Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations.
John Nye. 328 pp. Institute of Physics Publishing, Bristol and Philadelphia, 1999.
Price: ??? (cloth) ISBN 0 7503 0610 6. (Reviewed by Francis J. Wright.)
The most obvious optical phenomenon is the focusing of light onto caustics, which are usually seen as bright lines on a surface in the path of the light. In fact, after reading this book, you will notice caustics everywhere, produced for example by reflection or refraction of light by domestic objects. Caustics can be understood in terms of geometrical optics, the simplest and oldest theory of light. But closer inspection shows that caustics are blurred by diffraction effects, which can be understood in terms of a scalar wave theory. This diffraction can be seen with the naked eye in the caustics produced by a raindrop on glass close to the observer. Much of the structure of these diffraction patterns can be understood in terms of wave dislocations, which are dark lines that weave through the spatial patterns. Yet more detailed investigation of dislocation lines in light or any other electromagnetic wavefield reveals that they are only an approximation to a much richer polarization structure, the understanding of which requires a full vector wave theory.
The usual approach to modelling physical phenomena is to rely on symmetries to simplify the problem. An alternative approach is to consider what typically happens in the absence of all such constraints. The latter is the drive behind this book and the reason for the word “natural” in its title. A great deal of profound pure mathematics (which is referenced but not repeated in the book) underlies a rigorous proof that unconstrained caustics can take only a small number of surprisingly simple local forms. They are examples of the so-called “elementary catastrophes”, which have also been applied to model many other phenomena, although not always with the same rigor as the application to wave propagation. One is then led naturally back to symmetry because the standard elementary catastrophe models have symmetries that correspond to approximate local symmetries in caustics. Judicious abuse of the pure mathematics even helps to explain more symmetric caustics, including such highly symmetric examples as the spherical aberrations in a rotation-symmetric lens. Each typical class of local caustic is decorated with a characteristic local diffraction pattern with a specific dislocation structure.
Typicality is one of the guiding themes throughout this book; another is “singularity”. Singularities of mappings are important because they provide a framework for the detailed structure of the mapping. For example, consider a cube. It can be specified in terms of its corners alone. However, a more complete representation would include the edges, which then imply the faces uniquely. You can make a cardboard cube by cutting out a figure consisting of six adjoined squares and then folding it appropriately. This folding maps the planar cardboard figure into three dimensions and the fold lines are singularities of an otherwise smooth map. The corners of the cube are points where the cardboard is folded in two different directions, which are therefore “more singular” than the folds themselves. Thus there is a hierarchy of singularities: the highest order (corner) singularities give the essential framework, the disposition of the lower order (edge) singularities around them gives them substance, and so on. In this way, singularities act as “organising centres”. In optics, higher order caustics organise lower order caustics. Pushing the analogy further, caustic singularities organise phase singularities of a scalar wavefield (wave dislocations) and polarization singularities of a vector wavefield. These wavefield singularities to a large extent describe the structure of the wavefield.
Wavefield singularities occur on all size scales. On the largest, recent pictures from the Hubble Space Telescope show multiple images of the same galaxy due to gravitational lensing. On the smallest, nano-machines, built from small numbers of molecules, are a topic of current interest and speculation and the prospect of trapping and manipulating micron-sized particles in wave dislocations is tantalizing.
John Nye has been one of the principal investigators of natural focusing and fine structure of electromagnetic waves over a period of about 25 years, and this book is a review of the discoveries made over that time, many of which he had a hand in. Not surprisingly, therefore, it is the definitive summary of a large body of this work. It is very carefully written and excellently illustrated, and the publishers have been equally conscientious in the overall presentation of the book.
The flavour of this book is descriptive and there are plenty of experimental photographs to keep the reader in touch with reality, although mathematics is used where appropriate to make the description precise. The book should be accessible to students in the latter part of an undergraduate physics course. The reader needs to be familiar with mathematical models of wave propagation and electromagnetism, and with elementary mathematical techniques such as Taylor series expansions and determinants, although quite a lot of the mathematics could be skipped without losing the overall picture. There is enough material in the book to use it as the basis for an exciting and original course on wave phenomena, and the more descriptive parts could be used at pre-university level. There must be considerable advantage in having experiments that illustrate subtle focusing and diffraction phenomena but which nevertheless require that there be no special set-up and which benefit from dusty equipment!
Francis J. Wright is Reader in Mathematics at Queen Mary and Westfield College, University of London, England. His current research interests are principally mathematical computation, applied in particular to exact solution of differential equations and to optimum experimental design. His former research interests include wavefield singularities and catastrophe theory.