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Corner states of light in photonic waveguides

Abstract

The recently established paradigm of higher-order topological states of matter has shown that not only edge and surface states1,2 but also states localized to corners, can have robust and exotic properties3,4,5,6,7,8,9. Here we report on the experimental realization of novel corner states made out of visible light in three-dimensional photonic structures inscribed in glass samples using femtosecond laser technology10,11. By creating and analysing waveguide arrays, which form two-dimensional breathing kagome lattices in various sample geometries, we establish this as a platform for corner states exhibiting a remarkable degree of flexibility and control. In each sample geometry we measure eigenmodes that are localized at the corners in a finite frequency range, in complete analogy with a theoretical model of the breathing kagome7,8,9,12,13,14. Here, measurements reveal that light can be ‘fractionalized,’ corresponding to simultaneous localization to each corner of a triangular sample, even in the presence of defects.

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Fig. 1: Experimental set-up and breathing kagome lattice.
Fig. 2: Observation of light output from the output facet of rhombic and triangular lattices of waveguide arrays.
Fig. 3: Observation of corner states in the rhombic lattice of a waveguide array (d1 = 12 μm and d2 = 7 μm) with edges consisting of 11 waveguides.
Fig. 4: Observation of the ‘fractionalized’ corner states in a triangular lattice of a waveguide array with d1 = 11 μm and d2 = 6 μm with edges consisting of six waveguides.
Fig. 5: Observation of the ‘fractionalized’ corner states in a triangular lattice with a defect in a waveguide array with d1 = 11 μm and d2 = 6 μm with edges consisting of six waveguides.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  2. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  ADS  Google Scholar 

  3. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  4. Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F. & Brouwer, P. W. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett. 119, 246401 (2017).

    Article  ADS  Google Scholar 

  5. Song, Z., Fang, Z. & Fang, C. (d − 2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017).

    Article  ADS  Google Scholar 

  6. Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).

    Article  ADS  Google Scholar 

  7. Ezawa, M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Phys. Rev. Lett. 120, 026801 (2018).

    Article  ADS  Google Scholar 

  8. Kunst, F. K., van Miert, G. & Bergholtz, E. J. Lattice models with exactly solvable topological hinge and corner states. Phys. Rev. B 97, 241405(R) (2018).

    Article  ADS  Google Scholar 

  9. Kunst, F. K., van Miert, G. & Bergholtz, E. J. Boundaries of boundaries: a systematic approach to lattice models with solvable boundary states of arbitrary codimension. Phys. Rev. B 99, 085426 (2019).

    Article  ADS  Google Scholar 

  10. Davis, K. M., Miura, K., Sugimoto, N. & Hirao, K. Writing waveguides in glass with a femtosecond laser. Opt. Lett. 21, 1729–1731 (1996).

    Article  ADS  Google Scholar 

  11. Chen, G. Y. et al. Femtosecond-laser-written microstructured waveguides in BK7 glass. Sci. Rep. 8, 10377 (2018).

    Article  ADS  Google Scholar 

  12. Xu, Y., Xue, R. & Wan, S. Topological corner states on kagome lattice based chiral higher-order topological insulator. Preprint at https://arxiv.org/abs/1711.09202 (2017).

  13. Ezawa, M. Higher-order topological electric circuits and topological corner resonance on the breathing kagome and pyrochlore lattices. Phys. Rev. B 98, 201402(R) (2018).

    Article  ADS  Google Scholar 

  14. Araki, H., Mizoguchi, T. & Hatsugai, Y. Phase diagram of a disordered higher-order topological insulator: a machine learning study. Phys. Rev. B 99, 085406 (2019).

    Article  ADS  Google Scholar 

  15. Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).

    Article  ADS  Google Scholar 

  16. Imhof, S. et al. Topolectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).

    Article  Google Scholar 

  17. Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).

    Article  ADS  Google Scholar 

  18. Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).

    Article  ADS  Google Scholar 

  19. Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. Nat. Mater. 18, 108–112 (2019).

    Article  ADS  Google Scholar 

  20. Ni, X., Weiner, M., Alu, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).

    Article  ADS  Google Scholar 

  21. Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    Article  ADS  Google Scholar 

  22. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    Article  ADS  Google Scholar 

  23. Maczewsky, L. J., Zeuner, J. M., Nolte, S. & Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).

    Article  ADS  Google Scholar 

  24. Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).

    Article  ADS  Google Scholar 

  25. Mukherjee, S. et al. Observation of a localized flat-band state in a photonic Lieb lattice. Phys. Rev. Lett. 114, 245504 (2015).

    Article  ADS  Google Scholar 

  26. Vicencio, R. A. et al. Observation of localized states in Lieb photonic lattices. Phys. Rev. Lett. 114, 245503 (2015).

    Article  ADS  Google Scholar 

  27. Xie, B.-Y. et al. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals. Phys. Rev. Lett. 122, 233903 (2019).

    Article  ADS  Google Scholar 

  28. Chen, X.-D. et al. Direct observation of corner states in second-order topological photonic crystal slabs. Phys. Rev. Lett. 122, 233902 (2019).

    Article  ADS  Google Scholar 

  29. Mittal, S. et al. Photonic quadrupole topological phases. Nat. Photon. https://doi.org/10.1038/s41566-019-0452-0 (2019).

  30. Ota, Y. et al. Photonic crystal nanocavity based on a topological corner state. Optica 6, 786–789 (2019).

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the Swedish Research Council (VR) and the Knut and Alice Wallenberg Foundation. E.J.B. is a Wallenberg Academy Fellow.

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Contributions

E.J.B. initiated the research. F.K.K. and E.J.B. derived the theoretical results. A.E.H., A.M., G.A. and M.B. designed and carried out the experiment and performed the data analysis. M.B. supervised the experimental part. E.J.B. and F.K.K. wrote the main text. M.B. and A.E.H. wrote the experimental part. All authors discussed the results and contributed to the final version of the manuscript.

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Correspondence to Emil J. Bergholtz.

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Supplementary Information

Supplementary discussion and derivations, Figs. 1–11 and refs. 1–8.

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El Hassan, A., Kunst, F.K., Moritz, A. et al. Corner states of light in photonic waveguides. Nat. Photonics 13, 697–700 (2019). https://doi.org/10.1038/s41566-019-0519-y

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