The wave decorations exhibit interesting scaling laws, with implications for gravitational lensing 25. In wave optics, the caustic singularities of families of rays are smoothed by diffraction, which decorates them with rich and ubiquitous interference patterns 19, 20, described by a new class of special functions (‘diffraction catastrophes’ 21), represented by oscillatory integrals (chapter 36 of ref. This review includes some personal remarks. vector) waves (see the section ‘Polarisation’), there are singularities of polarisation: lines (in 3D) on which waves are purely circularly polarised or purely linearly polarised. In scalar wave optics (see the section ‘Phase’), there are phase singularities, also called wave vortices, wavefront dislocations, and nodal lines (in 3D). In geometrical optics (see the section ‘Rays’), the singularities are caustics: envelopes of families of rays. The emphasis will be on singularites that are natural, in the sense that they are stable under perturbation equivalent terms for this kind of naturalness are typicality, genericity, structural stability, and universality. In this review, three qualitatively different singularities will be described. Analogous considerations apply to other types of wave: quantum, acoustic, elastic, water…. From this perspective, there are different levels of description in optics, each characterised by different singularities. Geometry dominates modern optics, in particular through understanding light in terms of its singularities.
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