Diffraction Catastrophes

This chapter discusses the mathematics of diffraction catastrophes, focusing on the wavefront associated with a monochromatic wave with Q(x, y, f (x, y)) as a point on the wavefront surface and P (X, Y, Z) as a movable observation point. Optical caustics refer to surfaces (in space) and curves (in the plane) where light rays are focused. Caustics can be classified mathematically as the elementary catastrophes of singularity theory. In the plane, the classification gives two singularities: smooth caustic curves, which are “fold” catastrophes, and points where two fold caustics meet on opposite sides of a common tangent, which are “cusp” catastrophes. After providing an overview of the basic geometry of the fold and cusp catastrophes, the chapter considers the Fresnel integrals and the fold diffraction catastrophe. In particular, it examines the rainbow as a fold catastrophe, Airy integral in the complex plane, and the nature of the notation Ai(X).