A Quantum Graph Approach to Metamaterial Design

Lawrie, Tristan (2025) A Quantum Graph Approach to Metamaterial Design. PhD thesis, University of Nottingham.

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Abstract

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.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Tanner, Gregor
Keywords: Quantum Mechanics, Quantum Theory, Graph Theory, Quantum Graph Theory, Metric Graphs, Acoustics, Metamaterials, Material Science, Solid State Physics, Periodic Media, Filters.
Subjects: Q Science > QA Mathematics > QA150 Algebra
Q Science > QA Mathematics > QA299 Analysis
T Technology > TK Electrical engineering. Electronics Nuclear engineering > TK7800 Electronics
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 80281
Depositing User: Lawrie, Tristan
Date Deposited: 31 Jul 2025 04:40
Last Modified: 31 Jul 2025 04:40
URI: https://eprints.nottingham.ac.uk/id/eprint/80281

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