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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 3898–3917
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Dirac cone in two- and three-dimensional metamaterials

Kazuaki Sakoda  »View Author Affiliations


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 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

1. Introduction

Because photon is massless, its energy E (= h̄ω) is originally proportional to its momentum p (= h̄k):
E=cp,
(1)
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

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

, 3

3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn. Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

, 4

4. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).

, 5

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

] and metamaterials [6

6. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

, 7

7. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

, 8

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

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

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

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

12. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008). [CrossRef]

, 13

13. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley,2006).

]. Especially, its dispersion is parabolic in the Brillouin zone center. This feature is brought about by the mixing and repulsion of the linear dispersion lines folded back into the first Brillouin zone due to the presence of the periodic spatial modulation of the refractive index. If we denote the dispersion in the vicinity of the zone center by
ωω0+h¯k22m*,
(2)
we may regard that the photon has acquired an effective mass m* by interacting with the periodic structure.

This feature can be described from another point of view. Because Maxwell’s wave equation is of the second order of the time coordinate 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:
ωk=ωk.
(3)
Therefore, if we expand ω with respect to k, the linear term, and generally speaking, terms of odd orders of k are absent:
ωk=ω0+c2k2+c4k4+
(4)
Thus, the dispersion curves in the vicinity of the zone center should inevitably be parabolic.

However, this logic has a pitfall. It was shown by the transmission line theory that the dispersion curves can be linear if we have two modes and they coincide in the zone center as illustrated in Fig. 1 [13

13. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley,2006).

, 14

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

15. A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag. 5, 34–50 (2004). [CrossRef]

, 16

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]

]. Here, the two dispersion curves are symmetric with each other about the origin of 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.

Fig. 1 The linear dispersion in the zone center realized by two modes that are symmetric with each other about the origin of k.

On the other hand, I recently reformulated the same problem based on the tight-binding approximation and group theory, and found that the necessary and sufficient condition for this phenomenon is that the two modes have different spatial symmetries for 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

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

]. The two bands are mixed and repel each other for 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 C2v (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]

]. Thus the effective mass m* vanishes again due to the constraint imposed by the mode symmetry.

Now, the extension of this property to higher dimensions is quite an interesting problem, since the isotropic linear dispersion, or Dirac cone, in two and three dimensions brings about many exotic phenomena like unidirectional edge states [19

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

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

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]

], and pseudo-diffusive transmission [23

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

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]

]. In addition, it brings in a new prospect to the photon-dispersion engineering and may find applications in photonic and electromagnetic wave generation and propagation devices.

The presence of the Dirac cone on the Brillouin zone boundary was known for two-dimensional photonic crystals [25

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]

], and so far, those photonic crystals have been examined theoretically to predict the above-mentioned characteristics [19

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

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

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

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

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

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]

]. Quite recently, Chan and his collaborators showed by the multiple scattering theory that the Dirac cone can also be materialized in the zone center of a two-dimensional photonic crystal using the accidental degeneracy of a doubly degenerate mode and a non-degenerate mode [26

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]

].

In this paper, I present another way to materialize the isotropic Dirac cone in the Brillouin zone center of two- and three-dimensional metamaterials. 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. 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 a two-dimensional square-lattice metamaterial, is solved to show the presence of the isotropic Dirac cone. In Section 3, the theory is extended to three dimensions. We deal with the case of accidental degeneracy of a triply degenerate mode and a non-degenerate mode in the cubic lattice and show the presence of the Dirac cone that is isotropic in all directions. Lengthy calculations to derive and solve the secular equation for the square lattice are given in Appendix A. Those for the cubic lattice and detailed analysis of electromagnetic transfer integrals for the T1u and A1g modes are given in Appendix B.

2. Isotropic Dirac cone in two dimensions

Maxwell’s wave equation for the magnetic field H(r,t) is given by
𝒧H(r,t)×[1ε(r)×H(r,t)]=1c22t2H(r,t),
(5)
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

4. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).

] and assume, as an approximation to extract basic features of our problem, that ε(r) is real. Thus, Eq. (5) leads to eigenvalue problems.

The essential point to realize the Dirac cone in higher dimensions is mixing and repulsion of modes for all 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 A1 mode on the Γ point in the square lattice of unit structures with the C4v (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]

] and the presence of a resonant state of the E symmetry was shown by numerical calculation. In general, there are resonant states of all possible symmetries of the C4v point group.

Fig. 2 Left: Metallic unit structure with the C4v (regular square) symmetry. Right: Square array of the unit metallic structure on a uniform dielectric-slab waveguide with a back electrode.

For a single unit structure described by dielectric constant εs(r), we assume a doubly degenerate E resonant state and a non-degenerate A1 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:
×[1εs(r)×H(1,2)(r)]=ω12c2H(1,2)(r),
(6)
×[1εs(r)×H(0)(r)]=ω02c2H(0)(r),
(7)
where ω1 (ω0) is the resonance angular frequency of the E (A1) state. Without loss of generality, we can assume that these distributions are normalized as follows:
VdrH(i)*(r)H(j)(r)=Vδij,
(8)
where V is the volume on which we impose the periodic boundary condition.

We know from the group theory that the two eigenfunctions (magnetic field) of the E state (H(1) and H(2)) have the spatial symmetry as the x and y coordinates, respectively, while the eigenfunction of the A1 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]

]. When there are no other resonant states in the relevant frequency range, we can form the Bloch functions by linear combinations of these resonant states. Then, on the kx axis, where the parity with respect to the 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 ky axis.

In the following, I show that the mixing and repulsion take place for all two-dimensional directions to create the isotropic Dirac cone. Here we should note that the mixing does not take place on the Γ point, which is another necessary condition for the linear dispersion, since the Bloch functions of the Γ point have the full symmetry of the basis functions, and so the Bloch functions made of the E and A1 modes have different symmetries.

Thus I solve Maxwell’s wave equation by the tight-binding approximation to show these characters. This approximation is justified because the electromagnetic waves are well localized in each unit structure of metallic metamaterials [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

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

]. I describe the total magnetic field, according to the prescription of the tight-binding approximation in solid state physics, as a linear combination of 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.

Then, we can derive the secular equation of the electromagnetic eigenvalue problem, whose details are described in Appendix A. The problem is reduced to the diagonalization of a Hermitian matrix of the following form:
B=(B00B01B02B10B110B200B22).
(9)
When we keep the nearest neighbor terms, which are sufficient to analyze the dispersion in the vicinity of the Γ point, the matrix elements in Eq. (9) depend on kx and ky as
B00=ω02c2+M0+2M0(coskxa+coskya),
(10)
B11=ω12c2+M1+2M1coskxa+2M1coskya,
(11)
B22=ω12c2+M1+2M1coskya+2M1coskxa,
(12)
B01=B10*=2iM2sinkxa,
(13)
B02=B20*=2iM2sinkya,
(14)
where a is the lattice constant of the periodic metamaterial and Mn are electromagnetic transfer integrals whose definition and character are presented in Appendix A.

Now, we assume accidental degeneracy of the A1 and E mode frequencies in the zone center and denote the degenerate frequency by ωΓ. Such accidental degeneracy is generally possible by adjusting sample parameters. For example, it was attained by adjusting the lattice constant in Ref. [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]

]. By diagonalizing matrix B to the second order of kx and ky, we obtain the following dispersion relations:
ωk={ωΓ±|M2|ac2k/ωΓMa2c2k2/6ωΓ,ωΓMa2c2k2/6ωΓ,
(15)
where M = M1 + M2 + M0 and k=kx2+ky2 (See Appendix A). Figure 3 illustrates the three solutions. The dispersion relation of two of the three modes is linear to 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.

Fig. 3 Isotropic Dirac cone in two dimensions realized by accidental degeneracy of a doubly degenerate E mode and a non-degenerate A1 mode. The dashed square is the third parabolic band. The origin of the frequency ω is shifted to the degenerate frequency ωΓ.

3. Isotropic Dirac cone in three dimensions

In three dimensions, a similar situation can be realized by combining a T1u mode and an A1g mode in the simple cubic lattice of unit structures of the Oh symmetry, that is, the symmetry of a regular cube. Such a metamaterial is illustrated in Fig. 4. The T1u mode is triply degenerate and its three wave functions behave like the x, y, and z coordinates, respectively, whereas the A1g mode behaves like x2 + y2 + z2 [27

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

]. On each axis in the 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.

Fig. 4 Left: Metallic unit structure with the Oh (regular cube) symmetry. Right: Simple-cubic array of the unit metallic structures.

In this paper, we examined two cases, that is, the accidental degeneracy of the E and A1 mode of the square lattice and that of the T1u and A1g 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

It was shown by analytical calculation based on the tight-binding approximation and group theory 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, accidental degeneracy of a doubly degenerate E mode and a non-degenerate A1 mode of the square lattice of the C4v 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 T1u mode and a non-degenerate A1g 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.

The essential point for creating the Dirac cone is the crossing and anti-crossing properties caused by particular combinations of spatial symmetries of resonant states localized in a single unit structure of the metamaterial. That is, the two relevant modes must have different symmetries for 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.

To prove these properties, the dispersion relation of the periodic metamaterial was analytically calculated for small 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.

Appendix

A. E and A1 modes of two-dimensional square lattice

An essential part of the theory is evaluation of the electromagnetic transfer integral. In two dimensions, it is defined by
Llm(ij)1VVdrH(i)*(r)𝒧H(j)(rrlm),
(25)
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 A1 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 rlm are the lattice vectors of the square lattice given by
rlm=l(a00)+m(0a0),
(26)
where a is the lattice constant, and l and m are integers.

Because of the high symmetry of the unit structure (C4v) and the square lattice, we can derive various relations among Llm(ij). For the two eigenfunctions of the E mode, relations among their transfer integrals are given in Eqs. (14)(18) of Ref. [18

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

]. For the nearest neighbor lattice points, they are
L0,0(11)=L0,0(22)ω12c2+M1,
(27)
L±1,0(11)=L0,±1(22)M1,
(28)
L0,±1(11)=L±1,0(22)M1,
(29)
where ω1 is the resonance frequency of the E mode. In these equations, I use notations of M different from those 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]

] for the sake of convenience. All other Llm(ij) for the nearest neighbor lattice points are equal to zero. As for the A1 mode, it is invariant by any symmetry operation of the C4v point group. Using this property, we can easily prove
L0,0(00)ω02c2+M0,
(30)
L±1,0(00)=L0,±1(00)M0,
(31)
±L±1,0(01)=L±1,0(10)*=±L0,±1(02)=L0,±1(20)*M2,
(32)
where ω0 is the resonance frequency of the A1 mode.

According to the prescription of the tight-binding approximation, the Bloch eigenfunction is expressed by the linear combination of the three basis functions over all unit structures in V with k-dependent phase factors:
Hk(r)=1Vl,meikrlmi=02AiH(i)(rrlm).
(33)
Following the same procedure as described 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]

], the eigenvalue equation,
𝒧Hk(r)=ωk2c2Hk(r),
(34)
leads to the secular equation
|Bωk2c2I|=0,
(35)
where I is the rank-3 unit matrix. Elements of matrix B are defined by
Bij=lmeia(lkx+mky)Llm(ij).
(36)
When we keep the nearest neighbor terms, we obtain Eqs. (10)(14).

We consider the case of accidental degeneracy of the E and A1 modes at the Γ point in the following. So, we assume
ξΓ(1)=ξΓ(0)ξΓ.
(49)
Then, from Eqs. (39), (47), and (48),
b2=3ξΓ+Mk2a2,
(50)
p=4|M2|2k2a2,
(51)
q=0,
(52)
where
M=M0+M1+M1.
(53)
Thus, we obtain
ξ={ξΓ±2|M2|kaMk2a2/3,ξΓMk2a2/3.
(54)
In the vicinity of the Γ point where ckωΓcξΓ, we obtain
ωk=cξ={ωΓ±|M2|ac2k/ωΓMa2c2k2/6ωΓ,ωΓMa2c2k2/6ωΓ.
(55)

B. T1 and A1g modes of simple cubic lattice

In three dimensions, we define L by
Llmn(ij)1VVdrH(i)*(r)𝒧H(j)(rrlmn),
(56)
where rlmn are lattice vectors of the simple cubic lattice:
rlm=l(a00)+m(0a0)+n(00a).
(57)

A polynomial representation of the T1u 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]

]. So, we can choose three eigenfunctions of the T1u symmetry, H(1)(r), H(2)(r), and H(3)(r), such that they are transformed as the following three functions when any ROh, whose elements are illustrated in Fig. 5, is operated.
f1(r)=x,f2(r)=y,andf3(r)=z.
(58)
By definition, they are transformed as
Rfi(r)=fi(R1r).
(59)
So, for example,
σxf1(r)=x=f1(r),
(60)
σxf2(r)=y=f2(r),
(61)
σxf3(r)=z=f3(r).
(62)
We write these three equations in a matrix form:
σx=(f1f2f3)=(1000,100,01)(f1f2f3).
(63)

Fig. 5 Symmetry operations for the Oh point group

In this way, we can introduce the matrix representation of ROh:
σx:(1,0,00,1,00,0,1),C4x:(1,0,00,0,10,1,0),
(64)
σy:(1,0,00,1,00,0,1),C4y:(0,0,10,1,01,0,0),
(65)
σz:(1,0,00,1,00,0,1),C4z:(0,1,01,0,00,0,1),
(66)
C4x1:(1,0,00,0,10,1,0),C2x:(1,0,00,1,00,0,1),
(67)
C4y1:(0,0,10,1,01,0,0),C2y:(1,0,00,1,00,0,1),
(68)
C4z1:(0,1,01,0,00,0,1),C2z:(1,0,00,1,00,0,1),
(69)
C2(110):(0,1,01,0,00,0,1),C2(11¯0):(0,1,01,0,00,0,1),
(70)
C2(011):(1,0,00,0,10,1,0),C2(011¯):(1,0,00,0,10,1,0),
(71)
C2(101):(0,0,10,1,01,0,0),C2(1¯01):(0,0,10,1,01,0,0),
(72)
C3(111):(0,1,00,0,11,0,0),C3(111)1:(0,0,11,0,00,1,0),
(73)
C3(1¯11):(0,0,11,0,00,1,0),C3(1¯11)1:(0,1,00,0,11,0,0),
(74)
C3(11¯1):(0,0,11,0,00,1,0),C3(11¯1)1:(0,1,00,0,11,0,0),
(75)
C3(111¯):(0,0,11,0,00,1,0),C3(111¯)1:(0,1,00,0,11,0,0),
(76)
I:(1,0,00,1,00,0,1),E:(1,0,00,1,00,0,1),
(77)
IC4x:(1,0,00,0,10,1,0),IC4x1:(1,0,00,0,10,1,0),
(78)
IC4y:(0,0,10,1,01,0,0),IC4y1:(0,0,10,1,01,0,0),
(79)
IC4z:(0,1,01,0,00,0,1),IC4z1:(0,1,01,0,00,0,1),
(80)
σdx:(1,0,00,0,10,1,0),σdx:(1,0,00,0,10,1,0),
(81)
σdy:(0,0,10,1,01,0,0),σdy:(0,0,10,1,01,0,0),
(82)
σdz:(0,1,01,0,00,0,1),σdz:(0,1,01,0,00,0,1),
(83)
IC3(111):(0,1,00,0,11,0,0),IC3(111)1:(0,0,11,0,00,1,0),
(84)
IC3(1¯11):(0,0,11,0,00,1,0),IC3(1¯11)1:(0,1,00,0,11,0,0),
(85)
IC3(11¯1):(0,0,11,0,00,1,0),IC3(11¯1)1:(0,1,00,0,11,0,0),
(86)
IC3(111¯):(0,0,11,0,00,1,0),IC3(111¯)1:(0,1,00,0,11,0,0),
(87)

It is easily confirmed that this matrix representation gives the correct values of characters for the T1u 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 T1u representation, they transform like {fi}. But we should note that there is a difference due to the vector nature of the former. Their transformation is defined as
[RH(i)](r)RH(i)(R1r).
(88)

Table 1. Character table for the A1g and T1u modes of the Oh point group

table-icon
View This Table

Let us proceed to the derivation of relations among Llnm(ij) introduced by Eq. (56). As the first example, let us change the variable of integration from r to r′ = C4yr and evaluate L000(11). Since C4y does not change the size of volume elements, we have
VL000(11)=VdrH(1)*(C4y1r)[𝒧H(1)(C4y1r)]=Vdr[C4y1C4yH(1)*(C4y1r)][C4y1C4y𝒧C4y1C4yH(1)(C4y1r)],
(89)
where we introduced three pairs of C4y1C4y(E) for later use. Since C4y1 does not change the value of inner products, we obtain
VL000(11)=Vdr[C4yH(1)*(C4y1r)][𝒧C4yH(1)(C4y1r)],
(90)
where 𝒧′ is defined as
𝒧=C4y𝒧C4y1.
(91)
It is an operator in the r′ coordinate system equivalent to 𝒧 in the r coordinate system. Substituting the second relation of Eq. (65) and Eq. (88), we obtain
VL00(11)=Vdr[H(3)*(r)]𝒧[H(3)(r)]=VL000(33).
(92)
Similarly, by changing the variable from r to r′ = C4zr and using the second relation of Eq. (66), we obtain
L000(11)=L000(22).
(93)
So, we denote them by
L000(11)=L000(22)=L000(33)ω12c2+M1,
(94)
where the first term on the right-hand side is the original value for an isolated unit structure.

Next, let us examine L000(12). By changing the variable of integration from r to r′ = C4xr and using the second relation of Eq. (64), we obtain
VL000(12)=Vdr[C4xH(1)*(C4x1r)][𝒧C4xH(2)(C4x1r)]=VdrH(1)*(r)𝒧H(3)(r)=VL000(13).
(95)
Note that 𝒧=C4x𝒧C4x1 for this case. On the other hand, by changing the variable of integration from r to r=C4x1r and using the first relation of Eq. (67), we obtain
VL000(12)=Vdr[C4x1H(1)*(C4xr)][𝒧C4x1H(2)(C4xr)]=Vdr[C4x1H(1)*](r)[𝒧C4x1H(2)](r)=VdrH(1)*(r)𝒧[H(3)(r)]=VL000(13).
(96)
So, from Eqs. (95) and (96), we obtain
L000(12)=L000(13)=0.
(97)
By similar calculations, we also obtain
L000(21)=L000(31)=L000(23)=L000(32)=0.
(98)
Let us proceed to the evaluation of L100(11). Using r′ = C4yr,
VL100(11)=Vdr[C4yH(1)*](r)𝒧[C4yH(1)](rC4yr100)=Vdr[H(3)*(r)]𝒧[H(3)(rr00,1)]=VL00,1(33).
(99)
On the other hand, by changing the variable of integration from r to r=C4y1r, we obtain
VL100(11)=Vdr[C4y1H(1)*](r)𝒧[C4y1H(1)](rC4y1r100)=VL001(33).
(100)
These two relations and similarly calculations for C4z, C4z1, and C2y lead to
L±1,00(11)=L0,±1,0(22)=L00,±1(33)M1.
(101)
Application of all other ROh gives the same result as Eq. (101). As for L010(11), we may repeat similar calculations and obtain
L0,±1,0(11)=L00,±1(11)=L±1,00(22)=L00,±1(22)=L±1,00(33)=L0,±1,0(33)M1.
(102)
Next, let us examine L100(12). Using r′ = C4xr,
VL100(12)=Vdr[C4xH(1)*](r)𝒧[C4xH(2)](rC4xr100)=VdrH(1)*(r)𝒧H(3)(rr100)=VL100(13).
(103)
On the other hand, by changing the variable of integration from r to r=C4x1r, we obtain
VL100(12)=Vdr[C4x1H(1)*](r)𝒧[C4x1H(2)](rC4x1r100)=VL100(13).
(104)
So, we obtain
L100(12)=L100(13)=0.
(105)
This equation and similar calculations for all other ROh lead to
L±1,00(12)=L0,±1,0(12)=L±1,00(21)=L0,±1,0(21)=L±1,00(13)=L00,±1(13)=L±1,00(31)=L00,±1(31)=L0,±1,0(23)=L00,±1(23)=L0,±1,0(32)=L00,±1(32)=0,
(106)
where we used the following relation:
Llmn(ij)=Ll,m,n(ji)*.
(107)
As for L001(12), we change the variable of integration from r to r′ = σxr and obtain
VL001(12)=Vdr[σxH(1)*](r)𝒧[σxH(2)](rσxr001)=Vdr[H(1)*(r)]𝒧H(2)(rr001)=VL001(12).
(108)
So, we obtain L001(12)=0. Similar calculations lead to
L00,±1(12)=L00,±1(21)=L0,±1,0(13)=L0,±1,0(31)=L±1,00(23)=L±1,00(32)=0.
(109)
Equations (94), (97), (98), (101), (102), (106), and (109) give all relations among the three functions of the T1u mode.

Now let us proceed to the A1g mode. Its eigen function H(0) is invariant when any ROh is applied as can be found in Table 1:
[RH(0)](r)=H(0)(r).
(110)
First, let us denote L000(00) by
L000(00)ω02c2+M0.
(111)
Next, let us examine L100(00). By changing the variable of integration from r to r′ = σxr, we obtain
VL100(00)=VdrH(0)*(r)𝒧H(0)(rσxr100)=VL1,00(00).
(112)
In addition to this equation, by applying C3(111), C3(111)1, σy, and σz, we obtain
L±1,00(00)=L0,±1,0(00)=L00,±1(00)M0.
(113)
By using Eq. (107), we can prove
M0*=M0,
(114)
so M0 is a real number.

As for cross terms between the T1u and A1g modes, let us start with L000(01). By changing the variable of integration from r to r′ = σxr, we can easily prove
L000(01)=0.
(115)
Similar calculations lead to
L000(01)=L000(10)=L000(02)=L000(20)=L000(03)=L000(30)=0.
(116)
Also using σx, we obtain
L010(01)=0.
(117)
Similar calculations in this case result in
L0,±1,0(01)=L00,±1(01)=L0,±1,0(10)=L00,±1(10)=L±1,00(02)=L00,±1(02)=L±1,00(20)=L00,±1(20)=L±1,00(03)=L0,±1,0(03)=L±1,00(30)=L0,±1,0(30)=0.
(118)
Next, let us examine L100(01). By changing the variable of integration from r to r′ = σxr, we have
VL100(01)=Vdr[σxH(0)*](r)𝒧[σxH(1)](rσxr100)=VdrH(0)*(r)𝒧[H(1)(rr1,00)]=VL1,00(01).
(119)
Similarly, by using C3(111) and C3(111)1, and further applying σy and σz, we obtain
±L±1,00(01)=±L0,±1,0(02)=±L00,±1(03)M2.
(120)
Then, from Eq. (107), we also obtain
L±1,00(10)=L0,±1,0(20)=L00,±1(30)M2*.
(121)
In summary, we have the following relations:
L000(00)ω02c2+M0,
(122)
L±1,00(00)=L0,±1,0(00)=L00,±1(00)M0,
(123)
L000(11)=L000(22)=L000(33)ω12c2+M1,
(124)
L±1,00(11)=L0,±1,0(22)=L00,±1(33)M1,
(125)
L0,±1,0(11)=L00,±1(11)=L±1,00(22)=L00,±1(22)=L±1,00(33)=L0,±1,0(33)M1,
(126)
±L±1,00(01)=L±1,00(10)*=±L0,±1,0(02)=L0,±1,0(20)*=±L00,±1(03)=L00,±1(30)*M2.
(127)
All other Llmn(ij) for the nearest neighbor lattice points are equal to zero. So, the number of independent elements among 112 Llmn(ij) is six.

The Bloch function in the present case is given by a linear combination of the eigenfunctions of A1g mode (H(0)) and T1u mode (H(1), H(2), H(3)):
Hk(r)=1Vl,m,neikrlmni=03AiH(i)(rrlmn).
(128)
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
Bij=lmneia(lkx+mky+nkz)Llmn(ij).
(129)
When we keep the nearest neighbor terms, we obtain Eqs. (17)(23). We denote
ξ=ωk2c2,ξ0=ω02c2,ξ1=ω12c2
(130)
as before. Then the secular equation reduces to
ξ4+b3ξ3+b2ξ2+b1ξ+b0=0,
(131)
where
b3=(B00+B11+B22+B33),
(132)
b2=B00B11+B00B22+B00B33+B11B22+B11B33+B22B33(|B01|2+|B02|2+|B03|2),
(133)
b1=(B00B11B22+B11B22B33+B22B33B00+B33B00B11)+|B01|2(B22+B33)+|B02|2(B11+B33)+|B03|2(B11+B22),
(134)
b0=B00B11B22B33(|B01|2B22B33+|B02|2B11B33+|B03|2B11B22).
(135)
For the Γ point (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
ξ={ξ0+M0+6M0ξΓ(0),ξ1+M1+2M1+4M1ξΓ(1)(tripleroot).
(136)
To solve Eq. (131) for general k, we change the unknown variable from ξ to
η=ξ+b34.
(137)
Then, Eq. (131) is reduced to
η4+pη2+qη+r=0,
(138)
where
p=b23b328,
(139)
q=b1b2b32+b338,
(140)
r=b0b1b34+b2b32163b34256.
(141)
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 kx, ky and kz, respectively. The results are
p=38(ξΓ(1)ξΓ(0))2+14{(ξΓ(1)ξΓ(0))(M1+2M13M0)16|M2|2}k2a2
(142)
q=18(ξΓ(1)ξΓ(0))318{(ξΓ(1)ξΓ(0))2(M1+2M13M0)16(ξΓ(1)ξΓ(0))|M2|2}k2a2
(143)
r=3256(ξΓ(1)ξΓ(0))4+{164(ξΓ(1)ξΓ(0))3(M1+2M13M0)14(ξΓ(1)ξΓ(0))2|M2|2}k2a2+{1128(ξΓ(1)ξΓ(0))2[8(M1+2M1)M0+(M1+2M1+M0)(7M1+14M19M0)]12(ξΓ(1)ξΓ(0))(M1+2M1+M0)|M2|2}k4a4+{1768(ξΓ(1)ξΓ(0))3(M1+2M13M0)+(ξΓ(1)ξΓ(0))2[316(2M1+M1)M1+112|M2|2]+2(ξΓ(1)ξΓ(0))M1|M2|2}(kx4+ky4+kz4)a4+{316(ξΓ(1)ξΓ(0))2(M12+2M1M1+3M12)+2(ξΓ(1)ξΓ(0))(M1+M1)|M2|2}(kx2ky2+ky2kz2+kz2kx2)a4,
(144)
where k=kx2+ky2+kz2.

We consider the case of accidental degeneracy of the T1u and A1g modes at the Γ point in the following. So, we assume
ξΓ(1)=ξΓ(0)ξΓ.
(145)
Then, from Eqs. (142)(144),
p=4|M2|2k2a2,q=r=0.
(146)
Since
b3=4ξΓ+Mk2a2,
(147)
where M′ = M1 + 2M1 + M0, the solutions of the secular equation, Eq. (131), are
ξ={ξΓ±2|M2|kaMk2a2/4ξΓMk2a2/4(doubleroot).
(148)
So, in the vicinity of the Γ point where ckωΓcξΓ, we obtain
ωk={ωΓ±|M2|ac2k/ωΓMa2c2k2/8ωΓ,ωΓMa2c2k2/8ωΓ(doubleroot).
(149)

Acknowledgments

This study was supported by a Grant-in-Aid for Scientific Research on Innovative Areas from the Japanese Ministry of Education, Culture, Sports and Technology (Grant number 22109007).

References and links

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,” 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]

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. 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]

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]

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]

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]

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

  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]

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