  Since the turn of the century, metamaterials have gained a large amount of
attention due to their potential for possessing highly nontrivial and exotic
properties such as cloaking or perfect lensing. There has been a great push to
create reliable mathematical models that accurately describe the required
material composition. Here, we consider a quantum graph approach to
metamaterial design. An infnite square periodic quantum graph, constructed from
vertices and edges, acts as a paradigm for a 2D metamaterial. Wave transport
occurs along the edges with vertices acting as scatterers modelling
sub-wavelength resonant elements. These resonant elements are constructed with
the help of fnite quantum graphs attached to each vertex of the lattice with
customisable properties controlled by a unitary scattering matrix. The
metamaterial properties are understood and engineered by manipulating the band
diagram of the periodic structure. The engineered properties are then
demonstrated in terms of the refection and transmission behaviour of Gaussian
beam solutions at an interface between two diferent metamaterials. We extend
this treatment to N layered metamaterials using the Transfer Matrix Method. We
demonstrate both positive and negative refraction and beam steering. Our
proposed quantum graph modelling technique is very fexible and can be easily
adjusted making it an ideal design tool for creating metamaterials with exotic
band diagram properties or testing promising multi-layer set ups and wave
steering efects
