Alexander Uhlmann

We consider disordered finite systems of subwavelength resonators where the disorder is in the spatial arrangement of the resonators and their material parameters are perturbed randomly. We elucidate the mechanism of wave localisation in such systems. To do so, we employ the capacitance matrix formalism, make precise the notions of bandgap and localised eigenmodes, study their stabilities with respect to fluctuations in the system, and highlight the importance of defect hybridisation and level repulsion in creating localised eigenmodes for a given arrangement of the resonators by randomly changing their material parameters. We also introduce a method of localised eigenmode prediction based on the discrete Green function of the disordered system of resonators and show the existence of bound eigenmodes in disordered systems where local translation invariance is broken.

Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non-reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the \k\-Toeplitz matrix cases.