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Reconstructing the topology of optical polarization knots

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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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.

Author information

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.

Competing interests

The authors declare no competing interests.

Correspondence to Ebrahim Karimi.

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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.