## Double Dirac cones in triangular-lattice metamaterials |

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

http://dx.doi.org/10.1364/OE.20.009925

Acrobat PDF (965 KB)

### Abstract

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 *E*_{1} and *E*_{2} modes in triangular-lattice metamaterials of *C*_{6v} 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

## 1. Introduction

3. K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt. **54**, 271–317 (2010). [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. B **44**, 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]

11. 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*[12

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. A **75**, 063813 (2007). [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]

*Zitterbewegung*, which is an oscillatory transmission caused by the interference between modes above and below the Dirac point, may be disturbed by the auxiliary quadratic dispersion surface. In this paper, a new method is proposed to create a pair of identical Dirac cones, or double Dirac cone, without the auxiliary quadratic dispersion surface. This paper is organized as follows: In Section 2, the secular equation of the electromagnetic field, which is derived based on the resonant states of each unit structure of two-dimensional triangular-lattice metamaterials of the

*C*

_{6v}symmetry, is solved to show the presence of the double Dirac cone. We deal with the case of accidental degeneracy of doubly degenerate

*E*

_{1}and

*E*

_{2}modes. In Section 3, we examine the relation between the Dirac cone dispersion relation obtained by the tight-binding approximation and the conventional description of electromagnetic wave propagation by effective refractive indices. Lengthy calculations for deriving relations among electromagnetic transfer integrals and solving the secular equation to obtain the dispersion relation are given in Appendix A and Appendix B, respectively.

## 2. Theory

**H**(

**r**

*,t*) is given by where

*ε*(

**r**) is the position-dependent dielectric constant of the system,

*c*is the light velocity in free space, and operator

**is defined by the first equality of Eq. (1). The magnetic permeability is assumed to be unity, since we do not deal with magnetic materials. We impose the periodic boundary condition on

**H**and assume that

*ε*(

**r**) is real to make our problem well-defined [2]. Thus, Eq. (1) leads to an eigenvalue problem.

**k**except the Γ point. As will be shown below, this condition is fulfilled by accidental degeneracy of doubly degenerate

*E*

_{1}and

*E*

_{2}modes on the Γ point in the triangular lattice of unit structures of the

*C*

_{6v}(regular hexagon) symmetry. Figure 1 illustrates an example of such metamaterials. This metamaterial array was analyzed in Ref. [19

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

*E*

_{1}and

*E*

_{2}symmetries was shown by numerical calculation. We denote their magnetic field distributions by

**H**

^{(1)}and

**H**

^{(2)}for the

*E*

_{1}state and

**H**

^{(3)}and

**H**

^{(4)}for the

*E*

_{2}state. Thus, these magnetic fields satisfy the following eigenvalue equations: where

*ε*(

_{s}**r**) is the dielectric constant of a single unit structure and

*ω*

_{1}(

*ω*

_{2}) is the resonance angular frequency of the

*E*

_{1}(

*E*

_{2}) 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*

_{1}state (

**H**

^{(1)}and

**H**

^{(2)}) transform like

*x*and

*y*when any symmetry operation of

*C*

_{6v}is performed, while those of the

*E*

_{2}state (

**H**

^{(3)}and

**H**

^{(4)}) transform like 2

*xy*and

*x*

^{2}−

*y*

^{2}[21

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

*E*

_{1}state has a symmetry different from that made of the

*E*

_{2}state on the Γ point. Therefore, they do not mix with each other, but they cross each other when accidental degeneracy takes place. On the other hand, they are mixed and repel each other for non-zero

**k**. For example, on the

*k*axis,

_{x}*k*≠ 0 while

_{x}*k*= 0. So, the

_{y}**k**vector is conserved only by

*σ*symmetry operation (see Fig. 1(b)). Thus the Bloch function on the

_{y}*k*axis has the parity with respect of

_{x}*σ*, but has no other symmetry. Similarly, the Bloch function on the

_{y}*k*axis has the parity with respect of

_{y}*σ*.

_{x}*k*axis, Bloch functions made of

_{x}**H**

^{(1)}and

**H**

^{(4)}have the same even parity, so they repel each other. Bloch functions made of

**H**

^{(2)}and

**H**

^{(3)}also repel each other, since they have the same odd parity. Similarly, on the

*k*axis, Bloch functions made of

_{y}**H**

^{(1)}and

**H**

^{(3)}repel each other, and those made of

**H**

^{(2)}and

**H**

^{(4)}repel each other. By the balance of these crossing (

*k*= 0) and anti-crossing (

*k*≠ 0) properties, an isotropic linear dispersion relation, or Dirac cone, is realized.

*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 unique feature of this matrix is that all off-diagonal elements are equal to zero for

**k**= 0, which results in the crossing (

**k**= 0) and anti-crossing (

**k**≠ 0) properties mentioned previously.

*E*

_{1}and

*E*

_{2}mode frequencies in the zone center and denote the degenerate frequency by

*ω*

_{Γ}. Such accidental degeneracy is generally possible by adjusting sample parameters [18

18. 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*and

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

_{y}*M*=

*M*

_{12}+

*M*

_{13}+

*M*

_{22}+

*M*

_{23}and

*k*. Figure 2 illustrates the four solutions, which consist of two identical pairs of isotropic cones, or double Dirac cone.

## 3. Discussion

*ω*

_{Γ}obtained in the previous section is equivalent to a zero-index system and the effective refractive index is negative for

*ω*<

*ω*

_{Γ}, which implies that both effective dielectric constant and permeability in the conventional terminology are negative.

*k*. Then the relevant dispersion relation for the Dirac cone is where

*v*is the slope of the dispersion curve, or the group velocity at the Γ point, which is for our problem. Note that it is sufficient to deal with positive

*v*as far as we treat propagation in the positive

*k*direction. When we assume that we can introduce an effective refractive index

*n*

_{eff}to the present problem, it should satisfy the following equation by definition: At first glance, Eqs. (16) and (18) are not compatible with each other. But they are actually compatible if and only if

*n*

_{eff}has the following form: where

*A*(

*ω*) is a slowly varying function of

*ω*and is not divergent or equal to zero in the vicinity of

*ω*

_{Γ}. From Eqs. (16), (18), and (19), we obtain Thus, we can conclude that the metamaterials with the double Dirac cones described in this study are zero-index materials at

*ω*=

*ω*

_{Γ}and

*n*

_{eff}< 0 for

*ω*<

*ω*

_{Γ}.

*n*

_{eff}is described by a product of an effective dielectric constant

*ε*

_{eff}and permeability

*μ*

_{eff}such that both

*ε*

_{eff}and

*μ*

_{eff}are negative for

*ω*<

*ω*

_{Γ}. However, we should note that we do not need this decomposition of

*n*

_{eff}in the framework of the tight-binding approximation and

*n*

_{eff}is determined by

*ω*

_{Γ}and one of the electromagnetic transfer integrals,

*M*

_{3}.

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]

*ω*=

_{ep}*ω*≡

_{mp}*ω*

_{Γ}to realize a Dirac point. Thus which is equivalent to taking in Eq. (19).

*ω*=

*ω*

_{Γ}, we can expect several unique features of electromagnetic wave propagation such as constant-phase waveguides and lenses of arbitrary shapes [17

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]

*ω*

_{Γ}. Thus, Zitterbewegung, an oscillatory transmission caused by interference between modes above and below

*ω*

_{Γ}, is expected without disturbance due to the presence of the quadratic dispersion surface. Thirdly, mode symmetries are not fully taken into account in the conventional theory of the effective refractive index. As we showed in this paper, a particular combination of mode symmetries is necessary for materializing the Dirac cone without an auxiliary quadratic dispersion surface. This feature can only be clarified by theoretical analyses based on the symmetries of resonance states as presented in this paper.

*ω*≠

_{ep}*ω*, the effective refractive index is given by In this case, the electromagnetic eigenmodes consist of two branches and have the following dispersion relations in the vicinity of the Γ point: where we assumed

_{mp}*ω*>

_{ep}*ω*without loss of generality. The discussion of Wang et al. [16

_{mp}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]

*ω*=

_{ep}*ω*, takes place.

_{mp}*ω*

_{Γ}, we need to make two modes coincide with each other to form the isotropic linear dispersion. This is possible only for a particular combination of mode symmetries as we showed in the previous section. In general, their dispersions are not linear or isotropic, which can be verified by many counterexamples.

*A*

_{1}and

*B*

_{1}modes of two-dimensional square lattices of the

*C*

_{4v}(regular square) symmetry. We can derive relations among their electromagnetic transfer integrals as we do in the Appendix and easily solve the secular equations. The dispersion curves are quadratic in

**k**and anisotropic: where

*M*

_{1}and

*M*

_{2}are the nearest-neighbor transfer integrals for the

*A*

_{1}and

*B*

_{1}modes, respectively,

*M*

_{3}is their cross term, and

*ϕ*is the angle between

**k**and the

*x*axis. For the combination of

*A*

_{1}and

*B*

_{2}modes of the

*C*

_{4v}symmetry,

*M*

_{3}for the nearest-neighbor unit cells is equal to zero. When we take into account

*M*

_{3}for the second nearest unit cells, we can again show that the dispersion curves are quadratic in

**k**and anisotropic. These counterexamples clearly show that important information is missing in the description by the conventional effective refractive index and theoretical methods taking into account mode symmetries are necessary for the analysis of Dirac cones.

## 4. Conclusion

*Zitterbewegung*is also expected to be observed without disturbance due to an auxiliary quadratic dispersion surface, which is inevitably present for other combinations of mode symmetries to materialize Dirac cones. In addition, we reported two examples that do not give linear or isotropic dispersion by accidental degeneracy, which clearly shows the importance of mode symmetries that are not taken into account by the conventional analysis with effective refractive indices.

## A. Electromagnetic transfer integrals

*E*

_{1}and

*E*

_{2}irreducible representations of triangular-lattice metamaterials of the

*C*

_{6v}symmetry. The electromagnetic transfer integral is defined by where

**H**

^{(i)}(

**r**) (

*i*= 1,2,3,4) are the magnetic fields of the resonant states of a single unit structure of the

*E*

_{1}symmetry (

*i*= 1, 2) and the

*E*

_{2}symmetry (

*i*= 3, 4),

*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. (1), and {

**r**

*} are the lattice vectors of the triangular lattice. Since it is sufficient to deal with only the nearest-neighbor lattice points to evaluate the dispersion relation in the vicinity of the Γ point of the Brillouin zone, we denote the origin and the six nearest-neighbor lattice points by*

_{m}**r**

_{0}≡ (0,0,0) and

**r**

*(*

_{m}*m*= 1 to 6) as shown in Fig. 1(b).

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

*E*

_{1}mode transform like

*x*and

*y*when they are subjected to any symmetry operation of point group

*C*

_{6v}. Then, we can derive various relations among

*i, j*= 1

*,*2), which are given in Ref. [19

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

*ω*

_{1}is the resonance frequency of the

*E*

_{1}mode. We can further derive the following relations [19

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

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

*E*

_{1}and

*E*

_{2}modes.

*E*

_{2}mode whose two eigen functions can be assumed without loss of generality to transform like 2

*xy*and

*x*

^{2}−

*y*

^{2}when they are subjected to any symmetry operation of

*C*

_{6v}[21

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

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

*E*

_{2}mode have the same structure as the

*E*

_{1}mode: where

*ω*

_{2}is the resonance frequency of the

*E*

_{2}mode.

*E*

_{1}-mode eigen function and one

*E*

_{2}-mode eigen function like

*R*∈

*C*

_{6v}[19

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

20. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express **20**, 3898–3917 (2012). [CrossRef] [PubMed]

**r**to

**r**′ =

*R*

**r**leads to the following equation: since

*R*is an orthogonal transformation and does not change the size of the volume element. By introducing three

*R*

^{−1}

*R*(≡

*E*: identity operation) and noting that

*R*

^{−1}does not change the value of inner products, we obtain

**r′**coordinate system equivalent to

**in the

**r**coordinate system. We can prove

**′ =

**[2], although we do not use this relation in the following. To evaluate Eq. (54), we need to know transformation characteristics of {

**H**

^{(i)}}, which are derived and listed in Appendix B and Appendix C of Ref. [19

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

*E*

_{1}and

*E*

_{2}) of point groups (like

*C*

_{6v}),

*R*

**H**

^{(i)}can be expressed by a linear combination of eigen functions of each irreducible representation. Then must be a linear combination of {

*m*,

*i*,

*j*, and

*R*for transfer integrals that contain one

*E*

_{1}-mode eigen function and one

*E*

_{2}-mode eigen function, we obtain We can further prove To derive these equations, we used the following property: where −

*m*denotes the lattice vector given by −

**r**

*.*

_{m}## B. Solutions of secular equations

*V*with

**k**-dependent phase factors: where {

**H**

^{(1)},

**H**

^{(2)}} are resonance states of the

*E*

_{1}symmetry and {

**H**

^{(3)},

**H**

^{(4)}} are those of the

*E*

_{2}symmetry. Then, the eigenvalue equation, leads to the secular equation where I is the rank-4 unit matrix. Elements of matrix B are defined by When we keep the nearest neighbor terms, we obtain Eqs. (6) – (14).

*k*and

_{x}*k*were omitted, since they are not necessary in the following analysis.

_{y}**k**= 0) of the Brillouin zone, the secular equation, Eq. (75), can be solved easily because the off-diagonal elements of matrix B are all vanishing. Its four solutions are where we use Eqs. (39) and (50). To solve Eq. (75) for general

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

*ξ*to Then, Eq. (75) 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*and

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

_{y}*E*

_{1}and

*E*

_{2}modes at the Γ point in the following. So, we assume Then, by expanding sine functions in the limit of small

*k*and

_{x}*k*, we obtain from Eqs. (86) – (88) and (66) – (68), where

_{y}*M*=

*M*

_{12}+

*M*

_{13}+

*M*

_{22}+

*M*

_{23}, the solutions of the secular equation, Eq. (75), are So, in the vicinity of the Γ point where

## Acknowledgments

## References and links

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

2. | K. Sakoda, |

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

4. | C. Caloz and T. Itoh |

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 Symposium |

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

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

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

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. |

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

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

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

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. |

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. A |

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

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. |

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. |

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

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

20. | K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express |

21. | 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: February 21, 2012

Revised Manuscript: March 30, 2012

Manuscript Accepted: April 10, 2012

Published: April 17, 2012

**Citation**

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

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-9925

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, Berlin, 2004).
- K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt.54, 271–317 (2010). [CrossRef]
- C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).
- 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).
- A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag. 5, 34–50 (2004). [CrossRef]
- 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]
- M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991). [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]
- S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78, 033834 (2008). [CrossRef]
- 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]
- X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett.100, 113903 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18, 27371–27386 (2010). [CrossRef]
- K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express19, 13899–13921 (2011). [CrossRef] [PubMed]
- K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20, 3898–3917 (2012). [CrossRef] [PubMed]
- T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990). [CrossRef]

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