Introductory description

When waves propagate in heterogeneous and structured materials, an intricate combination of scattering, transmission and reflection can occur. These phenomena are the foundation of many modern wave control and manipulation devices, used in applications including telecommunications, noise control and medical imaging. These micro-structured materials are sometimes known as metamaterials. In this module, we will develop the mathematical tools needed to understand and design them.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The first portion of this course will be devoted to the study of waves in discrete mass-spring systems. This will be used as a toy platform to develop the key principles of wave propagation in structured materials: phase velocity, group velocity, dispersion curves and transmission and reflection coefficients.

The second portion of the course will study the one-dimensional wave equation. We will develop transfer matrix tools to study reflection and transmission by discontinuities in material parameters. Periodic materials will be considered in detail; we will use Floquet-Bloch’s theorem to characterise the spectrum and study the spectral band gaps. We will use asymptotic homogenisation to characterise effective properties of the system, both at low frequencies and arbitrary non-zero frequencies.

The third section of the course will cover multi-dimensional systems. Basic properties and asymptotics of eigenvalues of the Laplacian will be studied (motivated by the result of separation of variables). Homogenisation theories for multiple dimensions will be developed. Green’s functions and multiple scattering formulations will be introduced.

Finally, we will use the theory developed in this course to explore modern metamaterial applications. These will include the design of perfect (flat) lenses, gradient index lenses, graded energy harvesting devices and invisibility cloaks.

Learning outcomes

By the end of the module, students should be able to:

• Apply mathematical tools to describe wave propagation and scattering in heterogeneous media
• Use transfer matrices to describe wave propagation in one-dimensional systems
• Use asymptotic homogenisation to compute effective properties for linear waves in heterogeneous media
• Use Floquet-Bloch theory to compute spectra of periodic operators
• Understand how the eigenvalues of the Laplacian on a domain describe wave behaviour and understand their basic properties
• Understand how mathematical methods are used to design wave control devices

Indicative reading list

• “Wave propagation: from electrons to photonic crystals and left-handed materials” by Markos and Soukoulis
• “Oscillations and waves: an introduction” by Fitzpatrick
• “Partial differential equations” by Evans
• “Mathematical Theories for Metamaterials: From Condensed-Matter Physics to Subwavelength Physics” by Ammari, Davies and Hiltunen

Interdisciplinary

This module highlights some uses of mathematical modelling and approximation techniques in wave physics and engineering.

Subject specific skills

Students will develop an understanding of how mathematical approximation techniques can be used to model physical problems and design devices for applications. They will learn a suite of new mathematical tools, such as asymptotic homogenisation, transfer matrices, Floquet-Bloch analysis and Green’s functions. They will also learn new aspects of wave physics and how they relate to mathematical models.

Transferable skills

This module provides technical competence in characterising and approximating solutions to partial differential equations, with an emphasis on informing applications. This teaches students how to adapt a theoretical framework to real-world problems. It also develops problem solving abilities. The applications to metamaterials in the final part of the course will provide students with a case study of how mathematics is used in current cutting-edge physical applications.