Knotting fractional-order knots with the polarization state of light

This article has been updated

Abstract

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

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 https://doi.org/10.5281/zenodo.2649391.

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 https://doi.org/10.5281/zenodo.2649391.

Change history

  • 21 August 2019

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

References

  1. 1.

    Torres, J. P. & Torner, L. (eds) Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley, 2011).

  2. 2.

    Andrews, D. L. & Babiker, M. (eds) The Angular Momentum of Light (Cambridge University Press, 2012).

  3. 3.

    Gbur, G. Singular Optics (CRC Press, 2016).

  4. 4.

    Rubinsztein-Dunlop, H. et al. Roadmap on structured light. J. Opt. 19, 013001 (2017).

    ADS  Article  Google Scholar 

  5. 5.

    Nye, J. F. & Berry, M. V. Dislocations in wave trains. Proc. R. Soc. A 336, 165–190 (1974).

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photon. 6, 488–491 (2012).

    ADS  Article  Google Scholar 

  7. 7.

    Fürhapter, S., Jesacher, A., Bernet, S. & Ritsch-Marte, M. Spiral phase contrast imaging in microscopy. Opt. Express 13, 689–694 (2005).

    ADS  Article  Google Scholar 

  8. 8.

    Garcés-Chávez, V., Volke-Sepulveda, K., Chávez-Cerda, S., Sibbett, W. & Dholakia, K. Transfer of orbital angular momentum to an optically trapped low-index particle. Phys. Rev. A 66, 063402 (2002).

    ADS  Article  Google Scholar 

  9. 9.

    Padgett, M. & Bowman, R. Tweezers with a twist. Nat. Photon 5, 343–348 (2011).

    ADS  Article  Google Scholar 

  10. 10.

    Hernández-García, C. et al. Generation and applications of extreme-ultraviolet vortices. Photonics 4, 28 (2017).

    Article  Google Scholar 

  11. 11.

    Barnett, S. M. et al. On the natures of the spin and orbital parts of optical angular momentum. J. Opt. 18, 064004 (2016).

    ADS  Article  Google Scholar 

  12. 12.

    Molina-Terriza, G., Torres, J. P. & Torner, L. Twisted photons. Nat. Phys. 3, 305–310 (2007).

    Article  Google Scholar 

  13. 13.

    Van Enk, S. & Nienhuis, G. Spin and orbital angular momentum of photons. Eur. Phys. Lett. 25, 497–501 (1994).

    ADS  Article  Google Scholar 

  14. 14.

    Kedia, H., Bialynicki-Birula, I., Peralta-Salas, D. & Irvine, W. T. M. Tying knots in light fields. Phys. Rev. Lett. 111, 150404 (2013).

    ADS  Article  Google Scholar 

  15. 15.

    Leach, J., Dennis, M. R., Courtial, J. & Padgett, M. J. Laser beams: knotted threads of darkness. Nature 432, 165 (2004).

    ADS  Article  Google Scholar 

  16. 16.

    Dennis, M. R., King, R. P., Jack, B., O’Holleran, K. & Padgett, M. J. Isolated optical vortex knots. Nat. Phys. 6, 118–121 (2010).

    Article  Google Scholar 

  17. 17.

    Sugic, D. & Dennis, M. R. Knotted hopfion in tightly focused light. In Proceedings of the 4th International Conference on Optical Angular Momentum (ed. Marrucci, L.) 137 (Jean Gilder, 2017); http://www.jeangilder.it/icoam2017/abstract-book/

  18. 18.

    Larocque, H. et al. Reconstructing the topology of optical polarization knots. Nat. Phys. 14, 1079–1082 (2018).

    Article  Google Scholar 

  19. 19.

    Freund, I. Cones, spirals and Möbius strips, in elliptically polarized light. Opt. Commun. 249, 7–22 (2005).

    ADS  Article  Google Scholar 

  20. 20.

    Nye, J. F. & Hajnal, J. V. The wave structure of monochromatic electromagnetic radiation. Proc. R. Soc. Lond. A 409, 21–36 (1987).

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    Dennis, M. R. Polarization singularities in paraxial vector fields: morphology and statistics. Opt. Commun. 213, 201–221 (2002).

    ADS  Article  Google Scholar 

  22. 22.

    Bauer, T. et al. Multi-twist polarization ribbon topologies in highly-confined optical fields. Preprint at http://arxiv.org/abs/1901.11337 (2019).

  23. 23.

    Freund, I. Optical Möbius strips in three-dimensional ellipse fields: I. Lines of circular polarization. Opt. Commun. 283, 1–15 (2010).

    ADS  Article  Google Scholar 

  24. 24.

    Freund, I. Optical Möbius strips in three-dimensional ellipse fields: II. Lines of circular polarization. Opt. Commun. 283, 16–28 (2010).

    ADS  Article  Google Scholar 

  25. 25.

    Bauer, T. et al. Observation of optical polarization Möbius strips. Science 347, 964–966 (2015).

    ADS  Article  Google Scholar 

  26. 26.

    Bauer, T., Neugebauer, M., Leuchs, G. & Banzer, P. Optical polarization Möbius strips and points of purely transverse spin density. Phys. Rev. Lett. 117, 013601 (2016).

    ADS  Article  Google Scholar 

  27. 27.

    Galvez, E. J. et al. Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams. Sci. Rep. 7, 13653 (2017).

    ADS  Article  Google Scholar 

  28. 28.

    Garcia-Etxarri, A. Optical polarization Möbius strips on all-dielectric optical scatterers. ACS Photonics 4, 1159–1164 (2017).

    Article  Google Scholar 

  29. 29.

    Cardano, F. & Marrucci, L. Spin–orbit photonics. Nat. Photon 9, 776–778 (2015).

    ADS  Article  Google Scholar 

  30. 30.

    Bliokh, K. Y., Rodríguez-Fortuño, F. J., Nori, F. & Zayats, A. V. Spin–orbit interactions of light. Nat. Photon. 9, 796–808 (2015).

    ADS  Article  Google Scholar 

  31. 31.

    Bliokh, K. Y. Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium. J. Opt. A Pure Appl. Opt. 11, 094009 (2009).

    ADS  Article  Google Scholar 

  32. 32.

    Ballantine, K. E., Donegan, J. F. & Eastham, P. R. There are many ways to spin a photon: half-quantization of a total optical angular momentum. Sci. Adv. 2, e1501748 (2016).

    ADS  Article  Google Scholar 

  33. 33.

    Leach, J. et al. Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon. Phys. Rev. Lett. 92, 013601 (2004).

    ADS  Article  Google Scholar 

  34. 34.

    Galvez, E. J., Coyle, L. E., Johnson, E. & Reschovsky, B. J. Interferometric measurement of the helical mode of a single photon. New J. Phys. 13, 053017 (2011).

    ADS  Article  Google Scholar 

  35. 35.

    Kessler, D. A. & Freund, I. Lissajous singularities. Opt. Lett. 28, 111–113 (2003).

    ADS  Article  Google Scholar 

  36. 36.

    Freund, I. Bichromatic optical Lissajous fields. Opt. Commun. 226, 351–376 (2003).

    ADS  Article  Google Scholar 

  37. 37.

    Fleischer, A., Kfir, O., Diskin, T., Sidorenko, P. & Cohen, O. Spin angular momentum and tunable polarization in high-harmonic generation. Nat. Photon. 8, 543–549 (2014).

    ADS  Article  Google Scholar 

  38. 38.

    Freund, I. Polychromatic polarization singularities. Opt. Lett. 28, 2150–2152 (2003).

    ADS  Article  Google Scholar 

  39. 39.

    Freund, I. Polarization critical points in polychromatic optical fields. Opt. Commun. 227, 61–71 (2003).

    ADS  Article  Google Scholar 

  40. 40.

    Yan, H. & Lü, B. Dynamical evolution of Lissajous singularities in free-space propagation. Phys. Lett. A 374, 3695–3700 (2010).

    ADS  Article  Google Scholar 

  41. 41.

    Haitao, C., Gao, Z. & Wang, W. Propagation of the Lissajous singularity dipole emergent from non-paraxial polychromatic beams. Opt. Commun. 393, 17–24 (2017).

    ADS  Article  Google Scholar 

  42. 42.

    Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, 2004).

  43. 43.

    Rolfsen, D. Knots and Links 17–18 (American Mathematical Society, 2003).

  44. 44.

    Götte, J. B. et al. Light beams with fractional orbital angular momentum and their vortex structure. Opt. Express 16, 993–1006 (2008).

    ADS  Article  Google Scholar 

  45. 45.

    Freund, I. Coherency matrix description of optical polarization singularities. J. Opt. A Pure Appl. Opt. 6, S229 (2004).

    ADS  Article  Google Scholar 

  46. 46.

    Samim, M., Krouglov, S. & Barzda, V. Nonlinear Stokes–Mueller polarimetry. Phys. Rev. A 93, 013847 (2016).

    ADS  Article  Google Scholar 

  47. 47.

    Maucher, F., Skupin, S., Gardiner, S. A. & Hughes, I. G. Creating complex optical longitudinal polarization structures. Phys. Rev. Lett. 120, 163903 (2018).

    ADS  Article  Google Scholar 

  48. 48.

    Wilczek, F. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982).

    ADS  MathSciNet  Article  Google Scholar 

  49. 49.

    Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984).

    ADS  Article  Google Scholar 

  50. 50.

    Lin, Y.-J., Jiménez-García, K. & Spielman, I. B. Spin–orbit-coupled Bose–Einstein condensates. Nature 471, 83–86 (2011).

    ADS  Article  Google Scholar 

  51. 51.

    Maucher, F., Gardiner, S. A. & Hughes, I. G. Excitation of knotted vortex lines in matter waves. New J. Phys. 18, 063016 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  52. 52.

    Bloembergen, N. Conservation laws in nonlinear optics. J. Opt. Soc. Am. 70, 1429–1436 (1980).

    ADS  Article  Google Scholar 

  53. 53.

    Hickstein, D. D. et al. Non-collinear generation of angularly isolated circularly polarized high harmonics. Nat. Photon. 9, 743–750 (2015).

    ADS  Article  Google Scholar 

  54. 54.

    Pisanty, E. et al. Conservation of torus-knot angular momentum in high-order harmonic generation. Phys. Rev. Lett. 122, 203201 (2019).

    ADS  Article  Google Scholar 

  55. 55.

    Pisanty, E. LISSAFIRE: Lissajous-figure reconstruction for nonlinear polarization tomography of bichromatic fields https://github.com/episanty/LISSAFIRE (GitHub); v1.0.2 https://doi.org/10.5281/zenodo.2649390 (Zenodo, 2019).

  56. 56.

    Pisanty, E. et al. Code and data for ‘Knotting fractional-order knots with the polarization state of light’ https://doi.org/10.5281/zenodo.2649391 (Zenodo, 2019).

Download references

Acknowledgements

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

Affiliations

Authors

Contributions

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.

Corresponding author

Correspondence to Emilio Pisanty.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1038/s41566-019-0450-2

Download citation

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing