A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators

TL;DR

This paper develops a rigorous mathematical framework to explain super-resolution phenomena using a system of Helmholtz resonators, analyzing resonances and Green functions to demonstrate super-resolution in deterministic media.

Contribution

It introduces a systematic method for calculating resonant frequencies and Green functions in complex media, providing the first mathematical theory of super-resolution in deterministic settings.

Findings

Resonances are rigorously derived using boundary integral equations.

Green function asymptotics explain super-resolution capabilities.

Methodology applies to various space and frequency regimes.

Abstract

A rigorous mathematical theory is developed to explain the super-resolution phenomenon observed in the experiment by F.Lemoult, M.Fink and G.Lerosey (Acoustic resonators for far-field control of sound on a subwavelength scale, Phys. Rev. Lett., 107 (2011)). A key ingredient is the calculation of the resonances and the Green function in the half space with the presence of a system of Helmholtz resonators in the quasi-stationary regime. By using boundary integral equations and generalized Rouche's theorem, the existence and the leading asymptotic of the resonances are rigorously derived. The integral equation formulation also yields the leading order terms in the asymptotics of the Green function. The methodology developed in the paper provides an elegant and systematic way for calculating resonant frequencies for Helmholtz resonators in assorted space settings as well as in various frequency regimes. By using the asymptotics of the Green function, the analysis of the imaging functional of the time-reversal wave fields becomes possible, which clearly demonstrates the super-resolution property. The result provides the first mathematical theory of super-resolution in the context of a deterministic medium and sheds light to the mechanism of super-resolution and super-focusing for waves in deterministic complex media.