*(English)*Zbl 0984.78002

In this book optical caustics and their fine structure are studied. It presents the theory and describes experiments designed to produce and investigate caustics. Caustics arise by focusing of light. Focusing happens everywhere in nature, but very seldom light is focused into a point, since a point focus is not structurally stable when the wave front forming the focus is perturbed. Rather, what one sees in nature are certain types of caustic surfaces. These are the focusing phenomena which arise generically and are structurally stable. The investigation of this type of focusing is the theme of the book, and the title “natural focusing” refers to it. To study caustic surfaces one uses the model of geometrical optics, which assumes that light is propagating along rays. In this model caustic surfaces are singular surfaces of mappings from three-dimensional space into itself generated by the gradient of the distance function. To investigate and classify the singularities of such mappings is the aim of catastrophe theory. Therefore geometrical optics is one of the important applications of catastrophe theory.

The book starts by explaining these facts and by presenting examples of caustics arising in nature. These examples include swimming pool caustics, caustics generated by rippled glass, by reflections in rippling water, scintillation of stars, and by gravitational lensing. Subsequentially the celebrated theorem of R. Thom is cited, which assures that there are only seven types of such structurally stable caustic surfaces, called elementary catastrophes, namely the fold, the cusp, the swallowtail, the butterfly, the hyperbolic umbilic, the elliptic umbilic and the parabolic umbilic. The shape of these catastrophes is explored analytically and experiments are described, which allow to realize these caustic surfaces. For example, it is theoretically derived that the hyperbolic umbilic can be realized by sending light through a water drop in a circular hole of a vertical plane, and the elliptic umbilic by sending light through a water drop in a triangular hole on a horizontal plane. To show this, the boundary value problems for wave fronts of light passing through such water drops are derived and explicit solutions are given. The book contains many photos of caustics generated by these and other experiments.

In later sections higher catastrophes, which do not arise generically, are discussed similarly. Moreover, network patterns of catastrophes and statistical questions of caustics and twinkling are treated. Examples of network patterns of caustics arising in nature are swimming pool caustics and caustics generated by reflections of sun light in rippling water. Other topics discussed in the book are dislocations in scalar wave fields and diffraction. The idea behind a dislocation is as follows: Assume that a scalar wave field, a continuous complex function, is represented in the form

where the real function $\rho $ is the amplitude and the real function $\varphi $ is the phase. If there can be identified a closed path, along which the phase changes by $2\pi $, then, since $\psi $ is continuous, topological properties imply that on every surface bounded by this curve there must be a point where $\rho $ and therefore $\psi $ vanish. From this one concludes that the set of points where $\psi $ vanishes must lie on a curve which forms a closed loop or which ends at the boundary. This is a purely topological property of continuous complex valued functions, and not specific to wave fields. Because of this, analogies exist to dislocation fields in crystals, hence the name. The book even introduces edge and screw dislocations as in the theory of crystals. Because of their definition, the presence of dislocation curves has topological implications for the wave fields.

The geometrical optics model of light rays does not allow to compute the wave field and the intensity of light near caustic surfaces. Computation of the wave field is studied in the chapter on diffraction. To compute this field, a high frequency approximation of the wave field must be derived. The standard WKB-approximation (actually going back to Sommerfeld and Runge) for the stationary wave field

where $\ell \left(r\right)$ is the distance of the point $r\in {\mathbb{R}}^{3}$ from the caustic, cannot be used, since it becomes infinite at the caustic. This approximation is not valid on the caustic. However, a valid approximation, the diffraction integral, can be obtained by superposition of WKB-expressions. Of course, the Fourier integral operators of Hörmander are based on a similar idea.

This diffraction integral is derived in the book. By insertion of the respective phase functions for the different catastrophes into the integral, the wave fields of these diffraction catastrophes are computed and compared to optical experiments. Diffraction associated with the fold catastrophe of the rainbow is closely examined. The book concludes with the investigation of polarization singularities. This book contains a wealth of ideas, and the value of the book is not reduced by some reservations, which the referee has about the chapter on dislocations. This is a very general notion, for which it would be important to see more precisely when it really applies to wave fields and what it means for these fields. Since this is a minor point, which also can stimulate new research; the book can be warmly recommended to anyone who wants to learn about caustics in optical fields, who wants to learn about experiments in this field, and who wants to get new ideas for further theoretical and practical investigations.