![]() In electromagnetic waves, the polarisation singularities that are stable under perturbations are lines in three dimensions. Incorporating the vector (electromagnetic) nature of light brings further singularities, corresponding to the new physical property thus introduced, namely polarisation. Allen excitedly explaining OAM to me during a train journey in the early 1990s 70. We are celebrating 30 years since the discovery of OAM as a practical resource in optics 69. optical beams that are eigenstates of OAM, but in general the concepts are distinct, as counterexamples demonstrate 68. The association holds for the simplest cases, e.g. There has been a confusion in which wave vortices are regarded as inevitably associated with orbital angular momentum (OAM). ![]() In three dimensions, phase singularity lines can be linked and knotted 62, 63, 64, 65, 66, 67. acoustic 53 and tide 54, 55, 56, 57, 58) waves and have been extensively reviewed 59, 60, 61. They occur in all types of quantum 46, 47, 48, 49, 50, 51, 52 or classical (e.g. They can organise the coloured interference patterns formed by white light 31, 32, 45. Phase singularities can form intricate patterns, for example as fine detail in diffraction catastrophes 19, 20, 42 and near spiral phase plates 43, 44. The optical applications of superoscillations depend on a fundamental fact: there is a diffraction limit for bright light, but there is no such limit for dark light (evoking the mathematician André Weil’s playful “…principle of anti-inteference, which would have light burst forth from two darknesses.” 41). Therefore phase singularities are powerful sources of superoscillations (band-limited functions that vary faster than their fastest Fourier component) 33, with many applications 34, 35, for example to sub-wavelength microscopy 36, 37, 38, 39 and in mathematics 40. 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. ![]()
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