A Mathematical Theory of Super-Resolution by Using a System of Sub-Wavelength Helmholtz Resonators

Abstract

A rigorous mathematical theory is developed to explain the super-resolution phenomenon observed in the experiment (Lemoult et al., Phys Rev Lett 107:064301, 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 Rouché’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 on the mechanism of super-resolution and super-focusing for waves in deterministic complex media.