Tying Knots in Light Fields

Kedia, Hridesh; Bialynicki‐Birula, Iwo; Peralta‐Salas, Daniel; Irvine, William T. M.

doi:10.1103/physrevlett.111.150404

Cited by 177 publications

(258 citation statements)

References 37 publications

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“…We are following the method in Ref. [11]. First we consider the case in which the poloidal winding number n p = 1 and the toroidal winding number n t is any positive integer.…”

Section: Stability Analysismentioning

confidence: 99%

“…The Hopf fibration can also be used in the construction of finite-energy radiative solutions to Maxwell's equations and linearized Einstein's equations [8]. Some examples are Ranada's null electromagnetic (EM) hopfion [9,10] and its generalization to torus knots [1,11,12].…”

Section: Introductionmentioning

confidence: 99%

“…The Hopf structure of the Poynting vector propagates at the speed of light without deformation. The EM hopfion has been generalized to a set of null radiative fields based on torus knots with an identical Poynting vector structure [11]. The electric and magnetic fields of these toroidal solutions have integral curves that are not single rings, but rather each field line fills out the surface of a torus.…”

Section: Introductionmentioning

confidence: 99%

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Constructing a class of topological solitons in magnetohydrodynamics

et al. 2014

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We present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, n t and n p , respectively. The special case of n t = 1 and n p = 1 corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked exactly once, and form the surfaces of nested tori. We show that for n t ∈ Z + and n p = 1, these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity, and we estimate the soliton lifetime in such a medium as a function of the toroidal winding number.

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“…We are following the method in Ref. [11]. First we consider the case in which the poloidal winding number n p = 1 and the toroidal winding number n t is any positive integer.…”

Section: Stability Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constructing a class of topological solitons in magnetohydrodynamics

et al. 2014

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“…Helicity is a measure of the knottedness of field lines, which is conserved under appropriate conditions, and as such it has been called a "topological invariant" of many flows [5]. These ideas [6][7][8][9] have found applications in areas beyond fluid dynamics, such as DNA biology [10], optics [11] and electromagnetism [12].…”

mentioning

confidence: 99%

Helicity, topology, and Kelvin waves in reconnecting quantum knots

Leoni

Mininni

Brachet³

2016

Phys. Rev. A

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Helicity is a topological invariant that measures the linkage and knottedness of lines, tubes and ribbons. As such, it has found myriads of applications in astrophysics and solar physics, in fluid dynamics, in atmospheric sciences, and in biology. In quantum flows, where topology-changing reconnection events are a staple, helicity appears as a key quantity to study. However, the usual definition of helicity is not well posed in quantum vortices, and its computation based on counting links and crossings of vortex lines can be downright impossible to apply in complex and turbulent scenarios. We present a new definition of helicity which overcomes these problems. With it, we show that only certain reconnection events conserve helicity. In other cases helicity can change abruptly during reconnection. Furthermore, we show that these events can also excite Kelvin waves, which slowly deplete helicity as they interact nonlinearly, thus linking the theory of vortex knots with observations of quantum turbulence. Although helicity is perfectly conserved in barotropic ideal fluids, in real fluids [13,14] and in superfluids [15,16] vortex reconnection events, which alter the topology of the flow, can take place. It is unclear how well helicity is preserved under reconnection. As a few examples, experiments of vortex knots in water have shown that center line helicity remains constant throughout reconnection events [17], while theoretical arguments indicate that writhe (one component of the helicity) should be conserved in anti-parallel reconnection events [18], a fact later confirmed in numerical simulations of a few specific quantum vortex knots [19]. However, numerical studies of Burgers-type vortices indicate that helicity is not conserved [20]. While experiments studying helicity in quantum flows have not been done yet, the recent experimental creation of quantum knots in a Bose-Einstein condensate in the laboratory [21] is a significant step in that direction.Recently, quantum flows have been used as a testbed for many of these ideas [17,19], as vorticity in a quantum flow is concentrated along vortex lines with quantized circulation, and as these vortex lines can reconnect without dissipation. However, the lack of a fluid-like definition of helicity for a quantum flow requires complex topological measurements of the linking and knottedness of vortex lines [17], or artificial filtering of the fields [19] to prevent spurious values of helicity resulting from the singularity near quantum vortices. Moreover, helicity in quantum flows has an interest per se, as reconnection events in superfluid turbulence can excite Kelvin waves [22]. These are helical perturbations that travel along the vortex lines first predicted for classical vortices by Lord Kelvin (see [23]). Kelvin waves are believed to be responsible for the generation of an energy cascade [24,25] leading to Kelvin wave turbulence [26]. Possible links between helicity and the development of Kelvin wave turbulence have remain obscure as a result of the difficulties i...

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“…Henceforth, we will assume a barotropic relation, and discuss the equation of motion in the form of (20); one may generalize the following formulations to a baroclinic fluid by replacing dθ by T dS.…”

Section: Barotropic Fluidmentioning

confidence: 99%

Relativistic helicity and link in Minkowski space-time

Yoshida

Kawazura

Yokoyama

2014

Journal of Mathematical Physics

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A relativistic helicity has been formulated in the four-dimensional Minkowski space-time.Whereas the relativistic distortion of space-time violates the conservation of the conventional helicity, the newly defined relativistic helicity conserves in a barotropic fluid or plasma, dictating a fundamental topological constraint. The relation between the helicity and the vortex-line topology has been delineated by analyzing the linking number of vortex filaments which are singular differential forms representing the pure states of Banach algebra. While the dimension of space-time is four, vortex filaments link, because vorticities are primarily 2-forms and the corresponding 2-chains link in four dimension; the relativistic helicity measures the linking number of vortex filaments that are proper-time cross-sections of the vorticity 2-chains. A thermodynamic force yields an additional term in the vorticity, by which the vortex filaments on a reference-time plane are no longer pure states. However, the vortex filaments on a proper-time plane remain to be pure states, if the thermodynamic force is exact (barotropic), thus, the linking number of vortex filaments conserves.

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Tying Knots in Light Fields

Cited by 177 publications

References 37 publications

Constructing a class of topological solitons in magnetohydrodynamics

Constructing a class of topological solitons in magnetohydrodynamics

Helicity, topology, and Kelvin waves in reconnecting quantum knots

Relativistic helicity and link in Minkowski space-time

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