Photonic topological insulators

Journal name:
Nature Materials
Year published:
Published online


Recent progress in understanding the topological properties of condensed matter has led to the discovery of time-reversal-invariant topological insulators. A remarkable and useful property of these materials is that they support unidirectional spin-polarized propagation at their surfaces. Unfortunately topological insulators are rare among solid-state materials. Using suitably designed electromagnetic media (metamaterials) we theoretically demonstrate a photonic analogue of a topological insulator. We show that metacrystals—superlattices of metamaterials with judiciously designed properties—provide a platform for designing topologically non-trivial photonic states, similar to those that have been identified for condensed-matter topological insulators. The interfaces of the metacrystals support helical edge states that exhibit spin-polarized one-way propagation of photons, robust against disorder. Our results demonstrate the possibility of attaining one-way photon transport without application of external magnetic fields or breaking of time-reversal symmetry. Such spin-polarized one-way transport enables exotic spin-cloaked photon sources that do not obscure each other.

At a glance


  1. Wave propagation in a 2D PTI.
    Figure 1: Wave propagation in a 2D PTI.

    a, Photonic analogue of Kramers partners in a spin-degenerate metamaterial. b, Band structure of a metacrystal comprising a hexagonal lattice of spin-degenerate metamaterials with (dashed lines, χ  =  0.5) and without (solid lines, χ  =  0) optical activity (right inset). The metacrystal becomes a PTI through the opening of the second (topological) bandgap near the K and K′ points of the Brillouin zone for finite optical activity. The left inset shows the Brillouin zone with the K and K′ valleys indicated by blue and red triangles, respectively. c, Eigen-frequency surfaces illustrating degeneracy removal at the K (K′) point for χ≠0. d, The hexagonal lattice of the metacrystals and two possible microscopic structures of its metamaterial constituent rods with desirable bi-anisotropic response. PTI parameters: circular rods of radius r0  =  0.34a0 arranged in a hexagonal lattice with period a0; each rod filled with spin-degenerate metamaterial with , εzz  =  μzz  =  1, and .

  2. One-way spin-polarized transport of photonic edge states.
    Figure 2: One-way spin-polarized transport of photonic edge states.

    a, Dispersion of the spin-up (green) and spin-down (red) helical edge states supported by a bi-anisotropic domain wall. Opaque and transparent bands correspond to two interfaces, with χxy < 0 to χxy > 0 and with χxy > 0 to χxy < 0 transitions, respectively. The blue lines illustrate the super-cell’s bulk photonic states corresponding to different wavenumbers in the direction perpendicular to the interface. b, The absolute value of |ψe±| for right/left-propagating edge states of a bi-anisotropic (χxy < 0 to χxy > 0) domain wall. The enlarged regions show the difference in the temporal evolution between the spin-up ψe+ and spin-down ψemodes. Although the field profiles Re(ψ+) and Re(ψ) are identical, the modes propagate in opposite directions and the power flux (black arrows) inside the rods rotates in opposite directions. Field dynamics can be found in the animations of the ψ± wavefunctions given in the Supplementary Information.

  3. Excitation of surface waves by a point dipole source at the interface between topologically trivial and non-trivial photonic insulators.
    Figure 3: Excitation of surface waves by a point dipole source at the interface between topologically trivial and non-trivial photonic insulators.

    a, Selective excitation of spin-up and spin-down photonic one-way edge states along a straight interface. Bottom metacrystal parameters: the same as in Fig. 1. Top metacrystal: the same as bottom, but all sizes are scaled by the factor η  =  0.76 to ensure the coincidence of photonic bandgaps in both metacrystals. bd, Robustness of the edge modes against different types of defect: sharp bending of the interface (b), a cavity obstacle (c) and a strongly disordered domain in both of the adjacent crystals (d). Colour scale in ad: local field intensity .

  4. Non-obstructing large photon antennas due to spin-cloaking.
    Figure 4: Non-obstructing large photon antennas due to spin-cloaking.

    Spin-cloaked electric-dipole antenna (indicated by the rectangular region) embedded into the cavity between topologically trivial and non-trivial metacrystals. a,b, The spin-polarized one-way transport of the spin-up and spin-down edge modes avoiding a silent dipole antenna placed in the cavity. c,d, Selective directional excitation of these modes by the electric-dipole antenna.


  1. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
  2. Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
  3. Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).
  4. Moore, J. E. & Balents, L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306 (2007).
  5. Roy, R. Z2 classification of quantum spin Hall systems: An approach using time-reversal invariance. Phys. Rev. B 79, 195321 (2009).
  6. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 30453067 (2010).
  7. Qi, X-L. & Zhang, S-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 10571110 (2011).
  8. König, M. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766770 (2007).
  9. Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970974 (2008).
  10. Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nature Phys. 5, 398402 (2009).
  11. Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3, Sb2Te3 with a single Dirac cone on the surface. Nature Phys. 5, 438442 (2009).
  12. Lindner, N. l. H., Refael, G. & Galitski, V. Floquet topological insulators in semiconductor quantum wells. Nature Phys. 7, 490495 (2011).
  13. Haldane, F. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).
  14. Wang, Z., Chong, Y., Joannopoulos, J. & Soljaćić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008).
  15. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907912 (2011).
  16. Smith, D. R., Padilla, W. J., Vier, D. C., Nemat-Nasser, S. C. & Schultz, S. Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84, 41844187 (2000).
  17. Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and negative refractive index. Science 305, 788792 (2004).
  18. Ulf, L. Optical conformal mapping. Science 312, 17771780 (2006).
  19. Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312, 17801782 (2006).
  20. Chen, H., Chan, C. T. & Sheng, P. Transformation optics and metamaterials. Nature Mater. 9, 387396 (2010).
  21. Veselago, V. G. The electrodynamics of substances with simultaneously negative values of permittivity and permeability. Soviet Phys. Usp. 10, 509514 (1968).
  22. Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 39663969 (2000).
  23. Schurig, D. et al. Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977980 (2006).
  24. Raghu, S. & Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008).
  25. Fang, K., Yu, Z. & Fan, S. Microscopic theory of photonic one-way edge mode. Phys. Rev. B 84, 075477 (2011).
  26. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljaćić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772775 (2009).
  27. Poo, Y., Wu, R., Lin, Z., Yang, Y. & Chan, C. T. Experimental realization of self-guiding unidirectional electromagnetic edge states. Phys. Rev. Lett. 106, 093903 (2011).
  28. Zharov, A. A., Shadrivov, I. V. & Kivshar, Y. S. Nonlinear properties of left-handed metamaterials. Phys. Rev. Lett. 91, 037401 (2003).
  29. Klein, M. W., Enkrich, C., Wegener, M. & Linden, S. Second-harmonic generation from magnetic metamaterials. Science 313, 502504 (2006).
  30. Poutrina, E., Huang, D. & Smith, D. R. Analysis of nonlinear electromagnetic metamaterials. New J. Phys. 12, 093010 (2010).
  31. Maciejko, J., Hughes, T. L. & Zhang, S-C. The quantum spin Hall effect. Annu. Rev. Condens. Matter Phys. 2, 3153 (2011).
  32. Kong, J. A. Theorems of bianisotropic media. Proc. IEEE 60, 10361046 (1972).
  33. Serdyukov, A. N., Semchenko, I. V., Tretyakov, S. A. & Sihvola, A. Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Science, 2001).
  34. Pendry, J. B., Holden, A. J., Robbins, D. J. & Stewart, W. J. Magnetism from conductors, and enhanced non-linear phenomena. Microw. Theory Technol. 47, 20752084 (1999).
  35. Shelby, R. A., Smith, D. R. & Schultz, S. Experimental verification of a negative index of refraction. Science 292, 7779 (2001).
  36. Marqués, R., Medina, F. & Rafii-El-Idrissi, R. Role of bianisotropy in negative permeability and left-handed metamaterials. Phys. Rev. B 65, 144440 (2002).
  37. Rill, M. S. et al. Negative-index bianisotropic photonic metamaterial fabricated by direct laser writing and silver shadow evaporation. Opt. Lett. 34, 1921 (2009).
  38. Li, Z., Aydin, K. & Ozbay, E. Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients. Phys. Rev. E 79, 026610 (2009).
  39. Saadoun, M. M. I. & Engheta, N. A reciprocal phase shifter using novel pseudochiral or ω medium. Microw. Opt. Technol. Lett. 5, 184188 (1992).
  40. Tretyakov, S. A., Simovski, C. R. & Hudlička, M. Bianisotropic route to the realization and matching of backward-wave metamaterial slabs. Phys. Rev. B 75, 153104 (2007).
  41. Tretyakov, S. A., Mariotte, F., Simovski, C. R., Kharina, T. G. & Heliot, J-P. Analytical antenna model for chiral scatterers: Comparison with numerical and experimental data. IEEE Trans. Antennas Propag. 44, 10061014 (1996).
  42. Plum, E., Schwanecke, V. A. F., Zheludev, A. S. & Chen, N. I. Giant optical gyrotropy due to electromagnetic coupling. Appl. Phys. Lett. 90, 223113 (2007).
  43. Gansel, J. K. et al. Gold helix photonic metamaterial as broadband circular polarizer. Science 325, 15131515 (2009).
  44. Saenz, E. et al. Modeling of spirals with equal dielectric, magnetic, and chiral susceptibilities. Electromagnetics 28, 476493 (2008).
  45. Urzhumov, Y. A. & Shvets, G. Extreme anisotropy of wave propagation in two-dimensional photonic crystals. Phys. Rev. E 72, 026608 (2005).
  46. Haldane, F. D. M. Model for a quantum Hall effect without landau levels: Condensed-matter realization of the parity anomaly. Phys. Rev. Lett. 61, 20152018 (1988).
  47. Bernevig, B. A., Hughes, T. L. & Zhang, S. C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 17571761 (2006).
  48. Murakami, S., Nagaosa, N. & Zhang, S-C. Dissipationless quantum spin current at room temperature. Science 301, 13481351 (2003).
  49. Sheng, D. N., Weng, Z. Y., Sheng, L. & Haldane, F. D. M. Quantum spin-Hall effect and topologically invariant Chern numbers. Phys. Rev. Lett. 97, 036808 (2006).

Download references

Author information

  1. These authors contributed equally to this work

    • Alexander B. Khanikaev &
    • S. Hossein Mousavi


  1. Department of Physics, The University of Texas at Austin, One University Station, C1500, Austin, Texas 78712, USA

    • Alexander B. Khanikaev,
    • S. Hossein Mousavi,
    • Wang-Kong Tse,
    • Mehdi Kargarian,
    • Allan H. MacDonald &
    • Gennady Shvets


All authors contributed extensively to the work presented in this paper.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (2.07 MB)

    Supplementary Information


  1. Supplementary Information (6.44 MB)

    Supplementary Movie S1

  2. Supplementary Information (6.44 MB)

    Supplementary Movie S2

  3. Supplementary Information (32.0 MB)

    Supplementary Movie S3

  4. Supplementary Information (32.0 MB)

    Supplementary Movie S4

  5. Supplementary Information (32.0 MB)

    Supplementary Movie S5

  6. Supplementary Information (32.0 MB)

    Supplementary Movie S6

  7. Supplementary Information (32.0 MB)

    Supplementary Movie S7

  8. Supplementary Information (32.0 MB)

    Supplementary Movie S8

Additional data