Mathematical Foundation for Symmetry Breaking in Subwavelength Physics

Abstract

The control and manipulation of wave-matter interactions at subwavelength scales constitutes an important realm within wave physics. Topological insulators with subwavelength resonators exhibit promising physical phenomena. In a periodic material strurcture, Bloch’s theorem predicts eigenstates to be modulated plane waves. Hence it is essential to break the symmetries in order to produce captivating effects. This thesis focuses on structures composed of subwavelength resonators with broken symmetries. In the first part of the thesis, we observe non-Hermitian configurations of both one and three dimensional systems of subwavelength resonators. We demonstrate effects such as localisations of states, in particular, skin effect, where the eigenstates of the system are localised on the edge of a structure. In the second part of the thesis, we break the time-reversal symmetry of the wave equations in one or three dimensions. By setting the material parameters to be periodic in time, we observe interesting changes in band structures such as the formation of band gaps and k-gaps. This exploration serves a dual purpose - on one hand, it is driven by the multitude of relevant applications including wave manipulation, guiding of waves, superresolution sensing and the design of metamaterials. On the other hand, solving the mathematical models involves a broad set of techniques originating from PDE theory and the implementation of numerical analysis. Our approach involves the linearisation of eigenvalue problems using boundary integral equations, yielding compelling physical insights, as outlined below. The results are accompanied by rigorous mathematical proofs alongside efficient numerical implementations.