and the curvature corrections, proportional to the geometric invariants $H$ (mean curvature) and $K$ (Gaussian curvature), are derived explicitly. The formalism is illustrated on a comprehensive set of analytically tractable cases (planar, spherical, cylindrical, and ellipsoidal interfaces) for several kernel choices (Gaussian, Yukawa, tensorial Lorentz). Generalized Maxwell boundary conditions are established and compared with the classical Fresnel results.