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Linked and knotted beams of light

Nature Physics volume 4, pages 716–720 (2008)Cite this article

An Erratum to this article was published on 01 October 2008

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 little-known solutions, constructed by Rañada1, 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 eigenstates2, 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.

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Figure 1: Construction of the Hopf fibration.
Figure 2: Time evolution of the field lines and energy density of the Hopf knot.
Figure 3: Field-line structure of single-frequency CK curl eigenstates.
Figure 4: A generalization of the Hopf fields on the basis of the VSPHs, obtained using l=2,m=2 multipole fields and the spectrum of equation (5).

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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 EXT-CT-2006-042580.

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Authors and Affiliations

  1. Department of Physics, Center for Soft Condensed Matter Research, New York University, New York 10003, USA

    William T. M. Irvine

  2. Department of Physics, University of California, Santa Barbara, California 93106, USA

    William T. M. Irvine & Dirk Bouwmeester

  3. Huygens Laboratory, Leiden University, 2300 RA Leiden, PO Box 9504, The Netherlands

    Dirk Bouwmeester

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  1. William T. M. Irvine
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  2. Dirk Bouwmeester
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Correspondence to William T. M. Irvine.

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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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