A quantum graph approach to metamaterial design

To model the metamaterial, we establish one-dimensional connections between resonant elements rather than using discrete scatterers. This approach involves treating the system as a graph, where a regular lattice underlies a more complex structure, which may include resonators or intricate tree graphs. The properties of these resonating elements are utilized to analyze the characteristics of the graph.

We require information on wave propagation along the edges and the scattering behavior at the vertices. Typically, we consider incoming and outgoing waves, with a scattering matrix representing their interaction. In a simplified scenario with two vertices, we assume that the wave function is uniform across each edge, subject to specific derivative conditions. Introducing the parameter λ allows us to derive the corresponding scattering matrix, applicable in general cases.

In this context, V denotes the valency of the vertex, which is 2 in this instance and will increase to 4 later. As λ approaches 0, we observe flux conservation, corresponding to Neumann-Kirchhoff boundary conditions. Conversely, as λ approaches infinity, we encounter hard boundaries. These variations in λ influence the reflection and transmission of waves at the vertices. Our analysis will encompass one-dimensional, two-dimensional, and potentially three-dimensional lattices, typically employing Kirov-Neumann boundary conditions for the scattering matrices at the vertices.