## Proof of the universality of mode symmetries in creating photonic Dirac cones |

Optics Express, Vol. 20, Issue 22, pp. 25181-25194 (2012)

http://dx.doi.org/10.1364/OE.20.025181

Acrobat PDF (1226 KB)

### Abstract

We formulate a degenerate perturbation theory for the vector electromagnetic field of periodic structures and apply it to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. We derive a necessary condition by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. We analyze the structure of a matrix that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solution of dispersion curves in the vicinity of the zone center and can judge the presence of the Dirac cone. All these findings clearly show that the presence or absence of the Dirac cone in the zone center is solely determined by the spatial symmetry of the two modes.

© 2012 OSA

## 1. Introduction

1. 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]

12. K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B **29**, 2770–2778 (2012). [CrossRef]

1. 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]

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

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]

*Zitterbewegung*, or trembling motion, in particle physics by propagating an optical pulse of the Dirac point frequency [4

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

5. 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]

6. 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]

7. 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]

7. 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]

13. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using *ε*-near-zero materials,” Phys. Rev. Lett. **97**, 157403 (2006). [CrossRef] [PubMed]

14. A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B **75**, 155410 (2007). [CrossRef]

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

12. K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B **29**, 2770–2778 (2012). [CrossRef]

*A*

_{1}and

*B*

_{1}modes and that of

*A*

_{2}and

*B*

_{2}modes [8

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

*controlled symmetry reduction*of doubly degenerate modes of two-dimensional square-lattice metamaterials [9

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

*A*

_{1}and

*E*modes of square-lattice metamaterials and the combination of

*A*

_{1g}and

*T*

_{1u}modes of simple-cubic-lattice metamaterials create an isotropic Dirac cone with quadratic dispersion surfaces [10

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

*E*

_{1}and

*E*

_{2}modes of triangular-lattice metamaterials yields two-dimensional double Dirac cones, or a pair of identical Dirac cones [11

11. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express **20**, 9925–9939 (2012). [CrossRef] [PubMed]

12. K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B **29**, 2770–2778 (2012). [CrossRef]

*C*

_{2v}symmetry, the square lattice of

*C*

_{4v}symmetry, the triangular lattice of

*C*

_{6v}symmetry, and the simple-cubic lattice of

*O*symmetry. The condition for obtaining Dirac cones and/or double Dirac cones is clarified for each case. A summary of the present study is given in Section 7.

_{h}## 2. Theory

*n*and

**k**denote the band index and a wave vector in the first Brillouin zone, respectively,

*ε*is the periodic dielectric constant, and operator ℒ is defined by the first equality. We assume that the magnetic permeability of the system is equal to that of free space, since we do not deal with magnetic materials. Then the

**k**dependent eigenvalue is given by where

*c*is the speed of light in free space and

*ω*

_{kn}denotes the eigen angular frequency.

*ε*should be real for

*ω*

_{kn}to be real. According to Bloch’s theorem, the eigen function,

**H**

_{kn}, is a product of an exponential factor and a vector field with the lattice-translation symmetry: where

**a**is the elementary translation vector.

*ε*(

**r**) is real and independent of frequency and impose a periodic boundary condition to make our problem well-defined, ℒ is a Hermitian operator in the Hilbert space of complex vector fields, for which the inner product of two vector fields is defined as where

*V*is the volume on which the periodic boundary condition is imposed [15]. We normalize the eigen functions as Then,

**u**

_{kn}is an eigen function of operator ℒ

**defined by which is a Hermitian operator in the Hilbert space of complex vector fields with the lattice-translation symmetry. So, {**

_{k}**u**

_{kn}|

*n*= 1, 2,…} is a complete set. We normalize it as where

*V*

_{0}denotes the volume of the unit cell. Thus, in particular, for

**k**= 0, is an orthonormal complete set. Therefore, we can express any eigen function

**u**

_{kl}of operator ℒ

**by a linear combination of eigen functions {**

_{k}**u**

_{0n}} of operator ℒ

_{0}. Thus, for small

**k**in the vicinity of the Γ point, we can calculate

*λ*

_{kl}perturbatively using {

**u**

_{0n}} as a basis set. Because we are interested only in the presence or absence of terms linear in

**k**, we neglect the quadratic term of the perturbation operator: where

**u**

_{0l}|

*l*= 1, 2, … ,

*M*} are degenerate and denote their eigenvalue by

**u**

_{kl}(

*l*= 1, 2, … ,

*M*) is obtained by diagonalizing the matrix whose

*ij*(1 ≤

*i*,

*j*≤

*M*) element is given by which can bring about eigenvalue corrections linear in

**k**. Thus, our problem on the creation of photonic Dirac cones is reduced to examining whether the eigenvalues of matrix

**using the spatial symmetry of {**

_{k}**u**

_{0}

*}. We assume that the periodic structure that we deal with is invariant by symmetry operations of point group 𝒢. We denote the symmetry operations and their matrix representations by ℛ and R, respectively. First, we should note that since R is an orthogonal matrix. Second, from the nature of inner and outer products, we have where*

_{l}**P**

*is defined as We also have*

_{ij}*S*

_{0}denotes the surface of

*V*

_{0}and the first integral on the right-hand side is the surface integral of the normal component of the integrand. This surface integral is equal to zero because of the periodicity of

*ε*and

**u**. By the same transformation as in Eq. (15), we finally obtain So, we have Thirdly, we can prove [15] By a similar calculation, we can also prove

^{−1}ℛ (= identity operator) to the definition of

**P**

*, we obtain*

_{ij}**r**′ = R

**r**and by definition, In Eq. (22), we used the facts that

*ε*is invariant by ℛ and that R does not change the size of the volume element, since it is an orthogonal matrix.

**k**,

**u**

_{0i}, and

**u**

_{0j}contains a term invariant for all ℛ ∈ 𝒢,

**k**·

**P**

*, and consequently,*

_{ij}**u**

_{0i}and

**u**

_{0j}are eigen functions of ℒ

_{0}, which commutes with

^{∀}ℛ ∈ 𝒢, they are irreducible representations of group 𝒢, so their transformation by ℛ is well known. The transformation of vector

**k**by ℛ can also be found easily. Thus, we can select the candidates of combinations of mode symmetries that enable the creation of Dirac cones by using this necessary condition. The exact shapes of dispersion curves can also be clarified by examining the structure of C

**. Some examples will be given in next four sections.**

_{k}## 3. One-dimensional lattice of *C*_{2v} symmetry

*C*

_{2v}symmetry. An example of such structures was previously analyzed by tight-binding approximation in Ref. [8

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

*A*

_{1},

*A*

_{2},

*B*

_{1},

*B*

_{2}) for the

*C*

_{2v}point group [16

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

**k**, which should be regarded as a one-dimensional vector (or scalar), has the

*B*

_{1}symmetry. We can examine by the well-known reduction procedure [16

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

**k**,

**u**

_{0}

*, and*

_{i}**u**

_{0j}contains an invariant term, or the totally symmetric

*A*

_{1}representation. As is summarized in Table 1, {

*A*

_{1},

*B*

_{1}) modes and (

*A*

_{2},

*B*

_{2}) modes.

*C*

_{2v}, we can prove that matrix C

**has the following form for the combination of the**

_{k}*A*

_{1}(

**u**

_{01}) and

*B*

_{1}(

**u**

_{02}) modes. In Eq. (24), where

**e**is a unit vector in the direction of the one-dimensional alignment of unit cells. Then, the secular equation to determine the first-order correction, Δ

*λ*, to the eigenvalue of Eq. (1) is given by Thus, its solutions are given by which leads to the following dispersion relation in the vicinity of the Γ point: So, there is a one-dimensional Dirac cone. This result is consistent with Ref. [8

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

*A*

_{1}mode and a

*B*

_{1}mode by tight-binding approximation based on the resonant states of unit cells of periodic metamaterials. For the combination of the

*A*

_{2}and

*B*

_{2}modes, we have the same relations as Eq. (24), so there is also a Dirac cone.

**k**.

## 4. Square lattice of *C*_{4v} symmetry

*C*

_{4v}symmetry. An example of such structures was previously analyzed by tight-binding approximation in Ref. [10

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

*A*

_{1},

*A*

_{2},

*B*

_{1},

*B*

_{2}) and one two-dimensional representation (

*E*) [16

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

**k**, which should be regarded as a two-dimensional vector for this case, has the

*E*symmetry. The nature of

*A*

_{1},

*A*

_{2},

*B*

_{1}, or

*B*

_{2}mode and an

*E*mode.

*E*mode and an

*A*

_{1}mode. From group theory [16

*Group Theory and Its Applications in Physics* (Springer, Berlin, 1990). [CrossRef]

*E*mode is transformed like the

*x*coordinate and the other like the

*y*coordinate by the symmetry operation ℛ ∈

*C*

_{4}

*. We denote the first and second functions by*

_{v}**u**

_{01}and

**u**

_{02}, respectively. The eigen function of the

*A*

_{1}mode is denoted by

**u**

_{03}. By examining all transformation ℛ ∈

*C*

_{4v}, we can prove that matrix C

**has the following form: The secular equation for the first-order correction is given by whose solutions are where**

_{k}10. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express **20**, 3898–3917 (2012). [CrossRef] [PubMed]

**20**, 3898–3917 (2012). [CrossRef] [PubMed]

*E*,

*A*

_{2}), (

*E*,

*B*

_{1}), and (

*E*,

*B*

_{2}) modes, we can prove that C

**has the following forms, respectively:**

_{k}**29**, 2770–2778 (2012). [CrossRef]

## 5. Triangular lattice of *C*_{6v} symmetry

*C*

_{6v}symmetry. There are four one-dimensional representations (

*A*

_{1},

*A*

_{2},

*B*

_{1},

*B*

_{2}) and two two-dimensional representations (

*E*

_{1},

*E*

_{2}) [16

*Group Theory and Its Applications in Physics* (Springer, Berlin, 1990). [CrossRef]

**k**has the

*E*

_{1}symmetry. The nature of

*E*

_{1}mode and an

*E*

_{2}mode, we can prove by examining all transformations ℛ ∈

*C*

_{6v}that matrix C

**has the following form: where we assumed without loss of generality that the two eigen functions of the**

_{k}*E*

_{1}mode (

**u**

_{01},

**u**

_{02}) are transformed like

*x*and

*y*and those of the

*E*

_{2}mode (

**u**

_{03},

**u**

_{04}) are transformed like 2

*xy*and

*x*

^{2}−

*y*

^{2}[16

*Group Theory and Its Applications in Physics* (Springer, Berlin, 1990). [CrossRef]

11. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express **20**, 9925–9939 (2012). [CrossRef] [PubMed]

11. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express **20**, 9925–9939 (2012). [CrossRef] [PubMed]

*E*

_{1},

*A*

_{1}) and (

*E*

_{2},

*B*

_{2}) combinations, matrix C

**has the following form: On the other hand, for the (**

_{k}*E*

_{1},

*A*

_{2}) and (

*E*

_{2},

*B*

_{1}) combinations, For these cases, the solutions of the secular equation are the same and given by So, there are an isotropic Dirac cone and a quadratic dispersion surface. These conclusions are consistent with Ref. [12

**29**, 2770–2778 (2012). [CrossRef]

## 6. Simple-cubic lattice of *O*_{h} symmetry

_{h}

*O*symmetry. There are four one-dimensional representations (

_{h}*A*

_{1g},

*A*

_{1u},

*A*

_{2g},

*A*

_{2u}), two two-dimensional representations (

*E*,

_{g}*E*), and four three-dimensional representations (

_{u}*T*

_{1g},

*T*

_{1u},

*T*

_{2g},

*T*

_{2u}) [16

*Group Theory and Its Applications in Physics* (Springer, Berlin, 1990). [CrossRef]

**k**has the

*T*

_{1u}symmetry. The nature of

*T*

_{1u}mode and an

*A*

_{1g}mode [10

**20**, 3898–3917 (2012). [CrossRef] [PubMed]

*T*

_{1u}mode are transformed like the

*x*,

*y*, and

*z*coordinates [16

*Group Theory and Its Applications in Physics* (Springer, Berlin, 1990). [CrossRef]

**u**

_{01},

**u**

_{02}, and

**u**

_{03}, respectively. The eigen function of the

*A*

_{1g}mode is denoted by

**u**

_{04}. We can prove by examining all transformations ℛ ∈

*O*that matrix C

_{h}**has the following form: Because matrix C**

_{k}**is sufficiently sparse, we can easily diagonalize it to obtain the first-order correction to the eigenvalues. The result is where**

_{k}**20**, 3898–3917 (2012). [CrossRef] [PubMed]

*T*

_{1g},

*A*

_{1u}), (

*T*

_{2u},

*A*

_{2g}), and (

*T*

_{2g},

*A*

_{2u}), we can also prove that C

**has the same form as in Eq. (38). So, all these combinations yield an isotropic Dirac cone and two quadratic dispersion surfaces.**

_{k}*T*

_{1u}mode (

**u**

_{01},

**u**

_{02},

**u**

_{03}) and an

*E*mode (

_{g}**u**

_{04},

**u**

_{05}). We can assume that two eigen functions of the

*E*mode are transformed like 2

_{g}*z*

^{2}−

*x*

^{2}−

*y*

^{2}and

*Group Theory and Its Applications in Physics* (Springer, Berlin, 1990). [CrossRef]

**has the following form: Then, the secular equation is When we express the solutions by the spherical coordinates, (**

_{k}*k*,

*θ*,

*ϕ*), where Thus, there are two anisotropic Dirac cones and a quadratic dispersion surface unless

*F*(

*θ*,

*ϕ*) ≠ 0. When

*F*(

*θ*,

*ϕ*) = 0, which happens for

**k**in the (1,0,0) direction and its equivalent directions, two of the four

*k*-linear eigenvalues vanish, so the corresponding dispersion curves are quadratic in

*k*. For the combination of (

*T*

_{1g},

*E*), we can prove that C

_{u}**has the same structure as Eq. (40), so it also yields two anisotropic Dirac cones and a quadratic dispersion surface.**

_{k}*T*

_{2u},

*E*) and (

_{g}*T*

_{2g},

*E*), C

_{u}**has the following form: This C**

_{k}**leads to exactly the same secular equation as Eq. (41). So, the combinations of (**

_{k}*T*

_{2u},

*E*) and (

_{g}*T*

_{2g},

*E*) also yield two anisotropic Dirac cones and a quadratic dispersion surface.

_{u}*T*

_{1g}mode (

**u**

_{01},

**u**

_{02},

**u**

_{03}) and a

*T*

_{1u}mode (

**u**

_{04},

**u**

_{05},

**u**

_{06}), C

**has the following structure: which is sufficiently sparse so that we can easily obtain its analytical solutions: So, there are isotropic double Dirac cones with the same slope and two quadratic dispersion surfaces. We can prove that the combination of**

_{k}*T*

_{2g}and

*T*

_{2u}modes has C

**of the same structure, so this combination also yields double Dirac cones and two quadratic dispersion surfaces.**

_{k}*T*

_{1g},

*T*

_{2u}) and (

*T*

_{2g},

*T*

_{1u}), C

**has the following form: which leads to the following secular equation for the first-order correction. In general, three solutions of are given by where**

_{k}*l*= 1, 2, 3. So, six solutions of Eq. (48) are where So, there are three anisotropic Dirac cones.

**) by which we can select candidates of mode combinations that enable the creation of single and double Dirac cones. In addition, by examining the transformation of C**

_{k}**by symmetry operations for four periodic systems of different spatial symmetries, we could determine the structure of C**

_{k}**and obtain the shapes of dispersion curves in the vicinity of the zone center by purely analytical calculations. All these findings clearly show that the presence or absence of the Dirac cone by accidental degeneracy does not depend on the details of the periodic structure but is solely determined by the spatial symmetry of the two modes.**

_{k}**and obtain the shapes of dispersion curves in quite a similar manner. We should also note that if the periodic structure does not have a spatial symmetry, any two modes are mixed and repel each other when their eigen frequencies are close, so they do not create Dirac cones.**

_{k}**18**, 27371–27386 (2010). [CrossRef]

**20**, 3898–3917 (2012). [CrossRef] [PubMed]

**20**, 9925–9939 (2012). [CrossRef] [PubMed]

**29**, 2770–2778 (2012). [CrossRef]

**k**·

**p**theory in semiconductor physics [17

17. H. Haug and S. W. Koch, *Quantum Theory of the Optical and Electronic Properties of Semiconductors*, 5th Edition (World Scientific, Singapore, 2009). [CrossRef]

**k**·

**p**theory, the first-order term is absent and an effective mass is derived by the second-order perturbation, so the energy bands are quadratic in the Brillouin-zone center. However, in the present theory, the first-order term can be non-zero because of the accidental degeneracy of two modes, which resulted in the creation of Dirac cones.

## 7. Conclusion

**) that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solutions of the dispersion curves in the vicinity of the zone center and judge the presence of the Dirac cone.**

_{k}*C*

_{2v}symmetry, the square lattice of

*C*

_{4v}symmetry, the triangular lattice of

*C*

_{6v}symmetry, and the simple-cubic lattice of

*O*symmetry. We succeeded in examining all possible combinations of mode symmetries and obtaining the analytical solution of dispersion curves for each case. Thus, we fully clarified the conditions required to obtain Dirac cones and double Dirac cones in these four systems. These results are consistent with our previous calculations for periodic metamaterials by tight-binding approximation and for photonic crystals by plane-wave expansion.

_{h}**29**, 2770–2778 (2012). [CrossRef]

## Acknowledgments

## References and links

1. | F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. |

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

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

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

5. | 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 |

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

7. | 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. |

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

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

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

11. | K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express |

12. | K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B |

13. | M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using |

14. | A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B |

15. | K. Sakoda, |

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

17. | H. Haug and S. W. Koch, |

**OCIS Codes**

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: September 11, 2012

Revised Manuscript: October 13, 2012

Manuscript Accepted: October 13, 2012

Published: October 19, 2012

**Citation**

Kazuaki Sakoda, "Proof of the universality of mode symmetries in creating photonic Dirac cones," Opt. Express **20**, 25181-25194 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-25181

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### References

- 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]
- 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]
- 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]
- K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express20, 9925–9939 (2012). [CrossRef] [PubMed]
- K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B29, 2770–2778 (2012). [CrossRef]
- M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006). [CrossRef] [PubMed]
- A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75, 155410 (2007). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).
- T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1990). [CrossRef]
- H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition (World Scientific, Singapore, 2009). [CrossRef]

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