Abstract

It is shown by analytical calculation based on the tight-binding approximation that the isotropic Dirac cone in the Brillouin zone center can be created in two- and three-dimensional periodic metamaterials by accidental degeneracy of two modes. In the case of two dimensions, the combination of a doubly degenerate E mode and a non-degenerate A1 mode of the square lattice of the C4v symmetry is examined. For three dimensions, the combination of a triply degenerate T1u mode and a non-degenerate A1g mode of the cubic lattice of the Oh symmetry is examined. The secular equation of the electromagnetic field is derived and solved with detailed analysis of electromagnetic transfer integrals by group theory. This is the first theoretical prediction of the presence of the Dirac cone in the three-dimensional periodic structure.

© 2012 OSA

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn. Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
  4. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).
  5. K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. 54, 271–317 (2010).
    [CrossRef]
  6. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  7. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  8. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [CrossRef] [PubMed]
  9. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  10. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
    [CrossRef] [PubMed]
  11. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  12. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008).
    [CrossRef]
  13. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley,2006).
  14. C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Int. Symp. Antennas Propag. Dig. 2, 412–415 (2002).
  15. A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. 5, 34–50 (2004).
    [CrossRef]
  16. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Components Lett. 14, 68–70 (2004).
    [CrossRef]
  17. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18, 27371–27386 (2010).
    [CrossRef]
  18. K. Sakoda and H.-F. Zhou, “Analytical study of degenerate metamaterial steerable antennas,” Opt. Express 19, 13899–13921 (2011).
    [CrossRef] [PubMed]
  19. 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, 013904 (2008).
    [CrossRef] [PubMed]
  20. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
    [CrossRef]
  21. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
    [CrossRef]
  22. X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
    [CrossRef] [PubMed]
  23. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
    [CrossRef]
  24. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
    [CrossRef]
  25. M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  26. 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, 582–586 (2011).
    [CrossRef] [PubMed]
  27. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).
    [CrossRef]

2011 (2)

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, 582–586 (2011).
[CrossRef] [PubMed]

K. Sakoda and H.-F. Zhou, “Analytical study of degenerate metamaterial steerable antennas,” Opt. Express 19, 13899–13921 (2011).
[CrossRef] [PubMed]

2010 (3)

K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18, 27371–27386 (2010).
[CrossRef]

K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. 54, 271–317 (2010).
[CrossRef]

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[CrossRef]

2009 (1)

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

2008 (3)

X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (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, 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, 033834 (2008).
[CrossRef]

2007 (1)

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

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

2004 (2)

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. 5, 34–50 (2004).
[CrossRef]

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Components Lett. 14, 68–70 (2004).
[CrossRef]

2002 (2)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Int. Symp. Antennas Propag. Dig. 2, 412–415 (2002).

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

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. A 75, 063813 (2007).
[CrossRef]

Beenakker, C. W. J.

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

Caloz, C.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Components Lett. 14, 68–70 (2004).
[CrossRef]

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. 5, 34–50 (2004).
[CrossRef]

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Int. Symp. Antennas Propag. Dig. 2, 412–415 (2002).

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley,2006).

Chan, C. T.

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, 582–586 (2011).
[CrossRef] [PubMed]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[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 B 405, 2990–2995 (2010).
[CrossRef]

Grzegorczyk, T. M.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008).
[CrossRef]

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 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, 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, 582–586 (2011).
[CrossRef] [PubMed]

Haus, J. W.

K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. 54, 271–317 (2010).
[CrossRef]

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, 582–586 (2011).
[CrossRef] [PubMed]

Inui, T.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).
[CrossRef]

Ito, T.

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Int. Symp. Antennas Propag. Dig. 2, 412–415 (2002).

Itoh, T.

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. 5, 34–50 (2004).
[CrossRef]

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Components Lett. 14, 68–70 (2004).
[CrossRef]

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley,2006).

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn. Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[CrossRef]

Lai, A.

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. 5, 34–50 (2004).
[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, 582–586 (2011).
[CrossRef] [PubMed]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn. Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

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, 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, 155103 (2009).
[CrossRef]

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[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, 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, 033834 (2008).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008).
[CrossRef]

Sakoda, K.

Sanada, A.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Components Lett. 14, 68–70 (2004).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Sepkhanov, R. A.

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

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn. Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Zhang, X.

X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[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, 582–586 (2011).
[CrossRef] [PubMed]

Zhou, H.-F.

IEEE Int. Symp. Antennas Propag. Dig. (1)

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Int. Symp. Antennas Propag. Dig. 2, 412–415 (2002).

IEEE Microw. Mag. (1)

A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. 5, 34–50 (2004).
[CrossRef]

IEEE Microw. Wireless Components Lett. (1)

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Components Lett. 14, 68–70 (2004).
[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, 582–586 (2011).
[CrossRef] [PubMed]

Opt. Express (2)

Phys. Rev. A (2)

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

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[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. B 80, 155103 (2009).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Phys. Rev. Lett. (5)

X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (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, 013904 (2008).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

The linear dispersion in the zone center realized by two modes that are symmetric with each other about the origin of k.

Fig. 2
Fig. 2

Left: Metallic unit structure with the C4v (regular square) symmetry. Right: Square array of the unit metallic structure on a uniform dielectric-slab waveguide with a back electrode.

Fig. 3
Fig. 3

Isotropic Dirac cone in two dimensions realized by accidental degeneracy of a doubly degenerate E mode and a non-degenerate A1 mode. The dashed square is the third parabolic band. The origin of the frequency ω is shifted to the degenerate frequency ωΓ.

Fig. 4
Fig. 4

Left: Metallic unit structure with the Oh (regular cube) symmetry. Right: Simple-cubic array of the unit metallic structures.

Fig. 5
Fig. 5

Symmetry operations for the Oh point group

Tables (1)

Tables Icon

Table 1 Character table for the A1g and T1u modes of the Oh point group

Equations (149)

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E = c p ,
ω ω 0 + h ¯ k 2 2 m * ,
ω k = ω k .
ω k = ω 0 + c 2 k 2 + c 4 k 4 +
𝒧 H ( r , t ) × [ 1 ε ( r ) × H ( r , t ) ] = 1 c 2 2 t 2 H ( r , t ) ,
× [ 1 ε s ( r ) × H ( 1 , 2 ) ( r ) ] = ω 1 2 c 2 H ( 1 , 2 ) ( r ) ,
× [ 1 ε s ( r ) × H ( 0 ) ( r ) ] = ω 0 2 c 2 H ( 0 ) ( r ) ,
V d r H ( i ) * ( r ) H ( j ) ( r ) = V δ i j ,
B = ( B 00 B 01 B 02 B 10 B 11 0 B 20 0 B 22 ) .
B 00 = ω 0 2 c 2 + M 0 + 2 M 0 ( cos k x a + cos k y a ) ,
B 11 = ω 1 2 c 2 + M 1 + 2 M 1 cos k x a + 2 M 1 cos k y a ,
B 22 = ω 1 2 c 2 + M 1 + 2 M 1 cos k y a + 2 M 1 cos k x a ,
B 01 = B 10 * = 2 i M 2 sin k x a ,
B 02 = B 20 * = 2 i M 2 sin k y a ,
ω k = { ω Γ ± | M 2 | a c 2 k / ω Γ M a 2 c 2 k 2 / 6 ω Γ , ω Γ M a 2 c 2 k 2 / 6 ω Γ ,
B = ( B 00 B 01 B 02 B 03 B 10 B 11 0 0 B 20 0 B 22 0 B 30 0 0 B 33 ) ,
B 00 = ω 0 2 c 2 + M 0 + 2 M 0 ( cos k x a + cos k y a + cos k z a ) ,
B 11 = ω 1 2 c 2 + M 1 + 2 M 1 cos k x a + 2 M 1 ( cos k y a + cos k z a ) ,
B 22 = ω 1 2 c 2 + M 1 + 2 M 1 cos k y a + 2 M 1 ( cos k z a + cos k x a ) ,
B 33 = ω 1 2 c 2 + M 1 + 2 M 1 cos k z a + 2 M 1 ( cos k x a + cos k y a ) ,
B 01 = B 10 * = 2 i M 2 sin k x a ,
B 02 = B 20 * = 2 i M 2 sin k y a ,
B 03 = B 30 * = 2 i M 2 sin k z a .
ω k = { ω Γ ± | M 2 | a c 2 k / ω Γ M a 2 c 2 k 2 / 8 ω Γ , ω Γ M a 2 c 2 k 2 / 8 ω Γ ( double root ) ,
L l m ( i j ) 1 V V d r H ( i ) * ( r ) 𝒧 H ( j ) ( r r l m ) ,
r l m = l ( a 0 0 ) + m ( 0 a 0 ) ,
L 0 , 0 ( 11 ) = L 0 , 0 ( 22 ) ω 1 2 c 2 + M 1 ,
L ± 1 , 0 ( 11 ) = L 0 , ± 1 ( 22 ) M 1 ,
L 0 , ± 1 ( 11 ) = L ± 1 , 0 ( 22 ) M 1 ,
L 0 , 0 ( 00 ) ω 0 2 c 2 + M 0 ,
L ± 1 , 0 ( 00 ) = L 0 , ± 1 ( 00 ) M 0 ,
± L ± 1 , 0 ( 01 ) = L ± 1 , 0 ( 10 ) * = ± L 0 , ± 1 ( 02 ) = L 0 , ± 1 ( 20 ) * M 2 ,
H k ( r ) = 1 V l , m e i k r l m i = 0 2 A i H ( i ) ( r r l m ) .
𝒧 H k ( r ) = ω k 2 c 2 H k ( r ) ,
| B ω k 2 c 2 I | = 0 ,
B i j = l m e i a ( l k x + m k y ) L l m ( i j ) .
ξ = ω k 2 c 2 , ξ 0 = ω 0 2 c 2 , ξ 1 = ω 1 2 c 2 .
ξ 3 + b 2 ξ 2 + b 1 ξ + b 0 = 0 ,
b 2 = ( B 00 + B 11 + B 22 ) = { ξ 0 + 2 ξ 1 + M 0 + 2 M 1 + 2 ( M 0 + M 1 + M 1 ) ( cos k x a + cos k y a ) } ,
b 1 = B 00 B 11 + B 11 B 22 + B 22 B 00 | B 01 | 2 | B 12 | 2 | B 20 | 2 = ( ξ 1 + M 1 ) ( 2 ξ 0 + ξ 1 + 2 M 0 + M 1 ) + 2 { ( ξ 1 + M 1 ) ( 2 M 0 + M 1 + M 1 ) + ( ξ 0 + M 0 ) ( M 1 + M 1 ) } ( cos k x a + cos k y a ) + 4 { M 1 M 1 + M 0 ( M 1 + M 1 ) } ( cos 2 k x a + cos 2 k y a ) + 4 { ( M 1 2 + M 1 2 ) + 2 M 0 ( M 1 + M 1 ) } cos k x a cos k y a 4 | M 2 | 2 ( sin 2 k x a + sin 2 k y a ) ,
b 0 = 2 Re { B 01 B 12 B 20 } B 00 B 11 B 22 + ( B 00 | B 12 | 2 + B 11 | B 20 | 2 + B 22 | B 01 | 2 ) = { ( ξ 0 + M 0 ) ( ξ 1 + M 1 ) 2 + 2 { M 0 ( ξ 1 + M 1 ) 2 + ( ξ 0 + M 0 ) ( ξ 1 + M 1 ) ( M 1 + M 1 ) } ( cos k x a + cos k y a ) + 4 { ( ξ 0 + M 0 ) M 1 M 1 + M 0 ( ξ 1 + M 1 ) ( M 1 + M 1 ) } ( cos 2 k x a + cos 2 k y a ) + 4 { ( ξ 0 + M 0 ) ( M 1 2 + M 1 2 ) + 2 M 0 ( ξ 1 + M 1 ) ( M 1 + M 1 ) } cos k x a cos k y a + 8 M 0 ( M 1 2 + M 1 2 + M 1 M 1 ) ( cos 2 k x a cos k y a + cos k x a cos 2 k y a ) + 8 M 0 M 1 M 1 ( cos 3 k x a + cos 3 k y a ) } .
ξ = { ξ 0 + M 0 + 4 M 0 ξ Γ ( 0 ) , ξ 1 + M 1 + 2 ( M 1 + M 1 ) ξ Γ ( 1 ) ( double root ) .
η = ξ + b 2 3 .
η 3 + p η + q = 0 ,
p = b 1 b 2 2 3 ,
q = b 0 b 1 b 2 3 + 2 b 2 3 27 .
p = 1 3 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 + { 1 3 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) ( M 1 + M 1 2 M 0 ) 4 | M 2 | 2 } k 2 a 2 ,
q = 2 27 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 3 1 9 { ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 ( M 1 + M 1 2 M 0 ) 12 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) | M 2 | 2 } k 2 a 2 ,
ξ Γ ( 1 ) = ξ Γ ( 0 ) ξ Γ .
b 2 = 3 ξ Γ + M k 2 a 2 ,
p = 4 | M 2 | 2 k 2 a 2 ,
q = 0 ,
M = M 0 + M 1 + M 1 .
ξ = { ξ Γ ± 2 | M 2 | k a M k 2 a 2 / 3 , ξ Γ M k 2 a 2 / 3.
ω k = c ξ = { ω Γ ± | M 2 | a c 2 k / ω Γ M a 2 c 2 k 2 / 6 ω Γ , ω Γ M a 2 c 2 k 2 / 6 ω Γ .
L l m n ( i j ) 1 V V d r H ( i ) * ( r ) 𝒧 H ( j ) ( r r l m n ) ,
r l m = l ( a 0 0 ) + m ( 0 a 0 ) + n ( 0 0 a ) .
f 1 ( r ) = x , f 2 ( r ) = y , and f 3 ( r ) = z .
R f i ( r ) = f i ( R 1 r ) .
σ x f 1 ( r ) = x = f 1 ( r ) ,
σ x f 2 ( r ) = y = f 2 ( r ) ,
σ x f 3 ( r ) = z = f 3 ( r ) .
σ x = ( f 1 f 2 f 3 ) = ( 1 0 0 0 , 1 0 0 , 0 1 ) ( f 1 f 2 f 3 ) .
σ x : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) , C 4 x : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) ,
σ y : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) , C 4 y : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) ,
σ z : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) , C 4 z : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) ,
C 4 x 1 : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) , C 2 x : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) ,
C 4 y 1 : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) , C 2 y : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) ,
C 4 z 1 : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) , C 2 z : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) ,
C 2 ( 110 ) : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) , C 2 ( 1 1 ¯ 0 ) : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) ,
C 2 ( 011 ) : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) , C 2 ( 01 1 ¯ ) : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) ,
C 2 ( 101 ) : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) , C 2 ( 1 ¯ 01 ) : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) ,
C 3 ( 111 ) : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) , C 3 ( 111 ) 1 : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) ,
C 3 ( 1 ¯ 11 ) : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) , C 3 ( 1 ¯ 11 ) 1 : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) ,
C 3 ( 1 1 ¯ 1 ) : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) , C 3 ( 1 1 ¯ 1 ) 1 : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) ,
C 3 ( 11 1 ¯ ) : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) , C 3 ( 11 1 ¯ ) 1 : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) ,
I : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) , E : ( 1 , 0 , 0 0 , 1 , 0 0 , 0 , 1 ) ,
I C 4 x : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) , I C 4 x 1 : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) ,
I C 4 y : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) , I C 4 y 1 : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) ,
I C 4 z : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) , I C 4 z 1 : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) ,
σ d x : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) , σ d x : ( 1 , 0 , 0 0 , 0 , 1 0 , 1 , 0 ) ,
σ d y : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) , σ d y : ( 0 , 0 , 1 0 , 1 , 0 1 , 0 , 0 ) ,
σ d z : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) , σ d z : ( 0 , 1 , 0 1 , 0 , 0 0 , 0 , 1 ) ,
I C 3 ( 111 ) : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) , I C 3 ( 111 ) 1 : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) ,
I C 3 ( 1 ¯ 11 ) : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) , I C 3 ( 1 ¯ 11 ) 1 : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) ,
I C 3 ( 1 1 ¯ 1 ) : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) , I C 3 ( 1 1 ¯ 1 ) 1 : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) ,
I C 3 ( 11 1 ¯ ) : ( 0 , 0 , 1 1 , 0 , 0 0 , 1 , 0 ) , I C 3 ( 11 1 ¯ ) 1 : ( 0 , 1 , 0 0 , 0 , 1 1 , 0 , 0 ) ,
[ R H ( i ) ] ( r ) R H ( i ) ( R 1 r ) .
V L 000 ( 11 ) = V d r H ( 1 ) * ( C 4 y 1 r ) [ 𝒧 H ( 1 ) ( C 4 y 1 r ) ] = V d r [ C 4 y 1 C 4 y H ( 1 ) * ( C 4 y 1 r ) ] [ C 4 y 1 C 4 y 𝒧 C 4 y 1 C 4 y H ( 1 ) ( C 4 y 1 r ) ] ,
V L 000 ( 11 ) = V d r [ C 4 y H ( 1 ) * ( C 4 y 1 r ) ] [ 𝒧 C 4 y H ( 1 ) ( C 4 y 1 r ) ] ,
𝒧 = C 4 y 𝒧 C 4 y 1 .
V L 00 ( 11 ) = V d r [ H ( 3 ) * ( r ) ] 𝒧 [ H ( 3 ) ( r ) ] = V L 000 ( 33 ) .
L 000 ( 11 ) = L 000 ( 22 ) .
L 000 ( 11 ) = L 000 ( 22 ) = L 000 ( 33 ) ω 1 2 c 2 + M 1 ,
V L 000 ( 12 ) = V d r [ C 4 x H ( 1 ) * ( C 4 x 1 r ) ] [ 𝒧 C 4 x H ( 2 ) ( C 4 x 1 r ) ] = V d r H ( 1 ) * ( r ) 𝒧 H ( 3 ) ( r ) = V L 000 ( 13 ) .
V L 000 ( 12 ) = V d r [ C 4 x 1 H ( 1 ) * ( C 4 x r ) ] [ 𝒧 C 4 x 1 H ( 2 ) ( C 4 x r ) ] = V d r [ C 4 x 1 H ( 1 ) * ] ( r ) [ 𝒧 C 4 x 1 H ( 2 ) ] ( r ) = V d r H ( 1 ) * ( r ) 𝒧 [ H ( 3 ) ( r ) ] = V L 000 ( 13 ) .
L 000 ( 12 ) = L 000 ( 13 ) = 0.
L 000 ( 21 ) = L 000 ( 31 ) = L 000 ( 23 ) = L 000 ( 32 ) = 0.
V L 100 ( 11 ) = V d r [ C 4 y H ( 1 ) * ] ( r ) 𝒧 [ C 4 y H ( 1 ) ] ( r C 4 y r 100 ) = V d r [ H ( 3 ) * ( r ) ] 𝒧 [ H ( 3 ) ( r r 00 , 1 ) ] = V L 00 , 1 ( 33 ) .
V L 100 ( 11 ) = V d r [ C 4 y 1 H ( 1 ) * ] ( r ) 𝒧 [ C 4 y 1 H ( 1 ) ] ( r C 4 y 1 r 100 ) = V L 001 ( 33 ) .
L ± 1 , 00 ( 11 ) = L 0 , ± 1 , 0 ( 22 ) = L 00 , ± 1 ( 33 ) M 1 .
L 0 , ± 1 , 0 ( 11 ) = L 00 , ± 1 ( 11 ) = L ± 1 , 00 ( 22 ) = L 00 , ± 1 ( 22 ) = L ± 1 , 00 ( 33 ) = L 0 , ± 1 , 0 ( 33 ) M 1 .
V L 100 ( 12 ) = V d r [ C 4 x H ( 1 ) * ] ( r ) 𝒧 [ C 4 x H ( 2 ) ] ( r C 4 x r 100 ) = V d r H ( 1 ) * ( r ) 𝒧 H ( 3 ) ( r r 100 ) = V L 100 ( 13 ) .
V L 100 ( 12 ) = V d r [ C 4 x 1 H ( 1 ) * ] ( r ) 𝒧 [ C 4 x 1 H ( 2 ) ] ( r C 4 x 1 r 100 ) = V L 100 ( 13 ) .
L 100 ( 12 ) = L 100 ( 13 ) = 0.
L ± 1 , 00 ( 12 ) = L 0 , ± 1 , 0 ( 12 ) = L ± 1 , 00 ( 21 ) = L 0 , ± 1 , 0 ( 21 ) = L ± 1 , 00 ( 13 ) = L 00 , ± 1 ( 13 ) = L ± 1 , 00 ( 31 ) = L 00 , ± 1 ( 31 ) = L 0 , ± 1 , 0 ( 23 ) = L 00 , ± 1 ( 23 ) = L 0 , ± 1 , 0 ( 32 ) = L 00 , ± 1 ( 32 ) = 0 ,
L l m n ( i j ) = L l , m , n ( j i ) * .
V L 001 ( 12 ) = V d r [ σ x H ( 1 ) * ] ( r ) 𝒧 [ σ x H ( 2 ) ] ( r σ x r 001 ) = V d r [ H ( 1 ) * ( r ) ] 𝒧 H ( 2 ) ( r r 001 ) = V L 001 ( 12 ) .
L 00 , ± 1 ( 12 ) = L 00 , ± 1 ( 21 ) = L 0 , ± 1 , 0 ( 13 ) = L 0 , ± 1 , 0 ( 31 ) = L ± 1 , 00 ( 23 ) = L ± 1 , 00 ( 32 ) = 0.
[ R H ( 0 ) ] ( r ) = H ( 0 ) ( r ) .
L 000 ( 00 ) ω 0 2 c 2 + M 0 .
V L 100 ( 00 ) = V d r H ( 0 ) * ( r ) 𝒧 H ( 0 ) ( r σ x r 100 ) = V L 1 , 00 ( 00 ) .
L ± 1 , 00 ( 00 ) = L 0 , ± 1 , 0 ( 00 ) = L 00 , ± 1 ( 00 ) M 0 .
M 0 * = M 0 ,
L 000 ( 01 ) = 0.
L 000 ( 01 ) = L 000 ( 10 ) = L 000 ( 02 ) = L 000 ( 20 ) = L 000 ( 03 ) = L 000 ( 30 ) = 0.
L 010 ( 01 ) = 0.
L 0 , ± 1 , 0 ( 01 ) = L 00 , ± 1 ( 01 ) = L 0 , ± 1 , 0 ( 10 ) = L 00 , ± 1 ( 10 ) = L ± 1 , 00 ( 02 ) = L 00 , ± 1 ( 02 ) = L ± 1 , 00 ( 20 ) = L 00 , ± 1 ( 20 ) = L ± 1 , 00 ( 03 ) = L 0 , ± 1 , 0 ( 03 ) = L ± 1 , 00 ( 30 ) = L 0 , ± 1 , 0 ( 30 ) = 0.
V L 100 ( 01 ) = V d r [ σ x H ( 0 ) * ] ( r ) 𝒧 [ σ x H ( 1 ) ] ( r σ x r 100 ) = V d r H ( 0 ) * ( r ) 𝒧 [ H ( 1 ) ( r r 1 , 00 ) ] = V L 1 , 00 ( 01 ) .
± L ± 1 , 00 ( 01 ) = ± L 0 , ± 1 , 0 ( 02 ) = ± L 00 , ± 1 ( 03 ) M 2 .
L ± 1 , 00 ( 10 ) = L 0 , ± 1 , 0 ( 20 ) = L 00 , ± 1 ( 30 ) M 2 * .
L 000 ( 00 ) ω 0 2 c 2 + M 0 ,
L ± 1 , 00 ( 00 ) = L 0 , ± 1 , 0 ( 00 ) = L 00 , ± 1 ( 00 ) M 0 ,
L 000 ( 11 ) = L 000 ( 22 ) = L 000 ( 33 ) ω 1 2 c 2 + M 1 ,
L ± 1 , 00 ( 11 ) = L 0 , ± 1 , 0 ( 22 ) = L 00 , ± 1 ( 33 ) M 1 ,
L 0 , ± 1 , 0 ( 11 ) = L 00 , ± 1 ( 11 ) = L ± 1 , 00 ( 22 ) = L 00 , ± 1 ( 22 ) = L ± 1 , 00 ( 33 ) = L 0 , ± 1 , 0 ( 33 ) M 1 ,
± L ± 1 , 00 ( 01 ) = L ± 1 , 00 ( 10 ) * = ± L 0 , ± 1 , 0 ( 02 ) = L 0 , ± 1 , 0 ( 20 ) * = ± L 00 , ± 1 ( 03 ) = L 00 , ± 1 ( 30 ) * M 2 .
H k ( r ) = 1 V l , m , n e i k r l m n i = 0 3 A i H ( i ) ( r r l m n ) .
B i j = l m n e i a ( l k x + m k y + n k z ) L l m n ( i j ) .
ξ = ω k 2 c 2 , ξ 0 = ω 0 2 c 2 , ξ 1 = ω 1 2 c 2
ξ 4 + b 3 ξ 3 + b 2 ξ 2 + b 1 ξ + b 0 = 0 ,
b 3 = ( B 00 + B 11 + B 22 + B 33 ) ,
b 2 = B 00 B 11 + B 00 B 22 + B 00 B 33 + B 11 B 22 + B 11 B 33 + B 22 B 33 ( | B 01 | 2 + | B 02 | 2 + | B 03 | 2 ) ,
b 1 = ( B 00 B 11 B 22 + B 11 B 22 B 33 + B 22 B 33 B 00 + B 33 B 00 B 11 ) + | B 01 | 2 ( B 22 + B 33 ) + | B 02 | 2 ( B 11 + B 33 ) + | B 03 | 2 ( B 11 + B 22 ) ,
b 0 = B 00 B 11 B 22 B 33 ( | B 01 | 2 B 22 B 33 + | B 02 | 2 B 11 B 33 + | B 03 | 2 B 11 B 22 ) .
ξ = { ξ 0 + M 0 + 6 M 0 ξ Γ ( 0 ) , ξ 1 + M 1 + 2 M 1 + 4 M 1 ξ Γ ( 1 ) ( triple root ) .
η = ξ + b 3 4 .
η 4 + p η 2 + q η + r = 0 ,
p = b 2 3 b 3 2 8 ,
q = b 1 b 2 b 3 2 + b 3 3 8 ,
r = b 0 b 1 b 3 4 + b 2 b 3 2 16 3 b 3 4 256 .
p = 3 8 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 + 1 4 { ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) ( M 1 + 2 M 1 3 M 0 ) 16 | M 2 | 2 } k 2 a 2
q = 1 8 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 3 1 8 { ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 ( M 1 + 2 M 1 3 M 0 ) 16 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) | M 2 | 2 } k 2 a 2
r = 3 256 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 4 + { 1 64 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 3 ( M 1 + 2 M 1 3 M 0 ) 1 4 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 | M 2 | 2 } k 2 a 2 + { 1 128 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 [ 8 ( M 1 + 2 M 1 ) M 0 + ( M 1 + 2 M 1 + M 0 ) ( 7 M 1 + 14 M 1 9 M 0 ) ] 1 2 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) ( M 1 + 2 M 1 + M 0 ) | M 2 | 2 } k 4 a 4 + { 1 768 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 3 ( M 1 + 2 M 1 3 M 0 ) + ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 [ 3 16 ( 2 M 1 + M 1 ) M 1 + 1 12 | M 2 | 2 ] + 2 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) M 1 | M 2 | 2 } ( k x 4 + k y 4 + k z 4 ) a 4 + { 3 16 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) 2 ( M 1 2 + 2 M 1 M 1 + 3 M 1 2 ) + 2 ( ξ Γ ( 1 ) ξ Γ ( 0 ) ) ( M 1 + M 1 ) | M 2 | 2 } ( k x 2 k y 2 + k y 2 k z 2 + k z 2 k x 2 ) a 4 ,
ξ Γ ( 1 ) = ξ Γ ( 0 ) ξ Γ .
p = 4 | M 2 | 2 k 2 a 2 , q = r = 0.
b 3 = 4 ξ Γ + M k 2 a 2 ,
ξ = { ξ Γ ± 2 | M 2 | k a M k 2 a 2 / 4 ξ Γ M k 2 a 2 / 4 ( double root ) .
ω k = { ω Γ ± | M 2 | a c 2 k / ω Γ M a 2 c 2 k 2 / 8 ω Γ , ω Γ M a 2 c 2 k 2 / 8 ω Γ ( double root ) .

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