Knotting fractional-order knots with the polarization state of light

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The fundamental polarization singularities of monochromatic light are normally associated with invariance under coordinated rotations: symmetry operations that rotate the spatial dependence of an electromagnetic field by an angle θ and its polarization by a multiple γθ of that angle. These symmetries are generated by mixed angular momenta of the form Jγ = L + γS, and they generally induce Möbius-strip topologies, with the coordination parameter γ restricted to integer and half-integer values. In this work we construct beams of light that are invariant under coordinated rotations for arbitrary rational γ, by exploiting the higher internal symmetry of ‘bicircular’ superpositions of counter-rotating circularly polarized beams at different frequencies. We show that these beams have the topology of a torus knot, which reflects the subgroup generated by the torus-knot angular momentum Jγ, and we characterize the resulting optical polarization singularity using third- and higher-order field moment tensors, which we experimentally observe using nonlinear polarization tomography.

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Fig. 1: Coordinated-rotation invariance and torus-knot beam topology.
Fig. 2: Possible topologies of torus-knot beams.
Fig. 3: Experimental configuration and results.

Data availability

The data used to generate the experimental figures in this paper have been publicly archived in the Zenodo repository at

Code availability

The code used to generate the theoretical figures, and the scripts used to process the experimental data, have been publicly archived in the Zenodo repository at

Change history

  • 21 August 2019

    When this Article was originally published, the file for Supplementary Data 1 was missing; it has now been added.


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The authors thank M. Maffei and I. Freund for helpful conversations, and X. Menino for 3D-printing assistance. E.P. acknowledges Cellex-ICFO-MPQ fellowship funding. E.P., M.L. and A.C. acknowledge funding from the Spanish Ministry MINECO (National Plan 15 Grant: FISICATEAMO no. FIS2016-79508-P, SEVERO OCHOA no. SEV-2015-0522, FPI), the European Social Fund, Fundació Cellex, Generalitat de Catalunya (AGAUR grant no. 2017 SGR 1341 and CERCA/Program), ERC AdG OSYRIS, EU FETPRO QUIC and the National Science Centre, Poland-Symfonia grant no. 2016/20/W/ST4/00314. V.V.-H. acknowledges financial support from Secretaría de Ciencia, Tecnología e Innovación de la Ciudad de México. J.P.T. acknowledges support from Generalitat de Catalunya (Program ICREA Academia). G.J.M. was supported by the Secretaria d’Universitats i Recerca del Departament d’Economia i Coneixement de la Generalitat de Catalunya, as well as the European Social Fund—FEDER. A.P. acknowledges funding from Comunidad de Madrid through TALENTO grant ref. 2017-T1/IND-5432. A.C. acknowledges financial support from the ERC Synergy Grant UQUAM, the SFB FoQuS (FWF project no. F4016-N23), the UAB Talent Research programme and from the Spanish Ministry of Economy and Competitiveness under contract no. FIS2017-86530-P.

Author information




E.P. conceived the project and developed the theory. G.J.M., V.V.-H., E.P. and J.P.T. designed the experiment. G.J.M. and V.V.-H. conducted the experiment. A.P., A.C. and M.L. assisted with the theory. E.P. wrote the manuscript, with assistance from V.V.-H. for the Methods section. J.P.T. supervised the experimental work and M.L. oversaw the theory development. All authors contributed to the scientific discussion.

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Correspondence to Emilio Pisanty.

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Supplementary Information

This file contains more information about the work and Supplementary Figs. 1–11.

Supplementary Data 1

3D-printable models of Fig. 1d and e.

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Pisanty, E., Machado, G.J., Vicuña-Hernández, V. et al. Knotting fractional-order knots with the polarization state of light. Nat. Photonics 13, 569–574 (2019).

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