Defect modes in two-dimensional periodic photonic structures have found use in diverse optical devices. For example, photonic crystal cavities confine optical modes to subwavelength volumes and can be used for enhancement of nonlinearity, lasing and cavity quantum electrodynamics. Defect-core photonic crystal fibres allow for supercontinuum generation and endlessly single-mode fibres with large cores. However, these modes are notoriously fragile: small structural change leads to significant detuning of resonance frequency and mode volume. Here, we show that photonic topological crystalline insulator structures can be used to topologically protect the mode frequency at mid-gap and minimize the volume of a photonic defect mode. We experimentally demonstrate this in a femtosecond-laser-written waveguide array by observing the presence of a topological zero mode confined to the corner of the array. The robustness of this mode is guaranteed by a topological invariant that protects zero-dimensional states embedded in a two-dimensional environment—a novel form of topological protection that has not been previously demonstrated.
Subscribe to Journal
Get full journal access for 1 year
only $14.08 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).
Ozawa, T. et al. Topological photonics. Preprint at http://arXiv.org/abs/1802.04173 (2018).
Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).
Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).
Umucalılar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011).
Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).
Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).
Rechtsman, M. C. et al. Photonic Floquet topological insulator. Nature 496, 196–200 (2013).
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).
Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).
Wu, L.-H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).
von Klitzing, K., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
Kane, C. L. & Mele, E. J. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).
Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).
Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–640 (2017).
Shockley, W. On the surface states associated with a periodic potential. Phys. Rev. 56, 317–323 (1939).
Malkova, N., Hromada, I., Wang, X., Bryant, G. & Chen, Z. Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices. Phys. Rev. A 80, 043806 (2009).
Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).
Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131–136 (2001).
Fefferman, C. L., Lee-Thorp, J. P. & Weinstein, M. I. Topologically protected states in one-dimensional continuous systems and Dirac points. Proc. Natl Acad. Sci. USA 111, 8759–8763 (2014).
Poli, C., Bellec, M., Kuhl, U., Mortessagne, F. & Schomerus, H. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6, 6710 (2015).
Slobozhanyuk, A. P., Poddubny, A. N., Miroshnichenko, A. E., Belov, P. A. & Kivshar, Y. S. Subwavelength topological edge states in optically resonant dielectric structures. Phys. Rev. Lett. 114, 123901 (2015).
Blanco-Redondo, A. et al. Topological optical waveguiding in silicon and the transition between topological and trivial defect states. Phys. Rev. Lett. 116, 163901 (2016).
Iadecola, T., Schuster, T. & Chamon, C. Non-Abelian braiding of light. Phys. Rev. Lett. 117, 073901 (2016).
Asbóth, J. K., Oroszlány, L. & Pályi, A. Lecture Notes in Physics Vol. 919 (Springer Verlag, Berlin, 2016).
Mondragon-Shem, I., Hughes, T. L., Song, J. & Prodan, E. Topological criticality in the chiral-symmetric AIII class at strong disorder. Phys. Rev. Lett. 113, 046802 (2014).
Teo, J. C. Y. & Hughes, T. L. Existence of Majorana-fermion bound states on disclinations and the classification of topological crystalline superconductors in two dimensions. Phys. Rev. Lett. 111, 047006 (2013).
Benalcazar, W. A., Teo, J. C. Y. & Hughes, T. L. Classification of two-dimensional topological crystalline superconductors and majorana bound states at disclinations. Phys. Rev. B 89, 224503 (2014).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).
Englund, D. et al. Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal. Phys. Rev. Lett. 95, 013904 (2005).
Painter, O. et al. Two-dimensional photonic band-gap defect mode laser. Science 284, 1819–1821 (1999).
Russell, P. Photonic crystal fibers. Science 299, 358–362 (2003).
Liberal, I., Mahmoud, A. M. & Engheta, N. Geometry-invariant resonant cavities. Nat. Commun. 7, 10989 (2016).
Poli, C., Schomerus, H., Bellec, M., Kuhl, U. & Mortessagne, F. Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials. 2D Mater. 4, 025008 (2017).
Szameit, A. et al. Discrete nonlinear localization in femtosecond laser written waveguides in fused silica. Opt. Express 13, 10552–10557 (2005).
Kariyado, T. & Hu, X. Topological states characterized by mirror winding numbers in graphene with bond modulation. Sci. Rep. 7, 16515 (2017).
Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997).
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors. AIP Conf. Proc. 1134, 10–21 (2009).
Teo, J. C. Y. & Kane, C. L. Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B 82, 115120 (2010).
Keil, R. et al. Universal sign control of coupling in tight-binding lattices. Phys. Rev. Lett. 116, 213901 (2016).
Stützer, S. et al. Experimental realization of a topological Anderson insulator. In 2015 Conference on Lasers and Electro-Optics (CLEO) 1–2 (IEEE, 2015).
W.A.B. and T.L.H. are supported by the Office of Naval Research Young Investigator Program Award N00014-15-1-2383. M.C.R. acknowledges support from the National Science Foundation under grant number ECCS-1509546 and the Penn State Materials Research Science and Engineering Center, DMR-1420620 as well as from the Alfred P. Sloan Foundation under fellowship number FG-2016-6418. K.P.C. acknowledges the National Science Foundation under grant numbers ECCS-1509199 and DMS-1620218.
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary discussion; Supplementary Figures 1–8; Supplementary references 1–3. [In this file initially published, some characters did not display properly; this file has now been replaced.]
Excitation of zero mode.
Beating of two modes as wavelength is swept.
Injection of light at the corner.
Response due to detuning of refractive index at corner.
Response due to detuning of wavelength at corner.
About this article
Cite this article
Noh, J., Benalcazar, W.A., Huang, S. et al. Topological protection of photonic mid-gap defect modes. Nature Photon 12, 408–415 (2018). https://doi.org/10.1038/s41566-018-0179-3
Physical Review Letters (2020)
Symmetry-protected hierarchy of anomalous multipole topological band gaps in nonsymmorphic metacrystals
Nature Communications (2020)
Optics Communications (2020)
Optics Letters (2020)
Nature Communications (2020)