Abstract

By exploiting the accidental degeneracy of the doubly-degenerate dipolar and quadrupolar modes, we show that a two-dimensional dielectric photonic crystal (PC) can exhibit the double Dirac cone dispersion relation at the Γ point. Using a perturbation method and group theory, we demonstrate that the double cone is composed of two identical and overlapping Dirac cones with predictable linear slopes, and the linearity of the dispersion is guaranteed by the spatial symmetry of the Bloch eigenstates. Numerical simulations including wave-front shaping, unidirectional transmission and perfect tunneling show that the corresponding PC structure can be characterized by a zero effective refractive index.

© 2015 Optical Society of America

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  1. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
    [Crossref]
  2. R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
    [Crossref]
  3. M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
    [Crossref]
  4. Y. Wu, “A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal,” Opt. Express 22(2), 1906–1917 (2014).
    [Crossref] [PubMed]
  5. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
    [Crossref]
  6. S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
    [Crossref] [PubMed]
  7. J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
    [PubMed]
  8. F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
    [Crossref] [PubMed]
  9. R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
    [Crossref]
  10. S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
    [Crossref] [PubMed]
  11. S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
    [Crossref]
  12. 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]
  13. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
    [Crossref]
  14. 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]
  15. X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
    [Crossref] [PubMed]
  16. X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
    [Crossref] [PubMed]
  17. Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36(13), 2513–2515 (2011).
    [Crossref] [PubMed]
  18. 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 496(7444), 196–200 (2013).
    [Crossref] [PubMed]
  19. 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 (2012).
    [Crossref] [PubMed]
  20. M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
    [Crossref] [PubMed]
  21. G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
    [Crossref] [PubMed]
  22. D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
    [Crossref] [PubMed]
  23. D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
    [Crossref]
  24. J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
    [Crossref]
  25. 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]
  26. F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
    [Crossref]
  27. F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
    [Crossref]
  28. P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
    [Crossref]
  29. N. Mattiucci, M. J. Bloemer, and G. D’Aguanno, “Phase-matched second harmonic generation at the Dirac point of a 2-D photonic crystal,” Opt. Express 22(6), 6381–6390 (2014).
    [Crossref] [PubMed]
  30. X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
    [Crossref]
  31. Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
    [Crossref]
  32. Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
    [Crossref]
  33. 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]
  34. 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]
  35. K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
    [Crossref] [PubMed]
  36. K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic-crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
    [Crossref]
  37. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012).
    [Crossref] [PubMed]
  38. Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
    [PubMed]
  39. Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
    [Crossref] [PubMed]
  40. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
    [Crossref] [PubMed]
  41. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
    [Crossref]
  42. V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
    [Crossref] [PubMed]
  43. Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
    [Crossref]
  44. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
    [Crossref]
  45. F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
    [Crossref]

2014 (14)

R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
[Crossref]

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic-crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
[Crossref]

Y. Wu, “A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal,” Opt. Express 22(2), 1906–1917 (2014).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

N. Mattiucci, M. J. Bloemer, and G. D’Aguanno, “Phase-matched second harmonic generation at the Dirac point of a 2-D photonic crystal,” Opt. Express 22(6), 6381–6390 (2014).
[Crossref] [PubMed]

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

2013 (6)

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]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

2012 (7)

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 (2012).
[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]

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[Crossref]

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012).
[Crossref] [PubMed]

K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
[Crossref] [PubMed]

2011 (3)

Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36(13), 2513–2515 (2011).
[Crossref] [PubMed]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[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]

2010 (4)

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[Crossref] [PubMed]

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

2009 (4)

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]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

2008 (5)

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

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]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

2007 (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

2006 (1)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

Alù, A.

R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Barnes, W. L.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Beenakker, C. W. J.

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

Bellec, M.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

Bittner, S.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Bloemer, M. J.

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Chan, C. T.

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[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]

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (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]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Chang, K.

Chen, H.

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

Chen, L.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

Chen, X.

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]

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

Chen, Y.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Chen, Y.-F.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Chen, Z.-G.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

D’Aguanno, G.

de Dood, M. J. A.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[Crossref] [PubMed]

Deng, F.

Deng, F. S.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Dietz, B.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Dong, J.

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Engheta, N.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

Fleury, R.

R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
[Crossref]

Fu, L.

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

Fu, Y.

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

Geim, A. K.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[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]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

Halterman, K.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

Han, D.

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Hang, Z. H.

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[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]

Hao, J.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

He, C.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Hess, O.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Hu, N.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Huang, X.

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[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]

Jiang, H.

Jiang, H. T.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Joannopoulos, J. D.

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[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 (2012).
[Crossref] [PubMed]

Ke, M.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[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 (2012).
[Crossref] [PubMed]

Kuhl, U.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

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]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Li, Y.

Li, Y. H.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Liang, Q.

Liu, F.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

Liu, Z.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

Lu, J.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Lu, L.

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

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

Lu, M.-H.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Ma, Y. G.

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[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 (2012).
[Crossref] [PubMed]

Mariani, E.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Mattiucci, N.

Mayou, D.

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

McPhedran, R. C.

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

Mei, J.

Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

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]

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

Meylan, M. H.

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

Min, R.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Miski-Oglu, M.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Montambaux, G.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

Mortessagne, F.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

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 (2012).
[Crossref] [PubMed]

Nguyen, V. C.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

Ni, X.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Nilsson, J.

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

Nolte, S.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Novoselov, K. S.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[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]

Ong, C. K.

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (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]

Oria Iriarte, P.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Pei, L.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Peng, P.

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

Peres, N. M. R.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Qiu, C.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Qiu, M.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

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]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

Rechtsman, M. C.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Richter, A.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Sakoda, K.

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Sánchez-Dehesa, J.

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

Schäfer, F.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Sepkhanov, R. A.

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

Shi, Y.

Shi, Y. L.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[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 (2012).
[Crossref] [PubMed]

Silveirinha, M.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Skirlo, S. A.

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

Smith, M. J. A.

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

Soljacic, M.

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

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

Sun, X.-C.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Sun, Y.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

Szameit, A.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Torrent, D.

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (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 (2012).
[Crossref] [PubMed]

Wang, F.

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[Crossref]

Wang, P.

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

Wang, X.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

Wang, Z.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Wei, W.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Weick, G.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Woollacott, C.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Wu, Y.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

Y. Wu, “A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal,” Opt. Express 22(2), 1906–1917 (2014).
[Crossref] [PubMed]

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]

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

Xu, L.

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

Xu, S.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Xue, R.

Yan, C.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Yan, W.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

Yan, Y.

Ye, Y.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Zandbergen, S. R.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[Crossref] [PubMed]

Zeuner, J. M.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Zhang, X.

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

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]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

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]

Zheng, L.-Y.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Zi, J.

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Appl. Phys. Lett. (5)

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

Europhys. Lett. (2)

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[Crossref]

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

J. Appl. Phys. (1)

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Nat. Mater. (2)

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 (2012).
[Crossref] [PubMed]

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. Photonics (1)

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

Nature (1)

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 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. A (2)

K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic-crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
[Crossref]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

Phys. Rev. B (10)

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]

R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
[Crossref]

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[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]

Phys. Rev. Lett. (10)

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

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]

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[Crossref] [PubMed]

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Sci Rep (1)

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Waves Random Complex Media (1)

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Band structure of a two-dimensional PC consisting of a triangular array of dielectric cylindrical shells ( ε r =12 and μ r =1 ) embedded in an air host, whose structure is schematically shown in the center of (a). The outer and inner radii of dielectric shells are R 1 =0.45a and R 2 =0.2656a , respectively. Here, a is the lattice constant. A four-fold degenerate state is created at Point “A”. (b) Band structure for R 1 =0.45a and R 2 =0.3a . The four-fold degenerate state splits into two doubly-degenerate states. (c)-(f) Electric field distributions of the four degenerate eigenstates at Point “A”. The color patterns represent the electric field, whose positive and negative maxima are represented by dark red and dark blue, respectively. Arrows show the in-plane magnetic field, whose magnitude is proportional to the length of the arrows. The field patterns in (c) and (d) are dipolar modes, and those in (e) and (f) are quadrupolar modes.
Fig. 2
Fig. 2 (a) Enlarged view of band structure around Point “A”. Blue dots and red curves represent the results obtained by using COMSOL Multiphysics and predicted by the k p perturbation method, respectively. (b) Three-dimensional dispersion surfaces near Point “A”. Clearly, two identical and overlapping Dirac cones compose a double Dirac cone.
Fig. 3
Fig. 3 Electric field distributions produced by a Gaussian beam driven (a) at a frequency below the double Dirac point ω=0.3336( 2π c 0 /a ) , and (b) exactly at the double Dirac point frequency ω 0 =0.4044( 2π c 0 /a ) . Perfect-matched-layer boundary condition is used on the upper and lower walls of the waveguide channel. White arrows represent the direction of the incident waves. Panel (a) shows that the phase front is still curved at the exit port, and panel (b) shows that the incident phase front is re-shaped to a planar one.
Fig. 4
Fig. 4 Electric field distributions under the excitation of a plane wave at the double Dirac point frequency ω 0 =0.4044( 2π c 0 /a ) . (a) The plane wave is normally incident on the PC prism from the bottom. The impinging wave transmits through the prism and exits from the upper surface along the normal direction. (b) The plane wave is incident on the PC prism from the top. Obviously, the incident wave is not allowed to pass through the prism. Perfect-matched-layer boundary condition is used on the left-side and right-side walls. White arrows in (a) & (b) represent the direction of the incident plane waves. Black arrows in (a) denote the incident and refractive directions at the upper surface.
Fig. 5
Fig. 5 (a) Electric field distributions when a plane wave passes through an empty waveguide channel (incident from the left-side). Perfect-magnetic-conductor boundary condition is used on the upper and lower walls of the waveguide channel. H 2 / H 1 =0.2 , and the working frequency is at the double Dirac point ω 0 =0.4044( 2π c 0 /a ) . Most of the incident wave energy is reflected back and the planar wave-front is significantly disturbed. (b) The same as (a), except that the waveguide is filled with the PC structure. The plane wave squeezes through the narrow junction with little distortion and no phase variation is observed inside the narrow junction. Nevertheless, only 18% of the incident wave energy can pass through the narrow junction. (c) The same as (b), except that the cylindrical shells in the first/last rows are truncated with an appropriate cutting parameter L=1.866 R 1 , where perfect tunneling effect can be seen. The inset of (c) shows a schematic view of the cylindrical shell in the first row after truncation.

Equations (6)

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( 1 μ r ( r ) ψ n k ( r ))= ω 2 c 0 2 ε r ( r ) ψ n k ( r ),
det| H ω k 2 ω 0 2 c 0 2 I |=0,
H lj =( k k 0 ) p lj ,
p lj =i (2π) 2 Ω unit cell φ l * ( r )[ 2 φ j ( r ) μ r ( r ) +( 1 μ r ( r ) ) φ j ( r ) ] d r ,
Δ ω k = ω k ω 0 = γ β Δk,
n eff sinθ= n 1 sin0=0.

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