And because the phase changes by a multiple of 2π around a singularity, the local phase gradient vector rises to infinitely large values there. In white-light interference, universal colour patterns inhabit the near-darkness 31, 32. The darkness of a phase singularity can be regarded as perfect descructive interference 30, underlying several interesting features. Phase singularities are complementary to caustics, not only because the former are dark (zero intensity) and the latter are bright (infinite intensity), but also in the sense of Niels Bohr: caustics are prominent features in the short-wave asymptotic regime, in which phase singularities are too close to be clearly resolved-because these are fine-scale features, clearly discernable only in the opposite case of high magnification, where caustics are smoothed out and so are no longer distinct features. Geometrically, they are lines in space, or points in the plane, around which the phase changes by a multiple of 2 π (generically ☑) and the phase gradient vector circulates-hence the term optical vortices. On phase singularities, the light intensity is zero, so these are the singularities of dark light. Equivalent terms for phase singularities are optical vortices, nodal manifolds or wavefront dislocations 28, 29.

Wave optics, when represented by complex scalar wavefunctions, introduces the additional concept of phase, which has its own singularities. In white light, caustics display interesting colours when diffraction is incorporated 26, 27. 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.

Intensity equation polarized light