Oscillations, Waves & Optics 2014-15

*Dr John Richer*

## Prerequisites

The course assumes knowledge of the Part IA Physics and Part IA NST Mathematics courses. In addition, the mathematics of the Fourier Transform, which is taught in the Part IB NST Mathematics, or the Mathematical Methods course, is required.

## Learning Outcomes and assessment

At the end of this course, you should understand the motion of a harmonic oscillator, including both its free response from given starting conditions, and the response when driven at a single frequency. You will understand how to calculate the response to a general driving force using Fourier methods.

You will understand how to derive and solve the wave equation for a range of systems, and understand the general form of solution for non-dispersive waves in one, two and three dimensions. You will understand the concept of wave impedance, and be able to use it to calculate the reflection behaviour when waves encounter boundaries, and how media of different impedance can be matched. You will have an understanding of the properties of sound waves in gases, and on the possible acoustic waves that can propagate in solids. You will understand the effects on wave propagation caused by dispersion, and understand and define the concept of the group velocity for a wavepacket. You will understand the concept of guided waves, and be able to derive the dispersion relation for guided waves in simple cases.

The section on waves will enable you to understand how we can use Huygens' Principle to predict diffraction patterns caused when a wave encounters an obstacle. In particular, you will understand the nature of the diffraction pattern observed at large distances from an aperture and its relation to the Fourier transform, and how the pattern changes when it is observed close to the aperture. You will be able to calculate the far-field (Fraunhofer) diffraction patterns of simple one and two dimensional apertures, and the near-field (Fresnel) diffraction patterns caused by edges, slits and circular apertures. You will understand the imaging properties of a zone plate.

You will understand the processes which control the observed width and shape of spectral line emission, and how spectrometers can be used to measure lineshapes; in particular, you will be able to define and calculate the spectral resolving power of grating spectrometers. You will understand the effects of diffraction on imaging instruments such as microscopes and telescopes, and how they set the angular resolution obtained.

You will also understand the general conditions required for interference patterns to be observed, and be able to compute the interference patterns expected from thin films, and from cavities. You will be able to describe and understand the operation of a Michelson interferometer and a Fabry-Perot etalon, and how they can be used to measure the spectrum of light.

## Synopsis

**Oscillations:**Driven damped oscillations, frequency response, bandwidth, Q-factor. Impulse response and transient response.**Waves:**Revision of 1 dimensional wave equation. Waves on a stretched string. Polarisation. Wave impedance. Reflection and transmission. Impedance matching. Compression waves in a fluid. Waves in 2 and 3 dimensions. Standing waves in a box. Wave groups, group velocity, dispersion. Waveguides: cut-off and dispersion relation.**Fourier transforms**: Linear response and superposition in physics. Fourier series and Fourier transforms. Frequency response as Fourier transform of pulse response. Convolution. Applications to oscillating systems.**Optics and diffraction:**- Huygens' principle and solutions to the wave equation.
- Fraunhofer diffraction, Fraunhofer integral, relation to Fourier transform. Wide slit as example of extended source.
- The width of spectral lines.
- Gratings and spectroscopy.
- 2-D apertures, circular apertures, Babinet’s principle.
- Fresnel diffraction: edges and slit diffraction patterns, use of the Cornu spiral. , zone plate.

**Interference:**- General conditions for interference
- Thin-film interference.
- Fabry-Perot etalon.
- Michelson interferometer
- Fourier transform spectroscopy.

**BOOKS**

*Vibrations and Waves in Physics*, Main I G (3rd edn CUP 1993)

*The Physics of Vibrations and Waves*, Pain H J (5th edn Wiley 1999)

*Vibrations and Waves*, French A P (Chapman & Hall 1971)

*Optics*, Hecht E (4th edn Addison-Wesley 2001)