LECTURE

Making Light of Mathematics

Chaired by Victor Suchar

Sir Michael Berry FRS

University of Bristol,

10 April 2004

The Chairman introduced Sir Michael as the Royal Society Physics Professor at the University of Bristol and one of the leading and most influential scientists in the country.

Summary

Physics and Mathematics have evolved together and they remain deeply connected. It is important to look upon physics and mathematics as separate entities but in our scientific practice it can be very hard to tell them apart. The distinguished Oxford Chemist, Peter Atkins, put this succinctly when he stated that trying to tell where mathematics ends and physics begins is as pointless as determining the end of a morning mist. Nowhere is the connection between physics and mathematics more clearly seen than in optics.

Professor Berry assured us that in his lecture he would only give two mathematical equations both of which are very simple. His first equation was:

1+1=2

What indeed could be simpler! Suppose that one has two torch-lights then their intensities add. However, suppose that one uses instead two sources of laser light, which is pure light of one frequency, their intensities do NOT add. The reason is that one also has to take into consideration the phase of the laser light i.e. at what stage is the wave oscillation in its cycle. If one adds waves at different stages in their oscillation, i.e. of different phases, then their intensities do not add. Hence for the intensities it is possible to have zero intensity corresponding to destructive interference when the two waves cancel each other out or maximum intensity when the two waves reinforce each other.

It was Lord Rayleigh who first posed and answered the question as to why two flashlights are more intense than one. The phases of the waves in a flashlight are rapidly varying quantities and hence the eye only manages to see the average value.

Professor Berry has a great interest in drawing attention to natural phenomena, which can only be explained in terms of complex mathematics. These include mathematical singularities, which can be observed in rainbows and the brilliant point singularities occurring when bright sunlight is seen sparkling on a lake. He also demonstrated how it is possible to observe very simply subtle aspects of wave interference effects by shining a laser pointer through irregular bathroom glass. A further simple demonstration was using plastic sheets for transparent overhead projectors, which showed polarization singularities

Professor Berry devoted the concluding part of his talk to what has become known as the ‘Talbot effect’. It is named after William Henry Fox Talbot, a local hero who lived at Lacock Abbey some ten miles from Bath. Talbot epitomises the 19^th century polymath who is best remembered today for his invention of the modern photographic process where he created a negative image from which it was possible to obtain a large number of positive copies. However, he also invented the polarizing microscope used extensively by mineralogists, made researches in mathematics and helped in the transcription of Syrian and Chaldean inscriptions. His optical researches have recently received attention.

If one takes a diffraction grating such as fine silk or an umbrella material and shines a light through one side while observing from the other side with a magnifying glass, then one can obtain a clear image. If one then looks further away the image becomes blurs. However, if one looks further away still the image comes into sharp focus again. This sequence of sharp and blurred images continues for a distance of several metres and represents the ‘Talbot effect’ – the repeated self-imaging of a diffraction grating. The effect was rediscovered and extended by Lord Rayleigh in the early 1880s. It was then forgotten again before a further revival in the 1960s.. It is now appreciated that it is more than a curious optical effect but represents an important example of extreme interference of light waves. In addition it has an analogue in quantum physics known as ‘quantum revival’. This is an important aspect of physics in which an idea occurs in one area mysteriously seems to reappear in another. The essence of science is to, explain more and more phenomena in terms of similar concepts.

The mathematics of the Talbot effect is complex. At its heart it makes use of a simple identity:

(-1)ⁿ = (-1) ⁿ²

where ‘n’ is an integer representing the different contributing wavelets. This identity greatly simplifies the mathematical analysis.

Professor Berry’s lecture was well received by an enthusiastic audience who asked many questions. In a brief vote of thanks Dr Peter Ford drew attention to Professor Berry’s deep insight into physics and mathematics and his ability to communicate them in a lucid and clear manner.

Dr. Peter Ford