A quantum graph approach to metamaterial design

We have two length scales to consider: the difference between the underlying lattice constant and the connector length, denoted as ℓM.

Returning to our Graph Model, at a given vertex, we have four connections: left, right, upward, and downward. Each of these connections has an associated amplitude for incoming and outgoing waves. The wave solutions along these edges are governed by the Helmholtz equation. Additionally, due to the periodicity of the lattice at this interface, we introduce the Bloch wave number, κ_y.

The focus now is on how this interface scatters incoming waves, not just at a single vertex but across the entire interface. We can derive an expression for this scattering process.

Initially, we consider the amplitudes of incoming and outgoing waves at each vertex. The scattering matrix can be represented as having a valence of 4. For our analysis, we will set λ to zero, resulting in a term of 1/2 minus δ, PQ. The scattering matrix describes how the incoming wave is divided into transmitted and reflected components.

To simplify, we will disregard the degrees of freedom in the Y direction and derive the scattering matrix. This involves establishing connections between the incoming and outgoing amplitudes at each vertex using local scattering matrices, with λ set to zero for the left, right, up, and down connections.

Continuity conditions must be satisfied, meaning that the wave at position m plus μ, when set to zero, must equal the wave when moving along the line L μ downwards. This leads to a consistency condition, in addition to the Bloch condition.