IV Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns

This chapter discusses the morphologies of caustics and their diffraction patterns. In catastrophe optics, wave motion is viewed in terms of the contrast and interplay among the morphologies of three extreme regimes. Firstly, if the wavelength λ is small in comparison with scales of variation of diffracting objects or refracting media, the wavefield is dominated by the caustics and associated diffraction patterns. Secondly, when waves propagate in environments which can be modeled by a hierarchy of scales extending to the infinitely small, caustics cannot occur and the limit λ → 0 is not geometrical optics. And thirdly, when waves are explored on the scale of λ, the principal features are wavefronts, which are dominated by their singularities in the form of lines in space. The chapter also discusses the diffraction catastrophes that both clothe and underlie caustics. Each structurally stable caustic has its characteristic diffraction pattern, whose wave function has an integral representation in terms of the standard polynomial describing the corresponding catastrophe. The diffraction catastrophes constitute a new hierarchy of functions, different from the special functions of analysis. The newest application of catastrophe optics is to random short waves, whose statistical properties are determined by the random caustic structure.