OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 3898–3917

Dirac cone in two- and three-dimensional metamaterials

Kazuaki Sakoda  »View Author Affiliations

Optics Express, Vol. 20, Issue 4, pp. 3898-3917 (2012)

View Full Text Article

Enhanced HTML    Acrobat PDF (1749 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(350.3618) Other areas of optics : Left-handed materials
(160.3918) Materials : Metamaterials
(160.5298) Materials : Photonic crystals

ToC Category:

Original Manuscript: December 2, 2011
Revised Manuscript: January 20, 2012
Manuscript Accepted: January 27, 2012
Published: February 1, 2012

Kazuaki Sakoda, "Dirac cone in two- and three-dimensional metamaterials," Opt. Express 20, 3898-3917 (2012)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58, 2059–2062 (1987). [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58, 2486–2489 (1987). [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,” Science312, 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,” Science292, 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,” Science314, 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. B65, 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. Express18, 27371–27386 (2010). [CrossRef]
  18. K. Sakoda and H.-F. Zhou, “Analytical study of degenerate metamaterial steerable antennas,” Opt. Express19, 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. A78, 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. B80, 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. A75, 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 B405, 2990–2995 (2010). [CrossRef]
  25. M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B44, 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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited