A Quantum Graph Approach to Metamaterial Design - Lawrie, Tristan (2025)
Since the turn of the century, metamaterials have garnered significant attention for their ability to exhibit exotic properties such as cloaking and perfect lensing. This has led to a growing need for reliable mathematical models capable of describing these materials' complex behaviors. While various modeling techniques exist for studying and engineering metamaterials, this thesis introduces a novel approach based on the scattering formalism of quantum graph theory. The flexibility and mathematical simplicity of this framework make it an ideal tool for designing metamaterials with unique band structures and for exploring complex multi-layer configurations. This thesis begins by extending quantum graph theory's scattering formalism to study wave propagation in complex periodic and finite quantum systems. Green’s functions on quantum graphs are developed using a scattering approach, offering a powerful method for analyzing wave behavior on both closed and open graphs. Next, we apply this formalism to study acoustic metamaterials modeled as networks of interconnected waveguides, confirming the model's predictions through both simulations and experiments. Finally, the thesis explores the design of an angular Fourier filter using a periodic quantum graph with beyond-nearest-neighbor connections, demonstrating that quantum graphs can be used to model resonant wave transmission at discrete angles. The results were verified using COMSOL simulations in the acoustic regime, showing excellent agreement between theory, simulation, and experiment. This work establishes quantum graph theory as a new paradigm for metamaterial design, offering a versatile and intuitive framework for modeling wave behavior and guiding the development of future metamaterial technologies.