Abstract

Since the turn of the century, metamaterials have attracted significant attention due to their potential to exhibit highly nontrivial and exotic properties, such as cloaking and perfect lensing. A key challenge in their development is the creation of reliable mathematical models that accurately describe the required material compositions. In this work, we adopt a quantum graph approach to metamaterial design, in which an infinite square periodic quantum graph—constructed from vertices and edges—serves as a paradigm for a two-dimensional metamaterial. Wave transport occurs along the edges, while the vertices act as scatterers modelling subwavelength resonant elements.

The metamaterial properties are understood and engineered by manipulating the band diagram of the periodic structure, either by varying the vertex scattering properties or by altering the graph topology. These engineered features are demonstrated through the reflection and transmission behaviour of Gaussian beam solutions at an interface between two distinct metamaterials. The proposed quantum graph modelling framework is highly flexible and easily adjustable, making it an ideal design tool for creating metamaterials with exotic band structures and filtering capabilities.

The reliability of the model is verified through numerical simulations using COMSOL and confirmed experimentally in both the acoustic and microwave regimes. As an example, we conceptualise and numerically simulate a resonant metamaterial interface incorporating non-local—beyond nearest-neighbour—coupling, which acts as a discrete angular filter. This structure can be designed to achieve perfect transmission at customisable angles of incidence, without diffraction, enabling tailored transmission in arbitrarily narrow wavenumber windows.