Abstract
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. These littleknown solutions, constructed by Rañada^{1}, are based on the Hopf fibration. Here we analyse their physical properties to investigate how they can be experimentally realized. We study their time evolution and uncover, through a decomposition into a spectrum of spherical harmonics, a remarkably simple representation. 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. 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. 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.
Main
The concept of field lines whose tangents are the electric or magnetic field is typically used to visualize static solutions of Maxwell’s equations. Propagating solutions often have simple fieldline structures and so are not usually described in terms of field lines. In the present work, we study a propagating field whose defining and most striking property is the topological structure of its electric and magnetic field lines.
An intriguing configuration for field lines is to be linked and/or knotted. Two closed field lines c_{1}(τ), c_{2}(τ) are linked if they have nonvanishing Gauss linking integral^{3,4,5,6},
whereas for a single field line c(τ) the selflinking number, L(c,c), is a measure of knottedness. The linking integral L can also be computed visually by projecting the field lines onto a plane and subsequently counting the crossings in an oriented way^{3}. For example, the lines in Fig. 1a have linking number 1, but do not form a knot, whereas the blue and orange field lines in Fig. 4 below are knotted and linked to each other. In the case of magnetic or electric fields, averaging the linking integral over all fieldline pairs together with the selflinking number over all field lines gives rise to the magnetic and electric helicities^{4,5}:
where B:=∇×A and E:=∇×C in free space.
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}. More recently, an approach to knotted classical fields was proposed^{8} and further understood and developed^{9,10}. 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}.
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. The magnetic and Poynting field lines are similarly arranged. This structure is based on the Hopf fibration defined by the Hopf map (see Fig. 1)^{15,16}. Using stereographic projections, h can in turn be expressed as a complex function in (for example ζ(x,y,z,0) or η(x,y,z,0) below) whose lines of constant amplitude and phase are circles, and surfaces of constant amplitude are nested tori.
Electromagnetic fields derived from the Hopf fibration first appeared in ref. 17, and were extended to propagating solutions by Rañada in refs 1, 18, 19. The construction was cast in terms of differential forms, which provide a natural way to map fields between spaces of differing dimensions. The resulting electric and magnetic fields have simple expressions:
where , and x,y,z,t are dimensionless multiples of a length scale a. Since both ∇ η and are perpendicular to lines of constant η, the magnetic field is tangential to lines of constant η. A similar argument holds for the electric field and ζ. The corresponding field lines are shown in Fig. 2.
As a first step in our investigation, we present a numerical study of the evolution of the field lines and energy density, shown in Fig. 2. The initially spherical energy density expands like an ‘opening umbrella’ with a preferred propagation direction (z) while preserving the Hopf structure. The propagation direction is set by the crossproduct of the electric and magnetic n lines (as defined in Fig. 1). The s fibre twists around the centre of energy density. The n fibre cuts through the maximum of energy density and its tangent on the z axis undergoes a rotation with an angle analogous to the Gouy phase shift of Gaussian beam optics^{20}.
To further characterize the physical properties of the field configuration, we compute the full set of conserved quantities that correspond to the known (conformal) symmetries of electromagnetism in free space (see Table 1). Note that all currents, when scaled by the energy density and the scale factor a, are integer multiples of one another; that the fields carry angular momentum along the propagation axis and that the momentum is a fraction of the energy, so the Hopf fields can be transformed via a Lorentz transformation to a rest frame, or to a counterpropagating frame, making them even more beamlike.
Though the linking number is also a conserved quantity for the solution under consideration, it does not correspond to a spacetime symmetry; rather, it is a topological invariant^{4}. Indeed, the linking number is not conserved for a general freespace electromagnetic field, but only for fields that satisfy
In the case of the Hopf fields defined in equation (2), E · B=0 guarantees the conservation of linking number.
For problems with spherical symmetry, a natural basis for representing electromagnetic fields is that of the vector spherical harmonics (VSPHs). Labelled by angularmomentum integers l≥1 and −l≤m≤l, wavevector k and polarization TE/TM (electric/magnetic field transverse to the radial direction), the vector potential A_{l m}(k,r) for the VSPHs is^{21}
where L=−i r×∇, Y_{l m} are spherical harmonics and f_{l}(k r) is a linear combination of the spherical Bessel functions j_{l}(k r), n_{l}(k r), determined by boundary conditions. In free space . A general freespace vector potential A(r,t) can be expressed in the spherical harmonic basis as
We present the decomposition of the Hopf electromagnetic field in this basis (see Supplementary Information, Methods S1), revealing the following remarkably simple structure:
the Hopf field is a superposition of TE and TM vector spherical harmonics corresponding to a single multipole (l=m=1), a relative phase factor i and an energy spectrum
Strikingly, the superposition A^{TE}−i A^{TM} is an eigenstate of the curl operator, that is, it satisfies the eigenvalue equation
Such eigenstates, known as Chandrasekhar–Kendall (CK)^{2} states for constant k, are part of a family of fields known as forcefree fields and are of broad importance in plasma physics and fluid dynamics^{22,23,24}. The Hopf fields are therefore a pulsed version of the CK curl eigenstate fields with energy spectrum ωe^{−ω}.
To understand how such a simple superposition (equation (4)) gives rise to the remarkable fieldline structure of the Hopf field, we begin by studying, in Fig. 3, the field lines of the singlefrequency curl eigenstates, which have the unique property that the electric, magnetic and Afield lines have the same structure up to a rotation (). The A_{1,1}^{TE} and i A_{1,1}^{TM} field lines are symmetric under rotations about the centre axis and separate into sets of nested tori, with each set centred on zeros of the field. The A_{1,1}^{TE} field lines follow one and the i A_{1,1}^{TM} lines the other circumference of each torus. A superposition will therefore have field lines that stay confined and wrap around the tori with linking (winding ratio) that depends on the field strength (Fig. 3b).
To understand the step from the singlefrequency curl eigenstates to a Hopf configuration, it is necessary to take the energy spectrum into consideration. To nest all tori about the same s fibre, the spectrum must ‘eliminate’ all zeros of the radial function, effectively giving rise to a field without oscillations. This is achieved by the spectrum in equation (5). Interestingly, this spectrum is close to one used in research on singlecycle light ‘bullets’^{25} based on the fields of Ziolkowski^{26}, whose defining property is the absence of oscillations.
We now consider how the linking is preserved in time. A curl eigenstate has conserved helicity because its helicity integral is proportional to the norm of the state. A calculation of the helicity integral (equation (1)) for a general field in the CK basis gives
where of equation (3) are the coefficients of the CK eigenstates. Only the second term is time dependent; the absence of ((l,m),(l,−m)) pairs therefore guarantees the conservation of helicity. The conserved part (first term) of the linking number is proportional to the difference between the amplitudes of the + and − CK states.
The understanding gained above suggests a route to generalizing these solutions: taking superpositions of curl states with different values of l and m and a similar energy spectrum. The result for l=2, m=2, is shown in Fig. 4. Not all the field lines are closed and link in the same way, as is the case for the Hopf fields; however, fibres analogous to the s and p fibres of the Hopf knot can be found. These fibres close, follow the energy density and have a fixed linking structure in the form of two intertwined trefoil knots. In addition, we note that by varying the relative strength of the TE and TM components in the singlefrequency building blocks, p A^{TE}+i q A^{TM}, all possible torus knots with winding p,q can be produced at time t=0.
We now turn to the possibility of an experimental realization using laser fields. A simple argument suggests that the relative phase factor of the CK building blocks may be fairly robust: pure TE and TM freespace VSPHs are a simple superposition of outgoing and ingoing VSPHs; their timeaveraged Poynting vector is indeed purely azimuthal. The only way to construct a pure multipole field that propagates along the z axis in free space is by taking a superposition of TE and TM fields with a phase i; therefore, any propagating pure multipole is a Chandrasekhar–Kendall curl eigenstate.
The fact that the Hopf fields are built of CK states with only one value of l and m suggests Laguerre–Gaussian beams^{20} as a good starting point for their production; these are pure angularmomentum eigenstates of the paraxial wave equation and are used as a basis to model laser beams. Though studied extensively, only recently has the relation between strongly focused laser beams and VSPHs received some attention^{27,28}, motivated in part by optical tweezing. Using the code developed in ref. 28 provided to us by the authors, we found that a strongly focused zerothorder Gaussian beam with circular polarization converges towards a pure l=1,m=1 multipole field as the focusing angle increases toward 90^{∘}. This suggests that an experimental implementation of the CK basis states may be simpler than we might expect. To create the full Hopf field we would start with a single or fewcycle pulsed beam of circularly polarized light^{29} and focus it tightly. The pulse shape and spatial profile could be further controlled with a spatial light modulator using holographic techniques^{30}, which have been recently used to produce, for example, pure Airy beams^{31}.
In conclusion, we have investigated the physical properties of an exceptional solution of the chargefree Maxwell equations in which all field lines are linked once with one another. The decomposition into vector spherical harmonics has revealed the relation to eigenstates of the curl operator, led the way to new field configurations with multiple linking and given guidance on how to generate such special solutions in an experiment. Since the class of electromagnetic knots has both beamlike propagation and unique properties that have not been explored in this context, we predict a wide variety of potential applications and theoretical extensions in areas ranging from colloidal and atomic particle trapping to manipulating cold atomic ensembles and from generating solitonlike solutions in nonlinear media to helicity injection for plasma confinement.
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Acknowledgements
We gratefully acknowledge discussions with M. Srednicki, J. Hartle and K. Millett. We thank V. Vitelli, C. Simon and F. Azhar for comments on the manuscript. W.T.M.I. gratefully acknowledges support from the English Speaking Union through a Lindemann Fellowship. D.B. acknowledges support from Marie Curie EXTCT2006042580.
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Irvine, W., Bouwmeester, D. Linked and knotted beams of light. Nature Phys 4, 716–720 (2008). https://doi.org/10.1038/nphys1056
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DOI: https://doi.org/10.1038/nphys1056
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