Abstract

We developed a selection rule for Dirac-like points in two-dimensional dielectric photonic crystals. The rule is derived from a perturbation theory and states that a non-zero, mode-coupling integral between the degenerate Bloch states guarantees a Dirac-like point, regardless of the type of the degeneracy. In fact, the selection rule can also be determined from the symmetry of the Bloch states even without computing the integral. Thus, the existence of Dirac-like points can be quickly and conclusively predicted for various photonic crystals independent of wave polarization, lattice structure, and composition.

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  1. A. H. Castro Neto, 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. 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]
  3. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78(3), 033834 (2008).
    [CrossRef]
  4. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75(6), 063813 (2007).
    [CrossRef]
  5. R. A. Sepkhanov and C. W. J. Beenakker, “Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure,” Opt. Commun.281(20), 5267–5270 (2008).
    [CrossRef]
  6. 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]
  7. 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]
  8. Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett.36(13), 2513–2515 (2011).
    [CrossRef] [PubMed]
  9. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405(14), 2990–2995 (2010).
    [CrossRef]
  10. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
    [CrossRef] [PubMed]
  11. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80(15), 155103 (2009).
    [CrossRef]
  12. D. Torrent and J. Sánchez-Dehesa, “Acoustic analogue of graphene: observation of Dirac cones in acoustic surface waves,” Phys. Rev. Lett.108(17), 174301 (2012).
    [CrossRef] [PubMed]
  13. L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett.86(4), 47008 (2009).
    [CrossRef]
  14. 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]
  15. F. M. Liu, X. Q. Huang, and C. T. Chan, “Dirac cones at k→=0in acoustic crystals and zero refractive index acoustic materials,” Appl. Phys. Lett.100(7), 071911 (2012).
    [CrossRef]
  16. F. M. Liu, Y. Lai, X. Q. Huang, and C. T. Chan, “Dirac cones atk→=0in photonic crystals,” Phys. Rev. B84(22), 224113 (2011).
    [CrossRef]
  17. 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. B86(3), 035141 (2012).
    [CrossRef]
  18. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18(26), 27371–27386 (2010).
    [CrossRef] [PubMed]
  19. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20(4), 3898–3917 (2012).
    [CrossRef] [PubMed]
  20. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express20(9), 9925–9939 (2012).
    [CrossRef] [PubMed]
  21. J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (Addison-Wesley, 1970).
  22. P. M. Hui, W. M. Lee, and N. F. Johnson, “Theory of scalar wave propagation in periodic composites: ak→⋅p→, ” Solid State Commun.91(1), 65–69 (1994).
    [CrossRef]
  23. B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter12(34), R435–R461 (2000).
    [CrossRef]
  24. M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer-Verlag, 2008).

2012 (5)

D. Torrent and J. Sánchez-Dehesa, “Acoustic analogue of graphene: observation of Dirac cones in acoustic surface waves,” Phys. Rev. Lett.108(17), 174301 (2012).
[CrossRef] [PubMed]

F. M. Liu, X. Q. Huang, and C. T. Chan, “Dirac cones at k→=0in acoustic crystals and zero refractive index acoustic materials,” Appl. Phys. Lett.100(7), 071911 (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. B86(3), 035141 (2012).
[CrossRef]

K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20(4), 3898–3917 (2012).
[CrossRef] [PubMed]

K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express20(9), 9925–9939 (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. M. Liu, Y. Lai, X. Q. Huang, and C. T. Chan, “Dirac cones atk→=0in photonic crystals,” Phys. Rev. B84(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 (2)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405(14), 2990–2995 (2010).
[CrossRef]

K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18(26), 27371–27386 (2010).
[CrossRef] [PubMed]

2009 (3)

A. H. Castro Neto, 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]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80(15), 155103 (2009).
[CrossRef]

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett.86(4), 47008 (2009).
[CrossRef]

2008 (5)

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. A78(3), 033834 (2008).
[CrossRef]

R. A. Sepkhanov and C. W. J. Beenakker, “Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure,” Opt. Commun.281(20), 5267–5270 (2008).
[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]

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]

2007 (2)

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75(6), 063813 (2007).
[CrossRef]

2000 (1)

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter12(34), R435–R461 (2000).
[CrossRef]

1994 (1)

P. M. Hui, W. M. Lee, and N. F. Johnson, “Theory of scalar wave propagation in periodic composites: ak→⋅p→, ” Solid State Commun.91(1), 65–69 (1994).
[CrossRef]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

Bazaliy, Y. B.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75(6), 063813 (2007).
[CrossRef]

Beenakker, C. W. J.

R. A. Sepkhanov and C. W. J. Beenakker, “Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure,” Opt. Commun.281(20), 5267–5270 (2008).
[CrossRef]

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75(6), 063813 (2007).
[CrossRef]

Castro Neto, A. H.

A. H. Castro Neto, 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. M. Liu, X. Q. Huang, and C. T. Chan, “Dirac cones at k→=0in acoustic crystals and zero refractive index acoustic materials,” Appl. Phys. Lett.100(7), 071911 (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. B86(3), 035141 (2012).
[CrossRef]

F. M. Liu, Y. Lai, X. Q. Huang, and C. T. Chan, “Dirac cones atk→=0in photonic crystals,” Phys. Rev. B84(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]

Christodoulides, D. N.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405(14), 2990–2995 (2010).
[CrossRef]

Dong, J.

Foreman, B. A.

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter12(34), R435–R461 (2000).
[CrossRef]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

Geim, A. K.

A. H. Castro Neto, 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.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78(3), 033834 (2008).
[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]

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]

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]

Huang, X. Q.

F. M. Liu, X. Q. Huang, and C. T. Chan, “Dirac cones at k→=0in acoustic crystals and zero refractive index acoustic materials,” Appl. Phys. Lett.100(7), 071911 (2012).
[CrossRef]

F. M. Liu, Y. Lai, X. Q. Huang, and C. T. Chan, “Dirac cones atk→=0in photonic crystals,” Phys. Rev. B84(22), 224113 (2011).
[CrossRef]

Hui, P. M.

P. M. Hui, W. M. Lee, and N. F. Johnson, “Theory of scalar wave propagation in periodic composites: ak→⋅p→, ” Solid State Commun.91(1), 65–69 (1994).
[CrossRef]

Johnson, N. F.

P. M. Hui, W. M. Lee, and N. F. Johnson, “Theory of scalar wave propagation in periodic composites: ak→⋅p→, ” Solid State Commun.91(1), 65–69 (1994).
[CrossRef]

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405(14), 2990–2995 (2010).
[CrossRef]

Lai, Y.

F. M. Liu, Y. Lai, X. Q. Huang, and C. T. Chan, “Dirac cones atk→=0in photonic crystals,” Phys. Rev. B84(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]

Lee, W. M.

P. M. Hui, W. M. Lee, and N. F. Johnson, “Theory of scalar wave propagation in periodic composites: ak→⋅p→, ” Solid State Commun.91(1), 65–69 (1994).
[CrossRef]

Liang, Q.

Liu, F. M.

F. M. Liu, X. Q. Huang, and C. T. Chan, “Dirac cones at k→=0in acoustic crystals and zero refractive index acoustic materials,” Appl. Phys. Lett.100(7), 071911 (2012).
[CrossRef]

F. M. Liu, Y. Lai, X. Q. Huang, and C. T. Chan, “Dirac cones atk→=0in photonic crystals,” Phys. Rev. B84(22), 224113 (2011).
[CrossRef]

Liu, Z.

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]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

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. B86(3), 035141 (2012).
[CrossRef]

Novoselov, K. S.

A. H. Castro Neto, 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. B80(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. B80(15), 155103 (2009).
[CrossRef]

Peleg, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

Peres, N. M. R.

A. H. Castro Neto, 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]

Raghu, S.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78(3), 033834 (2008).
[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]

Sakoda, K.

Sánchez-Dehesa, J.

D. Torrent and J. Sánchez-Dehesa, “Acoustic analogue of graphene: observation of Dirac cones in acoustic surface waves,” Phys. Rev. Lett.108(17), 174301 (2012).
[CrossRef] [PubMed]

Segev, M.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

Sepkhanov, R. A.

R. A. Sepkhanov and C. W. J. Beenakker, “Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure,” Opt. Commun.281(20), 5267–5270 (2008).
[CrossRef]

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75(6), 063813 (2007).
[CrossRef]

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405(14), 2990–2995 (2010).
[CrossRef]

Torrent, D.

D. Torrent and J. Sánchez-Dehesa, “Acoustic analogue of graphene: observation of Dirac cones in acoustic surface waves,” Phys. Rev. Lett.108(17), 174301 (2012).
[CrossRef] [PubMed]

Wang, L.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett.86(4), 47008 (2009).
[CrossRef]

Wang, Z.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett.86(4), 47008 (2009).
[CrossRef]

Wu, Y.

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. B86(3), 035141 (2012).
[CrossRef]

Yan, Y.

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. B86(3), 035141 (2012).
[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]

Zhou, H.-F.

Zhu, S.-Y.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett.86(4), 47008 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

F. M. Liu, X. Q. Huang, and C. T. Chan, “Dirac cones at k→=0in acoustic crystals and zero refractive index acoustic materials,” Appl. Phys. Lett.100(7), 071911 (2012).
[CrossRef]

Europhys. Lett. (1)

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett.86(4), 47008 (2009).
[CrossRef]

J. Phys. Condens. Matter (1)

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter12(34), R435–R461 (2000).
[CrossRef]

Nat. Mater. (1)

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]

Opt. Commun. (1)

R. A. Sepkhanov and C. W. J. Beenakker, “Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure,” Opt. Commun.281(20), 5267–5270 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (2)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78(3), 033834 (2008).
[CrossRef]

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75(6), 063813 (2007).
[CrossRef]

Phys. Rev. B (3)

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80(15), 155103 (2009).
[CrossRef]

F. M. Liu, Y. Lai, X. Q. Huang, and C. T. Chan, “Dirac cones atk→=0in photonic crystals,” Phys. Rev. B84(22), 224113 (2011).
[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. B86(3), 035141 (2012).
[CrossRef]

Phys. Rev. Lett. (5)

D. Torrent and J. Sánchez-Dehesa, “Acoustic analogue of graphene: observation of Dirac cones in acoustic surface waves,” Phys. Rev. Lett.108(17), 174301 (2012).
[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]

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]

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]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett.98(10), 103901 (2007).
[CrossRef] [PubMed]

Physica B (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405(14), 2990–2995 (2010).
[CrossRef]

Rev. Mod. Phys. (1)

A. H. Castro Neto, 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]

Solid State Commun. (1)

P. M. Hui, W. M. Lee, and N. F. Johnson, “Theory of scalar wave propagation in periodic composites: ak→⋅p→, ” Solid State Commun.91(1), 65–69 (1994).
[CrossRef]

Other (2)

M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer-Verlag, 2008).

J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (Addison-Wesley, 1970).

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Figures (3)

Fig. 1
Fig. 1

(a) The band structure for TE-polarized electromagnetic waves propagating in a 2D PC consisting of a triangular lattice of cylindrical air boreholes in Si background ( ε r =12 ). The radius of the air holes is R=0.4429a , where a is the lattice constant. (b)-(d) Enlarged views of the band structures around the Γ point for different radii of air holes, with (b) R=0.4429a , (c) R=0.44a , and (d) R=0.445a . Blue dotted lines show the bands obtained by using COMSOL Multiphysics and the red solid curves give the results predicted by our perturbation method. (e)-(g) The field distributions of three degenerate Bloch states at the Γ point. Dark red and dark blue denote the positive and negative maxima of the magnetic field, H z , respectively. Arrows indicate the in-plane electric field vector, E , whose magnitude is proportional to the length of the arrows.

Fig. 2
Fig. 2

(a) The band structure for TE-polarized electromagnetic waves propagating in a 2D PC consisting of a triangular lattice of cylindrical air boreholes in Si background ( ε r =12 ). The radius of the air holes is R=0.3244a , where a is the lattice constant. (b)-(d) Enlarged views of the band structures around the Γ point for different radii of air holes, with (b) R=0.3244a , (c) R=0.33a , and (d) R=0.31a . (e)-(g) The field distributions of three degenerate Bloch states at the Γ point. Dark red and dark blue denote the positive and negative maxima of the magnetic field, H z , respectively. Arrows indicate the in-plane electric field vector, E , whose magnitude is proportional to the length of the arrows.

Fig. 3
Fig. 3

(a) The band structure for TM-polarized electromagnetic waves propagating in a 2D PC consisting of a triangular lattice of Si cylinders ( ε r =12 ) in air. The radius of the Si cylinders is R=0.4344a , where a is the lattice constant. (b) An enlarged view of the band structures around the Γ point. (c)-(e) The field distributions of three degenerate Bloch states at the Γ point. Dark red and dark blue denote the positive and negative maxima of the electric field, E z , respectively. Arrows indicate the in-plane magnetic field vector, H , whose magnitude is proportional to the length of the arrows.

Equations (26)

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( 1 μ r ( r ) E z )= ω 2 c 0 2 ε r ( r ) E z ,
Ψ n k ( r )= u n k ( r ) e i k r ,
(2π) 2 Ω unit cell Ψ l k * ( r ) ε r ( r ) Ψ j k ( r )d r = δ lj ,
Ψ n k ( r )= j A nj ( k ) e i( k k 0 ) r Ψ j k 0 ( r ).
j [ ω j0 2 ω n k 2 c 0 2 δ lj P lj ( k ) ] A nj ( k ) =0,
P lj ( k )=( k k 0 ) p lj ( k k 0 ) 2 q lj ,
p lj =i (2π) 2 Ω unit cell Ψ l k 0 * ( r )[ 2 Ψ j k 0 ( r ) μ r ( r ) +( 1 μ r ( r ) ) Ψ j k 0 ( r ) ] d r ,
q lj = (2π) 2 Ω unit cell Ψ l k 0 * ( r ) 1 μ r ( r ) Ψ j k 0 ( r )d r .
det| H ω n k 2 - ω j0 2 c 0 2 I |=0,
H lj = P lj .
Δ ω β Δk = γ β c 0 +O(Δk) β=1,2,3,...,s ,
φ ˜ l ( r )= C l φ l ( r ), (l=1,2,3),
1 a 2 unit cell φ ˜ l * ( r ) μ r ( r ) φ ˜ j ( r )d r = δ lj ,
p lj = i a 2 unit cell φ ˜ l * ( r ) [ 2 φ ˜ j ( r ) ε r ( r ) +( 1 ε r ( r ) ) φ ˜ j ( r ) ]d r .
1 ε r ={ 1 ε 1 r>R 1 ε 2 r<R ,
( 1 ε r ( r ) )= e ^ r ( 1 ε 1 1 ε 2 )δ(rR),
k = k x e x + k y e y =kcosβ e x +ksinβ e y ,
P lj ( k )= k p lj k 2 q lj =k p lj k 2 q lj ,
p lj =i cosβ a 2 [ unit cell φ ˜ l * ( r ) 2 ε r ( r ) φ ˜ j ( r ) x d r +( 1 ε 1 1 ε 2 ) L: x 2 + y 2 = R 2 φ ˜ l * ( r ) φ ˜ j ( r ) x x 2 + y 2 dL ], +i sinβ a 2 [ unit cell φ ˜ l * ( r ) 2 ε r ( r ) φ ˜ j ( r ) y d r +( 1 ε 1 1 ε 2 ) L: x 2 + y 2 = R 2 φ ˜ l * ( r ) φ ˜ j ( r ) y x 2 + y 2 dL ]
q lj = 1 a 2 unit cell φ ˜ l * ( r ) 1 ε r ( r ) φ ˜ j ( r )d r .
det| x ikd ikf ikd x 0 ikf 0 x |=0,
x= ( ω 0 2 ω β k 2 ) / c 0 2 d= 1.548 a cosβ+ 2.591 a sinβ f= 2.591 a cosβ 1.548 a sinβ.
x 1 =0 x 2,3 =± 3.019 a k,
ω β k = ω 0 ω β k = ω 0 1± 3.019 c 0 2 k ω 0 2 a ω 0 ± 3.019 c 0 2 k 2 ω 0 a ,
{ ω β k ω 0 Δk =0 ω β k ω 0 Δk =± 3.019 c 0 2 2 ω 0 a ,
{ γ β =0 γ β =± 3.019 c 0 2 ω 0 a =±0.340 ,

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