Abstract

A simple core-shell two-dimensional photonic crystal is studied where the triangular lattice symmetry and the C6 point group symmetry give rich physics in accidental touching points of photonic bands. We systematically evaluate different types of accidental nodal points at the Brillouin zone center for transverse-magnetic harmonic modes when the geometry and permittivity of the core-shell material are continuously tuned. The accidental nodal points can have different dispersions and topological properties (i.e., Berry phases). These accidental nodal points can be the critical states lying between a topological phase and a normal phase of the photonic crystal. They are thus very important for the study of topological photonic states. We show that, without breaking time-reversal symmetry, by tuning the geometry of the core-shell material, a phase transition into the photonic quantum spin Hall insulator can be achieved. Here the “spin” is defined as the orbital angular momentum of a photon. We study the topological phase transition as well as the properties of the edge and bulk states and their application potentials in optics.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Photonic Dirac monopoles and skyrmions: spin-1 quantization [Invited]

Todd Van Mechelen and Zubin Jacob
Opt. Mater. Express 9(1) 95-111 (2019)

Photonics meets topology

Bi-Ye Xie, Hong-Fei Wang, Xue-Yi Zhu, Ming-Hui Lu, Z. D. Wang, and Yan-Feng Chen
Opt. Express 26(19) 24531-24550 (2018)

References

  • View by:
  • |
  • |
  • |

  1. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
    [Crossref] [PubMed]
  2. K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
    [Crossref]
  3. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
    [Crossref]
  4. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
    [Crossref]
  5. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
    [Crossref]
  6. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
    [Crossref]
  7. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
    [Crossref]
  8. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
    [Crossref]
  9. L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
    [Crossref] [PubMed]
  10. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
    [Crossref] [PubMed]
  11. B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
    [Crossref]
  12. W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
    [Crossref] [PubMed]
  13. W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
    [Crossref]
  14. L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
    [Crossref] [PubMed]
  15. H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
    [Crossref]
  16. W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.
  17. L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
    [Crossref]
  18. X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
    [Crossref]
  19. R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
    [Crossref]
  20. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
    [Crossref] [PubMed]
  21. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
    [Crossref] [PubMed]
  22. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
    [Crossref]
  23. K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
    [Crossref]
  24. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2005).
  25. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).
  26. L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
    [Crossref]
  27. J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).
  28. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
    [Crossref]
  29. Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
    [Crossref] [PubMed]
  30. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
    [Crossref]
  31. J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.
  32. C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
    [Crossref]
  33. D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
    [Crossref]
  34. L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.
  35. M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
    [Crossref] [PubMed]
  36. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
    [Crossref] [PubMed]

2016 (2)

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

2015 (2)

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

2014 (6)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

2013 (6)

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

2012 (2)

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
[Crossref] [PubMed]

2011 (3)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

2009 (2)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

2008 (1)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

2007 (3)

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

2004 (1)

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

2002 (1)

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

1998 (1)

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

1980 (1)

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Barkeshli, M.

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

Bazaliy, Ya. B.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Beenakker, C. W. J.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Capasso, F.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

Chan, C. T.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

Chen, H.-Y.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Chen, W.

J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

Chen, W.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

Chen, X.

Chen, X.-D.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Dong, J.-W.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Dorda, G.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Economou, E. N.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Fang, C.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

Fu, L.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

Hafezi, M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Hang, Z. H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

He, W.-Y.

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

Hou, J.-M.

J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

Hsu, C. W.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

Hu, X.

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

Huang, X.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Jian, S.-K.

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

Jiang, J.-H.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Jiang, S.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Joannopoulos, J. D.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Johnson, S. G.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Kane, C. L.

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Klitzing, K. v.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Lai, Y.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Lee, D.-H.

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

Li, Y.

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Lu, L.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

Lubensky, T. C.

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Mei, J.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Mei., J.

Migdall, A.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Mittal, S.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Moroz, A.

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

Nagaosa, N.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

Pepper, M.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Qi, X.-L.

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Ran, L.

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Sakoda, K.

Segev, M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Sepkhanov, R. A.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Soljacic, M.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Stone, A. D.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Taylor, J. M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Tse, W. K.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

van Blaaderen, A.

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

Velikov, K. P.

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

Wang, H.-X.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Wang, L.

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

Wang, Z.

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Wu, L. H.

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

Wu, S.

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Wu, Y.

Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[Crossref] [PubMed]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Xiang, T.

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

Xiao, M.

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

Xu, L.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Yao, H.

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

Ye, D.

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Yu, N.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Zhang, G.-M.

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

Zhang, S.-C.

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

Zhang, Z.-Q.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Zhen, B.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

J. Phys.: Condens. Matter (1)

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Nat. Commun. (1)

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Nat. Mater. (3)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Nat. Photon. (3)

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Nat. Phys. (2)

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

Nature (London) (2)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Opt. Express (2)

Phys. Rev. A (1)

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Phys. Rev. B (4)

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Phys. Rev. Lett. (8)

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Rev. Mod. Phys. (1)

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

Sci. Rep. (1)

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

Science (1)

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Other (5)

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2005).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 (a) Schematic configuration of a triangle PhC using core-shell dielectric materials. The dielectric constant of the core-shell dielectric material and background material are ε2 and ε1, respectively. R1 and R2 are the outer and inner radii of the core-shell cylinder. a⃗1 and a⃗2 are the two basis vectors of the triangular lattice. We set a1 = a2 ≡ 1 as the lattice constant. The inverse structure is defined by exchange of the permittivity ε1 and ε2. (b) A typical band structure of the core-shell triangle lattice PhC with R1 = 0.16a1, R2 = 0.05a1, ε1 = 1 and ε2 = 12. We use the symbols s, p, d and f to label the modes at the Γ point.
Fig. 2
Fig. 2 Phase diagram of the p-d-inversion-induced photonic Z2 topological insulator in the R1-R2 parameter space. The diagonal line represents the homogeneous limit which separates the normal and reversed structures in the upper and lower triangular regions. The lower region is the parameter space for core-shell dielectric cylinder in air background (called as the normal structure), while the upper region is the parameter space for core-shell air cylinder in dielectric background. The dielectric constant for air and dielectric are ε1 = 1 and ε2 = 12, respectively. The contour color represents the value of Δωpd. The red region represents that the p bands are lower than the d bands at the Γ point, whereas the blue region stands for the photonic analog of the Z2 topological insulators with the p bands above the d bands at the Γ point. The p-d bands degeneracy at the Γ point is labeled by the black line. The s-p, p-f, s-d, f -d band inversions are also calculated and plotted. Several points in the phase diagram are labeled, which will be used for the discussions in the main text.
Fig. 3
Fig. 3 Band structures of different kinds of accidental degeneracy. (a) The band structure of the A point in the phase diagram with R1 = 0.1816a1 and R2 = 0. (b) The band structure of the B point in the phase diagram with R1 = 0.17a1 and R2 = 0.0947a1. (c) The band structure of the C point in the phase diagram with R1 = 0.5a1 and R2 = 0.4624a1. (d) The band structure of the D point in phase diagram with R1 = 0.28a1 and R2 = 0.0697a1. (e) The band structure of the E point in the phase diagram with R1 = 0.41a1 and R2 = 0.3844a1. (f) The band structure of the F point in the phase diagram with R1 = 0.45a1 and R2 = 0.2656a1. The geometry of the core-shell material in a unit cell is shown at the left-down corner of each figure. The colored curves labeled with s, p, d, and f represent the modes at the Γ point only (instead of the whole bands). We zoom in the dispersion near the nodal points with a dashed frame.
Fig. 4
Fig. 4 Relationship between ln(ε1/ε2) and the critical inner radius R2 where the double Dirac cone emerges. The outer radius of the hollow cylinder is fixed at R1 = 0.45a1.
Fig. 5
Fig. 5 Topology-induced edge states. (a) The normal band structure of the G point in phase diagram with R1 = 0.40a1 and R2 = 0.26a1. (b) The p-d reversed band structure of the H point in the phase diagram (R1 = 0.45a1 and R2 = 0.32a1). The common complete band gap is marked with the cyan ribbon. (c) The projected band structure of two PhCs with oblique line edge. A and B mostly comprise of the pseudo-spin-up and spin-down edge states, respectively. (d) and (e) are the Ez field pattern of A and B, respectively. The time-averaged Poynting vectors S⃗ = Re[E⃗ × H⃗*]/2 near the boundary (between the two dashed lines) are shown by the black arrows.
Fig. 6
Fig. 6 Photonic band structure and topological edge states. (a) The normal band structure with R1 = 0.45a1, R2 = 0.2a1, ε1 = 1, and ε2 = 9. (b) The p-d reversed band structure with R1 = 0.45a1, R2 = 0.3a1, ε1 = 1, and ε2 = 12. They have a common band gap marked with cyan ribbon. (c) Projected band structure of the two PhCs with a line boundary.
Fig. 7
Fig. 7 Optical property near double Dirac Cone. (a) The band structure near double Dirac cone with R1 = 0.45 and R2 = 0.2656. Four cone-like surfaces touch at ω0 of the Γ point. (b) The iso-frequency surfaces for the frequency of higher (in red) and lower (in blue) than ω0, respectively. In the middle it is the iso-frequency surface of light in air. The refraction law is derived from the conservation of frequency and the wave vector parallel to the interface, kx. Right panel: the property of positive refraction (upper plot) and negative refraction (lower plot).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Δ ω p d = 2 ω d ω p ω d + ω p ,
× 1 ε ( r ) × h n , k ( r ) = ω 2 c 2 h n , k ( r ) ,
n n ( k ) = ω n , 0 2 c 2 δ n n + k P n n u . c . d r ε ( r ) h n , 0 * ( r ) [ k × ( k × h n , 0 ( r ) ) ] ,
P n , n = u . c . d r ε ( r ) [ h n , 0 * ( r ) × ( i × h n , 0 * ( r ) ) + ( i × h n , 0 ( r ) ) × h n , 0 * ( r ) ] .
= ( ω p 2 c 2 0 A k + 0 0 ω p 2 c 2 0 A * k A * k 0 ω 2 d c 2 0 0 A k + 0 ω 2 d c 2 ) ,
n ( ω ) = sin ( θ 1 ) sin ( θ 2 ) = 2 ω 0 | A | c ( ω ω 0 ω ) ,

Metrics