Caustics, Catastrophes and Wave Fields in a sense continues the treatment of the earlier volume 6 "Geometrical Optics of Inhomogeneous Media" in the present book series, by analysing caustics and their fields on the basis of modern catastrophe theory. This volume covers the key generalisations of geometrical optics related to caustic asymptotic expansions: The Lewis-Kravtsov method of standard functions, Maslov's method of caonical operators, Orlov's method of interference integrals, as well as their modifications for penumbra, space-time, random and other types of caustics. All the methods are amply illustrated by worked problems concerning relevant wave-field applications.

Caustics, Catastrophes and Wave Fields in a sense continues the treatment of the earlier volume 6 "Geometrical Optics of Inhomogeneous Media" by analysing caustics and their fields on the basis of modern catastrophe theory. The present volume covers local and uniform caustic asymptotic expansions: The Lewis-Kravtsov method of standard functions, Maslov's method of canonical operators , Orlov's method of interference integrals, as well as their modifications for penumbra, space-time, random and other types of caustics. All the methods are amply illustrated by worked problems concerning relevant wave-field applications.

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A general formalism is presented for the study of the transient wave fields in the vicinity of caustics of arbitrary complexity. The caustics are classified using catastrophe theory. A general introduction to the subject is presented and explicit results (numerical and analytical) for the cuspoid family of catastrophes which show the smooth variation of the acoustic field as the control parameters are independently varied. These results vividly demonstrate the phenomenon of ray focusing, showing the number of rays which converge at each catastrophe, the resulting phase shifts they undergo, as well as their decay in shadow regions. Analytical results are given which describe the time-dependent acoustic field on and at large distances away from each of the cuspoid catastrophes. Four sets of numbers are introduced which describe the behavior of the wave field in the vicinity of each catastrophe. These are: (1) an exponent which describes the temporal decay of the acoustic field on the most singular point of each catastrophe; (2) a number which describes the asymmetry of the time-dependent acoustic field on the most singular point of each catastrophe; (3) an exponent which describes the time separation as a function of control parameter between individual ray arrivals as each of the control parameters are independently varied; and (4) an exponent which describes the asymptotic decay as a function of control parameter of each constituent ray arrival. Sets (1) and (3) are shown to be simply related to the singularity and fringe indices. It is argued that many features of the frequency domain representation of the acoustic field in the vicinity of the catastrophes are most easily understood by examining the time domain representation.

The caustics of high-frequency wave propagation may be classified using catastrophe theory. This paper examines the transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic catastrophes. Analytical results are presented which describes the time-dependent wave field due to an impulsive source on and at large distances in each control direction away from the most singular point of each of these transient wave fields as each control parameter is independently varied. These results are compared to previously derived results on the transient wave fields in the vicinity of the cuspoid catastrophes (the cuspoid and umbilic catastrophes have coranks 1 and 2, respectively). The transient wave fields in the vicinity of the cuspoid and umbilic catastrophes differ with regard to the temporal structure of the wave field on the most singular point of each catastrophe, the manner in which these temporal structures unfold, and the phase shifts which individual rays undergo as a result of touching one of the catastrophes.

Caustics are natural phenomena, forming light patterns in rainbows or through drinking glasses, and creating light networks at the bottom of swimming pools. Only in recent years have scientists started to artificially create simple caustics with laser light. However, these realizations have already contributed to progress in advanced imaging, lithography, and micro-manipulation. In this book, Alessandro Zannotti pioneers caustics in many ways, establishing the field of artificial caustic optics. He employs caustic design to customize high-intensity laser light. This is of great relevance for laser-based machining, sensing, microscopy, and secure communication. The author also solves a long standing problem concerning the origin of rogue waves which appear naturally in the sea and can have disastrous consequences. By means of a far-reaching optical analogy, he identifies scattering of caustics in random media as the origin of rogue waves, and shows how nonlinear light-matter interaction increases their probability.

Transient acoustic wave fields in the vicinity of some possible caustics, the time history of the wave field due to an impulsive source includes a precursor. This noncausal behavior is reconciled with the physical principle of causality by requiring that such caustics be a part of certain higher dimensional caustics. With this constraint all noncausal arrivals are preceded by a causal arrival and the total time history is causal. This constraint is used to derive some simple general properties of transient wave fields in the short wave limit, e.g., the number of arrivals must be an odd number.