## Dirac cone in two- and three-dimensional metamaterials |

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

http://dx.doi.org/10.1364/OE.20.003898

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### 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 *A*_{1} mode of the square lattice of the *C*_{4v} symmetry is examined. For three dimensions, the combination of a triply degenerate *T*_{1u} mode and a non-degenerate *A*_{1g} mode of the cubic lattice of the *O _{h}* 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

## 1. Introduction

*E*(=

*h̄ω*) is originally proportional to its momentum

*p*(=

*h̄k*): where

*c*is the velocity of light in free space. However, as is well known, the photon dispersion relation deviates quite a lot from this linear relation in periodic structured materials like photonic crystals [1

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]

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]

*m*

^{*}by interacting with the periodic structure.

*t*, it has a time-reversal symmetry in the absence of the static magnetic field, so its dispersion as a function of the wave number

*k*is symmetric: Therefore, if we expand

*ω*with respect to

*k*, the linear term, and generally speaking, terms of odd orders of

*k*are absent: Thus, the dispersion curves in the vicinity of the zone center should inevitably be parabolic.

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]

*k*, so Eq. (3) is satisfied in spite of their finite slopes at

*k*= 0. The coincidence of the two bands was realized by accidental degeneracy with appropriate choice of the sample structure.

*k*= 0 and have the same symmetry for

*k*≠ 0 [17

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]

*k*≠ 0 because they have the same symmetry, while they cross each other for

*k*= 0 because their symmetries are different. The balance of these crossing and anti-crossing behaviors results in the linear dispersion relation in the vicinity of the Γ point of the Brillouin zone. This property can actually be materialized, for example, by one-dimensional periodic metamaterials composed of metallic unit structures with the

*C*

_{2v}(rectangular) spatial symmetry [17

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]

*m*

^{*}vanishes again due to the constraint imposed by the mode symmetry.

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]

*Zitterbewegung*[22

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]

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]

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]

*electromagnetic transfer integrals*for the

*T*

_{1u}and

*A*

_{1g}modes are given in Appendix B.

## 2. Isotropic Dirac cone in two dimensions

**H**(

**r**,

*t*) is given by where

*ε*(

**r**) is the position-dependent dielectric constant of the system and operator is defined by the first equality of Eq. (5). The magnetic permeability is assumed to be unity, since we do not deal with magnetic materials. We impose the periodic boundary condition on

**H**to make our problem well-defined [4] and assume, as an approximation to extract basic features of our problem, that

*ε*(

**r**) is real. Thus, Eq. (5) leads to eigenvalue problems.

**k**except the Γ point. As will be proved in the following, this condition can be fulfilled by accidental degeneracy of a doubly degenerate

*E*mode and a non-degenerate

*A*

_{1}mode on the Γ point in the square lattice of unit structures with the

*C*

_{4v}(regular square) symmetry. Figure 2 illustrates an example of such metamaterials, which was analyzed in Ref. [18

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

*E*symmetry was shown by numerical calculation. In general, there are resonant states of all possible symmetries of the

*C*

_{4v}point group.

*ε*(

_{s}**r**), we assume a doubly degenerate

*E*resonant state and a non-degenerate

*A*

_{1}state, and denote their magnetic field distributions by

**H**

^{(1)}and

**H**

^{(2)}for the former and

**H**

^{(0)}for the latter. Thus, these magnetic fields satisfy the following eigenvalue equations: where

*ω*

_{1}(

*ω*

_{0}) is the resonance angular frequency of the

*E*(

*A*

_{1}) state. Without loss of generality, we can assume that these distributions are normalized as follows: where

*V*is the volume on which we impose the periodic boundary condition.

*E*state (

**H**

^{(1)}and

**H**

^{(2)}) have the spatial symmetry as the

*x*and

*y*coordinates, respectively, while the eigenfunction of the

*A*

_{1}state (

**H**

^{(0)}) has the symmetry of the

*z*coordinate [27

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

*k*axis, where the parity with respect to the

_{x}*y*coordinate is a good quantum number,

**H**

^{(1)}and

**H**

^{(0)}have the same parity so that their Bloch functions are mixed and repel each other. Similarly, the Bloch functions made of

**H**

^{(2)}and

**H**

^{(0)}are mixed and repel each other on the

*k*axis.

_{y}*E*and

*A*

_{1}modes have different symmetries.

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]

**H**

^{(0)},

**H**

^{(1)}, and

**H**

^{(2)}localized in each unit structure with a phase factor that is dependent on the wave vector

**k**to obtain a Bloch function.

*k*and

_{x}*k*as where

_{y}*a*is the lattice constant of the periodic metamaterial and

*M*are

_{n}*electromagnetic transfer integrals*whose definition and character are presented in Appendix A.

*A*

_{1}and

*E*mode frequencies in the zone center and denote the degenerate frequency by

*ω*

_{Γ}. Such accidental degeneracy is generally possible by adjusting sample parameters. For example, it was attained by adjusting the lattice constant in Ref. [17

**18**, 27371–27386 (2010). [CrossRef]

*k*and

_{x}*k*, we obtain the following dispersion relations: where

_{y}*M*=

*M*′

_{1}+

*M*″

_{2}+

*M*′

_{0}and

*k*in the vicinity of the zone center and the third one is parabolic and nearly horizontal. The slopes of the former two modes have the same amplitude but opposite signs, so we have the isotropic Dirac cone.

## 3. Isotropic Dirac cone in three dimensions

*T*

_{1u}mode and an

*A*

_{1g}mode in the simple cubic lattice of unit structures of the

*O*symmetry, that is, the symmetry of a regular cube. Such a metamaterial is illustrated in Fig. 4. The

_{h}*T*

_{1u}mode is triply degenerate and its three wave functions behave like the

*x*,

*y*, and

*z*coordinates, respectively, whereas the

*A*

_{1g}mode behaves like

*x*

^{2}+

*y*

^{2}+

*z*

^{2}[27

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

*k*space, the latter Bloch wave function has the same parity as one of the former three, so they are mixed and repel each other. Thus we can expect the generation of the Dirac cone that is isotropic in all directions.

*T*

_{1u}and

*A*

_{1g}modes are carefully examined based on the group theory. After a lengthy calculation, which is also described in detail in Appendix B, we finally obtain four dispersion curves, which are accurate to the second order of

*k*,

_{x}*k*, and

_{y}*k*: where

_{z}*M*′ =

*M*′

_{1}+ 2

*M*″

_{1}+

*M*′

_{0}and

*k*in the vicinity of the zone center and the third and fourth ones are parabolic and nearly horizontal. The slopes of the former two modes have the same amplitude and opposite signs, so we have the isotropic Dirac cone.

*E*and

*A*

_{1}mode of the square lattice and that of the

*T*

_{1u}and

*A*

_{1g}mode of the simple cubic lattice. Because the Dirac cone is realized by the balance of crossing and anti-crossing properties due to particular combinations of spatial symmetries of resonant states, we may be able to find still more different mode combinations to create the Dirac cone, which remains as an interesting future work. Numerical studies for the practical design of metamaterials to achieve the accidental degeneracy as described in this paper are also required in future.

## 4. Conclusion

*E*mode and a non-degenerate

*A*

_{1}mode of the square lattice of the

*C*

_{4v}symmetry was examined and the presence of the isotropic Dirac cone, or a pair of modes whose dispersion is linear to

**k**, in the Brillouin zone center was shown. For three dimensions, accidental degeneracy of a triply degenerate

*T*

_{1u}mode and a non-degenerate

*A*

_{1g}mode was examined and the presence of the Dirac cone in the zone center that was isotropic in all directions was shown. To the best of my knowledge, this is the first theoretical prediction of the presence of the Dirac cone in the three-dimensional periodic structure.

**k**= 0 so that they are not mixed and cross each other when the accidental degeneracy takes place, while the two modes must have the same symmetry for

**k**≠ 0 so that they are mixed and repel each other. The balance of these crossing and anti-crossing behaviors results in the linear dispersion in the vicinity of the Brillouin zone center.

**k**by tight-binding approximation with the localized resonance states in the single unit structure as basis functions. The mutual relations among different electromagnetic transfer integrals were derived based on the group theory and used to solve the secular equation.

## A. *E* and *A*_{1} modes of two-dimensional square lattice

*electromagnetic transfer integral*. In two dimensions, it is defined by where

**H**

^{(i)}(

**r**) (

*i*= 0, 1, 2) are the magnetic fields of the resonant states of the

*E*symmetry (

*i*= 1, 2) and the

*A*

_{1}symmetry (

*i*= 0),

*V*is the volume on which the periodic boundary condition is imposed, is the operator that defines the eigenvalue problem of the periodic metamaterial given in Eq. (5), and

**r**

*are the lattice vectors of the square lattice given by where*

_{lm}*a*is the lattice constant, and

*l*and

*m*are integers.

*C*

_{4v}) and the square lattice, we can derive various relations among

*E*mode, relations among their transfer integrals are given in Eqs. (14) – (18) of Ref. [18

**19**, 13899–13921 (2011). [CrossRef] [PubMed]

*ω*

_{1}is the resonance frequency of the

*E*mode. In these equations, I use notations of

*M*different from those in Ref. [18

**19**, 13899–13921 (2011). [CrossRef] [PubMed]

*A*

_{1}mode, it is invariant by any symmetry operation of the

*C*

_{4v}point group. Using this property, we can easily prove where

*ω*

_{0}is the resonance frequency of the

*A*

_{1}mode.

*V*with

**k**-dependent phase factors: Following the same procedure as described in Ref. [18

**19**, 13899–13921 (2011). [CrossRef] [PubMed]

**k**= 0) of the Brillouin zone, Eq. (38) can be solved easily. Its three solutions are To solve Eq. (38) for general

**k**, we change the unknown variable from

*ξ*to Then, Eq. (38) is reduced to where To evaluate the

*k*dependence of the solution in the vicinity of the Γ point, we calculate

*p*and

*q*to the third order of

*k*and

_{x}*k*. The results are where

_{y}## B. *T*_{1} and *A*_{1g} modes of simple cubic lattice

*L*by where

**r**

*are lattice vectors of the simple cubic lattice:*

_{lmn}*T*

_{1u}mode is given by {

*x*,

*y*,

*z*} [27

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

*T*

_{1u}symmetry,

**H**

^{(1)}(

**r**),

**H**

^{(2)}(

**r**), and

**H**

^{(3)}(

**r**), such that they are transformed as the following three functions when any

*R*∈

*O*, whose elements are illustrated in Fig. 5, is operated. By definition, they are transformed as So, for example, We write these three equations in a matrix form:

_{h}*R*∈

*O*:

_{h}*T*

_{1u}representation in Table 1, which are the trace (sum of the diagonal elements) of the above matrices. Because {

**H**

^{(i)}} (

*i*= 1, 2, 3) are also a

*T*

_{1u}representation, they transform like {

*f*}. But we should note that there is a difference due to the vector nature of the former. Their transformation is defined as

_{i}**r**to

**r**′ =

*C*

_{4y}

**r**and evaluate

*C*

_{4y}does not change the size of volume elements, we have

**r**′ coordinate system equivalent to in the

**r**coordinate system. Substituting the second relation of Eq. (65) and Eq. (88), we obtain Similarly, by changing the variable from

**r**to

**r**′ =

*C*

_{4z}

**r**and using the second relation of Eq. (66), we obtain So, we denote them by where the first term on the right-hand side is the original value for an isolated unit structure.

**r**to

**r**′ =

*C*

_{4x}

**r**and using the second relation of Eq. (64), we obtain Note that

**r**to

**r**′ =

*C*

_{4y}

**r**,

**r**to

*C*

_{4z},

*C*

_{2y}lead to Application of all other

*R*∈

*O*gives the same result as Eq. (101). As for

_{h}**r**′ =

*C*

_{4x}

**r**,

**r**to

*R*∈

*O*lead to

_{h}**r**to

**r**′ =

*σ*

_{x}**r**and obtain

*T*

_{1u}mode.

*A*

_{1g}mode. Its eigen function

**H**

^{(0)}is invariant when any

*R*∈

*O*is applied as can be found in Table 1: First, let us denote

_{h}**r**to

**r**′ =

*σ*

_{x}**r**, we obtain In addition to this equation, by applying

*C*

_{3(111)},

*σ*, and

_{y}*σ*, we obtain By using Eq. (107), we can prove so

_{z}*M*′

_{0}is a real number.

*T*

_{1u}and

*A*

_{1g}modes, let us start with

**r**to

**r**′ =

*σ*

_{x}**r**, we can easily prove Similar calculations lead to Also using

*σ*, we obtain Similar calculations in this case result in

_{x}**r**to

**r**′ =

*σ*

_{x}**r**, we have

*C*

_{3(111)}and

*σ*and

_{y}*σ*, we obtain Then, from Eq. (107), we also obtain In summary, we have the following relations: All other

_{z}*A*

_{1g}mode (

**H**

^{(0)}) and

*T*

_{1u}mode (

**H**

^{(1)},

**H**

^{(2)},

**H**

^{(3)}): Then, the secular equation is derived as before by keeping the nearest neighbor terms. The Hermitian matrix B that should be diagonalized is now a 4 × 4 matrix. Its elements are given by When we keep the nearest neighbor terms, we obtain Eqs. (17) – (23). We denote as before. Then the secular equation reduces to where

**k**= 0) of the Brillouin zone, the secular equation, Eq. (131), can be solved easily because the off-diagonal elements of matrix B are all vanishing. Its four solutions are To solve Eq. (131) for general

**k**, we change the unknown variable from

*ξ*to Then, Eq. (131) is reduced to where To evaluate the k dependence of the solution in the vicinity of the Γ point,

*p*,

*q*, and

*r*were evaluated to the second, third, and fourth order of

*k*,

_{x}*k*and

_{y}*k*, respectively. The results are

_{z}*T*

_{1u}and

*A*

_{1g}modes at the Γ point in the following. So, we assume Then, from Eqs. (142) – (144), Since where

*M*′ =

*M*′

_{1}+ 2

*M*″

_{1}+

*M*′

_{0}, the solutions of the secular equation, Eq. (131), are So, in the vicinity of the Γ point where

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn. |

4. | K. Sakoda, |

5. | K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. |

6. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. |

7. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

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. |

9. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

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 |

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 |

12. | S. A. Ramakrishna and T. M. Grzegorczyk, |

13. | C. Caloz and T. Itoh, |

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. |

15. | A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. |

16. | A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Components Lett. |

17. | K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express |

18. | K. Sakoda and H.-F. Zhou, “Analytical study of degenerate metamaterial steerable antennas,” Opt. Express |

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. |

20. | S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A |

21. | T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B |

22. | X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. |

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 |

24. | M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B |

25. | M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B |

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. |

27. | T. Inui, Y. Tanabe, and Y. Onodera, |

**OCIS Codes**

(350.3618) Other areas of optics : Left-handed materials

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Metamaterials

**History**

Original Manuscript: December 2, 2011

Revised Manuscript: January 20, 2012

Manuscript Accepted: January 27, 2012

Published: February 1, 2012

**Citation**

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

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3898

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### References

- 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]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn. Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
- K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).
- K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt.54, 271–317 (2010). [CrossRef]
- V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp.10, 509–514 (1968). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
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