Here we consider a state of light whose electric field lines are all closed and any two are linked to each other as described in Figs 1 and 2. Knotted vortex lines have also been considered in phases associated with the electron states of hydrogen 11, with the Riemann–Silberstein vector of the electromagnetic field 12 and in phases associated with lines of darkness in a monochromatic light field 13, with the latter predictions experimentally verified 14. More recently, an approach to knotted classical fields was proposed 8 and further understood and developed 9, 10. Since Kelvin proposed knotted field configurations as a model for atoms, knots and links have been studied in branches of physics as diverse as fluid dynamics 7, plasma 5 and polymer physics 6. In the present work, the fibres of two everywhere-orthogonal Hopf fibrations correspond to electric and magnetic field lines (see Fig. 2 for t=0). The circular (straight) special fibres are mapped to the north (south) pole and will be referred to as the n ( s) fibres. Each circle is mapped to a point, each torus in onto a (parallel) circle on. Right column: The Hopf map maps such circles in to points on the sphere. These two fibres will provide an economical way of characterizing the time evolution of the configuration. There are two ‘special’ fibres: the circle of unit radius that corresponds to the infinitely thin torus, and the straight line, or circle of infinite radius, that corresponds to an infinitely large torus. c, By nesting such tori into one another, the whole of three dimensional space, including the point at ( ) can be filled with linked circles. a, b, Each circle in such a configuration wraps once around each circumference of the torus. We predict theoretical extensions and potential applications, in fields ranging from fluid dynamics, topological optical solitons and particle trapping to cold atomic gases and plasma confinement.Ī– c, Left column: A torus can be constructed out of circles (fibres) in such a way that no two circles cross and each circle is linked to every other one. Second, we show how a new class of knotted beams of light can be derived, and third, we show that approximate knots of light may be generated using tightly focused circularly polarized laser beams. Using this representation, first, a connection is established to the Chandrasekhar–Kendall curl eigenstates 2, which are of broad importance in plasma physics and fluid dynamics. We study their time evolution and uncover, through a decomposition into a spectrum of spherical harmonics, a remarkably simple representation. Here we analyse their physical properties to investigate how they can be experimentally realized. These little-known solutions, constructed by Rañada 1, are based on the Hopf fibration. Maxwell’s equations allow for curious solutions characterized by the property that all electric and magnetic field lines are closed loops with any two electric (or magnetic) field lines linked.
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