A mathematical and numerical framework for near-field optics

TL;DR

This paper develops a mathematical and numerical method to achieve super-resolution in reconstructing small surface perturbations by analyzing plasmonic resonance shifts, using conformal mapping and Fourier analysis.

Contribution

It introduces a novel framework combining conformal mapping and Fourier analysis for super-resolved surface perturbation reconstruction in near-field optics.

Findings

Successful numerical reconstruction of surface perturbations.

Demonstrates the method's viability and limitations.

Provides a direct approach for inverse surface problems.

Abstract

This paper is concerned with the inverse problem of reconstructing small and local perturbations of a planar surface using the field interaction between a known plasmonic particle and the planar surface. The aim is to perform a super-resolved reconstruction of these perturbations from shifts in the plasmonic frequencies of the particle-surface system. In order to analyze the interaction between the plasmonic particle and the planar surface, a well chosen conformal mapping, which transforms the particle-surface system into a coated structure, is used. Then the even Fourier coefficients of the transformed domain are related to the shifts in the plasmonic resonances of the particle-surface system. A direct reconstruction of the perturbations of the planar surface is proposed. Its viability and limitations are documented by numerical examples.