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# MATH3071 Light and Waves

## Module Overview

The first part of the module introduces the multipole approximation for both scalar and vector fields in the context of electrostatics and radiating fields. To do this we recap Maxwell's equations and the derivation of the wave equation. The multipole approximation is a very powerful technique to determine the field produced by a distribution of charges. A standard example of this technique is the study of the field emitted by a radio antenna. We then show how its equation of motion has to be modified if we are interested in light propagation in a medium. The modification is in principle very simple, but opens up an impressive array of possibilities from the rather trivial bending of light as it goes through the surface of a water pool, to the rather more complicated dynamics of rays that wrap themselves around objects making them to all intents and purposes invisible. It is not enough to generate waves: we also need to propagate them. We will consider two different types of wave motion. The first is the propagation of waves in conduits. A very simple example is the flow of air in the pipe of wind musical instrument. The most well known example in the field of electromagnetism are optical fibres, thin “wires” of very pure glass that transmit light pulses across oceans and continents and that are the backbone of the world wide web. Another aspect of wave motion that we will consider is wave propagation in “dispersive” media, i.e. materials such that the wave speed depends on the wave frequency. This will require us to introduce an extremely powerful mathematical tool, the Fourier transform, the generalisation of the Fourier series to non-periodic functions: it is an essential tool in the analysis and study of the frequency-dependent behaviour of waves. All materials are dispersive: this can be a hindrance, but it is a feature that can also be harnessed to produce very stable waves called solitons (the Severn bore can be considered an example of these waves) or to produce very short pulses of light that last a millionth of a billionth of a second. The module ends by studying the connection between Maxwell’s equations and Einstein’s (special) theory of relativity. The link between the two is very deep: for example, Maxwell’s equations show that the speed of light is the same in all inertial reference frames. This observation was the birth of the theory of relativity. Conversely, space-time is the natural context for Maxwell’s equations: using the notation of special relativity we will see that the electric and magnetic field can indeed be expressed as a single four dimensional field and Maxwell’s equations acquire a very simple and elegant form.

### Aims and Objectives

#### Module Aims

To give an introduction to the mathematical description of wave phenomena in the context of electromagnetism. The mathematical techniques used are, however, very general and can be applied to analyse waves in all branches of applied mathematics, from fluid dynamics to acoustics and relativity.

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Understand the fundamentals of electro-magnetism related to light propagation in optical media
• Understand how to model wave propagation in conduits and dispersive media
• Understand the basic elements of relativistic electromagnetism; solve problems of light propagation

### Syllabus

1) Revision of Maxwell's equations 2) Derivation of the wave equation: 1. Wave equation in Cartesian, cylindrical and spherical coordinates; 2. Wave equation in 3D and 1D; 3. Plane wave solutions and the polarisation of waves. 3) Fields in linear isotropic materials: 1. Polarisation and magnetisation fields in an optical medium; 2. Boundary conditions; 3. Metamaterials and invisibility cloaks. 4) Multipole expansion of an electromagnetic field 5) Dielectric waveguides 6) Dispersion relations 1. Fourier transform; 2. Phase and group velocity; 3. Kramers-Kronig relation. 7) Relativistic electromagnetism

### Learning and Teaching

TypeHours
Independent Study108
Teaching42
Total study time150

Griffiths DJ (1999). Introduction to electrodynamics.

Cottingham WN & Greenwood DA (1991). Electricity and Magnetism.

Okamoto K (2000). Fundamental of optical waveguides.

Grant IS & Phillips WR (1990). Electromagnetism.

Lorrain P, Corson D & Lorrain F (1988). Electromagnetic Fields and Waves.

Jackson JD (1999). Classical Electrodynamics.

Hecht E (2002). Optics.

### Assessment

#### Summative

MethodPercentage contribution
Coursework and class tests 20%
Exam 80%

#### Referral

MethodPercentage contribution
Exam %

Pre-requisites: MATH2008 AND MATH2038 AND MATH2044 AND MATH2045

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.

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