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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 9 — Apr. 23, 2012
  • pp: 9925–9939

Double Dirac cones in triangular-lattice metamaterials

Kazuaki Sakoda  »View Author Affiliations

Optics Express, Vol. 20, Issue 9, pp. 9925-9939 (2012)

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It is shown by tight-binding approximation and group theory that a double Dirac cone, or a pair of two identical Dirac cones, of the electromagnetic dispersion relation can be created in the Brillouin zone center by accidental degeneracy of E1 and E2 modes in triangular-lattice metamaterials of C6v symmetry. The Dirac point thus obtained is equivalent to a zero-index system, so we can expect unique optical propagation phenomena such as constant-phase waveguides and lenses of arbitrary shapes. Zitterbewegung is also expected without disturbance due to an auxiliary quadratic dispersion surface, which is present for other combinations of mode symmetries to materialize the Dirac cones. To the best of the author’s knowledge, this is the first prediction of the presence of a double Dirac cone in metamaterials.

© 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: February 21, 2012
Revised Manuscript: March 30, 2012
Manuscript Accepted: April 10, 2012
Published: April 17, 2012

Kazuaki Sakoda, "Double Dirac cones in triangular-lattice metamaterials," Opt. Express 20, 9925-9939 (2012)

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, Berlin, 2004).
  3. K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt.54, 271–317 (2010). [CrossRef]
  4. C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).
  5. 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 Antennas and Propagation Society International Symposium2, 412–415 (2002).
  6. A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag. 5, 34–50 (2004). [CrossRef]
  7. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett.14, 68–70 (2004). [CrossRef]
  8. M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991). [CrossRef]
  9. 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]
  10. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78, 033834 (2008). [CrossRef]
  11. 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]
  12. X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett.100, 113903 (2008). [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, 47008 (2009). [CrossRef]
  14. 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]
  15. 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]
  16. L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett.34, 1510–1512 (2009). [CrossRef] [PubMed]
  17. 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,” Nature Mater.10, 582–586 (2011). [CrossRef]
  18. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18, 27371–27386 (2010). [CrossRef]
  19. K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express19, 13899–13921 (2011). [CrossRef] [PubMed]
  20. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20, 3898–3917 (2012). [CrossRef] [PubMed]
  21. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990). [CrossRef]

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