Light shaped as a smoke ring behaves like a particle. (IMAGE)
Caption
(a) The parameter-space sphere which represents spin: the longitude and latitude degrees (α and β) of a parametric 2-sphere are represented by hue color and its lightness (dark towards the south pole, where spin is down, and bright towards the north pole, where spin is up). Each point on a parametric 2-sphere corresponds to a closed iso-spin line located in a 3D Euclidean space. (b) The lines projected from the selected points of the same latitude β and different longitude α on the hypersphere (highlighted by the solid dots with the corresponding hue colors), form torus knots covering a torus (with different tori corresponding to different β). (c) The real-space visualization of a Hopf fibration as a full stereographic mapping from a hypersphere: torus knots arranged on a set of coaxially nested tori, with each torus corresponding to different latitude β of a parametric 2-sphere. The black circle corresponds to the south pole (spin down) and the axis of the nested tori corresponds to the north pole (spin up) in (a). (d) The 3D spin distribution in a hopfion, corresponding to the isospin contours in (c) with each spin vector colored by its α and β parameters of a parametric sphere in (a) as shown in the insert. (e, f) The cross-sectional view of the spin distribution in (d): (e) xy (z = 0) and (f) yz (x = 0) cross-sections show skyrmion-like structures with the grey arrows marking the vorticity of the skyrmions. Color scale is the same as that corresponding to the spin direction in (d).
Credit
Shen et al., doi 10.1117/1.AP.5.1.015001
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