Lawrie Tristan, Tanner Gregor, Chronopoulos Dimitrios
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK.
Department of Mechanical Engineering and Mecha(tro)nic System Dynamics (LMSD), KU Leuven, 9000, Leuven, Belgium.
Sci Rep. 2022 Oct 26;12(1):18006. doi: 10.1038/s41598-022-22265-2.
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 infinite 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 finite 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 reflection and transmission behaviour of Gaussian beam solutions at an interface between two different 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 flexible 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 effects.
自世纪之交以来，超材料因其具有诸如隐身或完美透镜等高度非平凡和奇异特性的潜力而备受关注。人们一直在大力推动创建能够准确描述所需材料成分的可靠数学模型。在此，我们考虑一种用于超材料设计的量子图方法。由顶点和边构成的无限方形周期量子图，作为二维超材料的范例。波沿着边传输，顶点充当模拟亚波长共振元件的散射体。这些共振元件借助附着在晶格每个顶点上的有限量子图构建而成，其可定制特性由酉散射矩阵控制。通过操纵周期结构的能带图来理解和设计超材料特性。然后根据高斯光束解在两种不同超材料界面处的反射和透射行为来展示所设计的特性。我们使用转移矩阵法将这种处理扩展到N层超材料。我们展示了正折射和负折射以及光束转向。我们提出的量子图建模技术非常灵活且易于调整，使其成为创建具有奇异能带图特性的超材料或测试有前景的多层结构和波转向效应的理想设计工具。
