A quantum graph approach to metamaterial design

In this section, we will explore the modeling of metamaterials. Instead of using traditional scatterers or resonant elements, we will represent them through one-dimensional connections between these elements. I will denote this special resonating element as γ, which may represent a single resonator or a more complex structure, such as a tree graph.

The foundational concept involves a regular lattice that serves as the underlying structure, with resonating elements layered on top. We will utilize the properties of these resonating elements to analyze the overall properties of the graph.

To proceed, we need to understand how waves propagate along the edges of this structure and how they scatter at the resonating elements. Typically, we analyze the interaction of incoming and outgoing waves through a scattering matrix. For illustration, consider a simple case with only two vertices, in contrast to the four vertices in a regular two-dimensional lattice.

At each vertex, we assume that the wave function on each edge maintains a consistent value, and we impose a condition on the derivative. By incorporating the parameter λ, we can derive the corresponding scattering matrix for this scenario, which represents the general case of our model.