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Reconstructing the topology of optical polarization knots

Nature Physics volume 14, pages 1079–1082 (2018)Cite this article

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Abstract

Knots are topological structures describing how a looped thread can be arranged in space. Although most familiar as knotted material filaments, it is also possible to create knots in singular structures within three-dimensional physical fields such as fluid vortices1 and the nulls of optical fields2,3,4. Here we produce, in the transverse polarization profile of optical beams, knotted lines of circular transverse polarization. We generate and observe both simple torus knots and links as well as the topologically more complicated figure-eight knot. The presence of these knotted polarization singularities endows a nontrivial topological structure on the entire three-dimensional propagating wavefield. In particular, the contours of constant polarization azimuth form Seifert surfaces of high genus5, which we are able to resolve experimentally in a process we call seifertometry. This analysis reveals a level of topological complexity, present in all experimentally generated polarization fields, that goes beyond the conventional reconstruction of polarization singularity lines.

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Fig. 1: Schematic of the experimental apparatus used to generate and characterize polarization singularity knots.
Fig. 2: Topological traits of torus structures.
Fig. 3: Topological characterization and seifertometry of an optical figure-eight knot.

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Acknowledgements

The authors thank F. Bouchard for his advice on using the SLM and P. Banzer for fruitful discussions. This work was supported by Canada Research Chair (CRC) and Canada Foundation for Innovation (CFI). R.F. acknowledges the financial support of the Banting postdoctoral fellowship of the NSERC. E.K. and R.W.B. acknowledge the support of the Canada Excellence Research Chairs (CERC) Program. D.S, A.J.T and M.R.D were supported by the Leverhulme Trust Research Programme grant no. RP2013-K-009, SPOCK: Scientific Properties of Complex Knots.

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Authors and Affiliations

  1. Department of Physics, University of Ottawa, Ottawa Ontario, Canada

    Hugo Larocque, Dominic Mortimer, Robert Fickler, Robert W. Boyd & Ebrahim Karimi

  2. H. H. Wills Physics Laboratory, University of Bristol, Bristol, UK

    Danica Sugic, Alexander J. Taylor & Mark R. Dennis

  3. Institute of Optics, University of Rochester, Rochester, New York, NY, USA

    Robert W. Boyd

  4. School of Physics and Astronomy, University of Birmingham, Birmingham, UK

    Mark R. Dennis

  5. Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

    Ebrahim Karimi

Authors
  1. Hugo Larocque
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  2. Danica Sugic
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  3. Dominic Mortimer
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  4. Alexander J. Taylor
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  5. Robert Fickler
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  7. Mark R. Dennis
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  8. Ebrahim Karimi
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Contributions

H.L. and D.S. designed the holograms used to generate the knots. H.L., D.M. and R.F. performed the experiment. D.S., A.J.T. and M.R.D. performed the topological analysis. R.W.B, M.R.D. and E.K. supervised all aspects of the project. All authors discussed the results and contributed to the text of the manuscript.

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Correspondence to Ebrahim Karimi.

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Larocque, H., Sugic, D., Mortimer, D. et al. Reconstructing the topology of optical polarization knots. Nature Phys 14, 1079–1082 (2018). https://doi.org/10.1038/s41567-018-0229-2

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