OSA's Digital Library

Optics Express

Optics Express

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

Double Dirac cones in triangular-lattice metamaterials

Kazuaki Sakoda  »View Author Affiliations


Optics Express, Vol. 20, Issue 9, pp. 9925-9939 (2012)
http://dx.doi.org/10.1364/OE.20.009925


View Full Text Article

Acrobat PDF (965 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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 E1 and E2 modes in triangular-lattice metamaterials of C6v symmetry. The Dirac point thus obtained is equivalent to a zero-index system, so we can expect unique optical propagation phenomena such as constant-phase waveguides and lenses of arbitrary shapes. Zitterbewegung is also expected without disturbance due to an auxiliary quadratic dispersion surface, which is present for other combinations of mode symmetries to materialize the Dirac cones. To the best of the author’s knowledge, this is the first prediction of the presence of a double Dirac cone in metamaterials.

© 2012 OSA

1. Introduction

Metamaterials have brought about new possibilities in photon dispersion engineering. As is well known for periodic structural materials [1

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

3

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

], peculiar dispersion relations of the electromagnetic field result in many novel optical properties such as inhibition of spontaneous emission due to photonic band gaps, enhancement of stimulated emission by small group velocity, extraordinary refraction due to anisotropic isofrequency surface, and enhancement of nonlinear optical processes by small group velocity and pseudo phase matching. In addition to these properties common to periodic structures, metamaterials have one more degree of freedom, that is, the electromagnetic resonance in unit structures. This last feature gives us larger flexibility in designing the electromagnetic field.

An example is the linear dispersion relation in the Brillouin zone center brought about by accidental degeneracy, or degeneracy without a symmetry reason, of two modes, which is realized by appropriately designed sample structures and their resonances. This property was demonstrated in the microwave frequency range and the linear dispersion relation was verified by the transmission line theory [4

4. C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).

7

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]

].

On the other hand, the presence of an isotropic linear dispersion relation, or Dirac cone, on the Brillouin zone boundary was known for two-dimensional photonic crystals [8

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]

]. The presence of a Dirac cone results in many exotic phenomena like unidirectional edge states [9

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

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

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]

], and pseudo-diffusive transmission [14

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

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]

]. 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 accidental degeneracy of two modes [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]

].

We recently analyzed the dispersion relation of metamaterials based on tight-binding approximation and group theory and clarified the condition needed to obtain the linear dispersion in the Brillouin zone center from a view point of spatial symmetries of resonance states [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]

, 19

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

]. We also showed that we can create the Dirac cone in two- and three-dimensional metamaterials by accidental degeneracy of degenerate and non-degenerate eigenmodes [20

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

]. In Ref. [20

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

], accidental degeneracy of a doubly degenerate E mode and a non-degenerate A1 mode was examined for two-dimensional square lattices and that of a triply degenerate T1u mode and a non-degenerate A1g mode was examined for three-dimensional simple cubic lattices. In both cases, the presence of the Dirac cone in the Brillouin zone center was shown in addition to an auxiliary quadratic dispersion surface. To the best of the author’s knowledge, this study was the first prediction of the presence of the Dirac cone in three-dimensional periodic structures.

However, the presence of the auxiliary quadratic dispersion surface may be an obstacle for various applications mentioned above. For example, 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 C6v symmetry, is solved to show the presence of the double Dirac cone. We deal with the case of accidental degeneracy of doubly degenerate E1 and E2 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

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),
(1)
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

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

]. Thus, Eq. (1) leads to an eigenvalue problem.

The essential point to materialize the Dirac cone in the Brillouin zone center is mixing and repulsion of modes for all k except the Γ point. As will be shown below, this condition is fulfilled by accidental degeneracy of doubly degenerate E1 and E2 modes on the Γ point in the triangular lattice of unit structures of the C6v (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]

], and the presence of resonant states of the E1 and E2 symmetries was shown by numerical calculation. We denote their magnetic field distributions by H(1) and H(2) for the E1 state and H(3) and H(4) for the E2 state. Thus, these magnetic fields satisfy the following eigenvalue equations:
×[1εs(r)×H(1,2)(r)]=ω12c2H(1,2)(r),
(2)
×[1εs(r)×H(3,4)(r)]=ω22c2H(3,4)(r),
(3)
where εs(r) is the dielectric constant of a single unit structure and ω1 (ω2) is the resonance angular frequency of the E1 (E2) state. Without loss of generality, we can assume that these distributions are normalized as follows:
VdrH(i)*(r)H(j)(r)=Vδij,
(4)
where V is the volume on which we impose the periodic boundary condition.

Fig. 1 (a) Top view of the unit structure with the C6v (regular hexagon) symmetry composed of a thin metallic film. (b) Symmetry operations of the C6v group. There are two sets of three equivalent mirror reflections that are denoted by (σx, σx, σx) and (σy, σy, σy). Seven lattice points (the origin and its nearest neighbors) are denoted by integers from 0 to 6. (c) Triangular array of the unit metallic structures on a uniform dielectric-slab waveguide with a back electrode. Localized electromagnetic resonance states are formed between the metallic unit structures and the back electrode [19].

We know from the group theory that the two eigen functions of the E1 state (H(1) and H(2)) transform like x and y when any symmetry operation of C6v is performed, while those of the E2 state (H(3) and H(4)) transform like 2xy and x2y2 [21

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

]. When there are no other resonant states in the relevant frequency range, we can form Bloch functions by linear combinations of these resonant states. Because a Bloch function has full symmetry of its basis function on the Γ point, the Bloch function made of the E1 state has a symmetry different from that made of the E2 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 kx axis, kx ≠ 0 while ky = 0. So, the k vector is conserved only by σy symmetry operation (see Fig. 1(b)). Thus the Bloch function on the kx axis has the parity with respect of σy, but has no other symmetry. Similarly, the Bloch function on the ky axis has the parity with respect of σx.

Parities of the four functions are summarized in Table 1. On the kx axis, Bloch functions made of 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 ky axis, Bloch functions made of 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.

Table 1. Parity of resonant states of a single unit structure.

table-icon
View This Table

We can derive the secular equation of the electromagnetic eigenvalue problem based on the tight-binding approximation, whose details are described in the Appendix. The problem is reduced to the diagonalization of a Hermitian matrix of the following form:
B=(B11B12B13B14B12*B22B23B13B13*B23*B33B34B14*B13*B34*B44).
(5)
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. (5) depend on kx and ky as
B11=ω12c2+2M12coskx+4M14coskxa2cos3kya2,
(6)
B22=ω12c2+2M13coskx+4M15coskxa2cos3kya2,
(7)
B33=ω22c2+2M22coskx+4M24coskxa2cos3kya2,
(8)
B44=ω22c2+2M23coskx+4M25coskxa2cos3kya2,
(9)
B12=4M16sinkxa2sin3kya2,
(10)
B13=4iM3coskxa2sin3kya2,
(11)
B14=2iM1sinkxa+4iM4sinkxa2cos3kya2,
(12)
B23=2iM2sinkxa+4iM5sinkxa2cos3kya2,
(13)
B34=4M26sinkxa2sin3kya2,
(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. 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.

Now, we assume accidental degeneracy of the E1 and E2 mode frequencies in the zone center and denote the degenerate frequency by ωΓ. 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]

]. By diagonalizing matrix B to the second order of kx and ky, we obtain the following dispersion relation:
ωkωΓ±3|M3|ac2kωΓ3Ma2c2k216ωΓ(doubleroots).
(15)
where M = M12 + M13 + M22 + M23 and k=kx2+ky2 (See Appendix B). So, in the vicinity of the Γ point, the dispersion relation is isotropic and linear in k. Figure 2 illustrates the four solutions, which consist of two identical pairs of isotropic cones, or double Dirac cone.

Fig. 2 Double Dirac cone realized by accidental degeneracy of E1 and E2 modes of triangular-lattice metamaterials of the C6v symmetry. The origin of the frequency ω is shifted to the degenerate frequency ωΓ. Although the two cones are illustrated as slightly separated from each other to emphasize the presence of two cones, their actual shapes are identical.

3. Discussion

We consider wave propagation in a certain direction and denote its wave number by k. Then the relevant dispersion relation for the Dirac cone is
ωωΓ=vk,
(16)
where v is the slope of the dispersion curve, or the group velocity at the Γ point, which is
v=3|M3|ac2ωΓ
(17)
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 neff to the present problem, it should satisfy the following equation by definition:
ω=cneffk.
(18)
At first glance, Eqs. (16) and (18) are not compatible with each other. But they are actually compatible if and only if neff has the following form:
neff=A(ω)(ωωΓ),
(19)
where A(ω) is a slowly varying function of ω and is not divergent or equal to zero in the vicinity of ωΓ. From Eqs. (16), (18), and (19), we obtain
v=cωA(ω)cωΓA(ωΓ).
(20)
Thus, we can conclude that the metamaterials with the double Dirac cones described in this study are zero-index materials at ω = ωΓ and neff < 0 for ω < ωΓ.

If we assume, as is often done in the conventional discussion on metamaterials, that neff is described by a product of an effective dielectric constant εeff and permeability μeff such that
neff=εeffμeff,
(21)
both εeff and μeff are negative for ω < ωΓ. However, we should note that we do not need this decomposition of neff in the framework of the tight-binding approximation and neff is determined by ωΓ and one of the electromagnetic transfer integrals, M3.

In this sense, the effective refractive index introduced by the tight-binding picture is a generalization of the conventional effective refractive index and the present theory does not need the ideas of effective dielectric constant and permeability. As an example, let us consider the Dirac point discussed by Wang et al. [16

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]

]. They assumed the following forms:
εeff(ω)=1ωep2ω2,
(22)
μeff(ω)=1ωmp2ω2.
(23)
They further assumed ωep = ωmpωΓ to realize a Dirac point. Thus
neff=1ωΓ2ω2,
(24)
which is equivalent to taking
A(ω)=ω+ωΓω2
(25)
in Eq. (19).

Let us make three remarks here. First, because the double Dirac cone is equivalent to a zero-index material at ω = ωΓ, 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]

]. Second, because we do not need an auxiliary quadratic dispersion surface to materialize the Dirac cone in our system, the wave propagation is solely carried out by modes with the linear dispersion relation in the vicinity of ωΓ. Thus, Zitterbewegung, an oscillatory transmission caused by interference between modes above and below ωΓ, 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.

To illustrate the third point clearly, let us examine the effective dielectric constant and permeability given in Eqs. (22) and (23) in more detail. When ωepωmp, the effective refractive index is given by
neff=(1ωep2ω2)(1ωmp2ω2).
(26)
In this case, the electromagnetic eigenmodes consist of two branches and have the following dispersion relations in the vicinity of the Γ point:
ω+ωep+ωepc2k22(ωep2ωmp2),
(27)
ωωmpωmpc2k22(ωep2ωmp2),
(28)
where we assumed ωep > ωmp without loss of generality. The discussion of Wang et al. [16

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]

] depends on the assumption that the two dispersion curves given in Eqs. (27) and (28) form the Dirac cone when accidental degeneracy, i.e. ωep = ωmp, takes place.

The simplest counterexample may be the combination of A1 and B1 modes of two-dimensional square lattices of the C4v (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:
ω=ωΓ+a2c2k24ωΓ{±(M1M2)2+4|M3|2cos22ϕ(M1+M2)},
(29)
where M1 and M2 are the nearest-neighbor transfer integrals for the A1 and B1 modes, respectively, M3 is their cross term, and ϕ is the angle between k and the x axis. For the combination of A1 and B2 modes of the C4v symmetry, M3 for the nearest-neighbor unit cells is equal to zero. When we take into account M3 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

It was shown by analytical calculation based on the tight-binding approximation and group theory that a double Dirac cone in the Brillouin zone center can be created in two-dimensional triangular-lattice metamaterials of the C6v symmetry by accidental degeneracy of eigenmodes of the E1 and E2 symmetries. The double Dirac cone consists of four isotropic linear dispersion relations with the same slopes, or in other words, a pair of identical Dirac cones. To prove this property, we examined electromagnetic transfer integrals for the nearest-neighbor lattice points and derived their mutual relations by fully using the spatial symmetry of the eigenmodes. The secular equation, which is a quartic equation of ω2, was solved analytically for small k in the Brillouin zone center. To the best of the author’s knowledge, this is the first paper proving the presence of a double Dirac cone in metamaterials.

By comparing the present results with conventional description by effective refractive indices, we showed that 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 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.

Appendix

A. Electromagnetic transfer integrals

In this section, we derive various relations among the electromagnetic transfer integrals for eigenmodes of the E1 and E2 irreducible representations of triangular-lattice metamaterials of the C6v symmetry. The electromagnetic transfer integral is defined by
Lm(ij)1VVdrH(i)*(r)𝒧H(j)(rrm),
(30)
where H(i)(r) (i = 1,2,3,4) are the magnetic fields of the resonant states of a single unit structure of the E1 symmetry (i = 1, 2) and the E2 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 {rm} 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 r0 ≡ (0,0,0) and rm (m = 1 to 6) as shown in Fig. 1(b).

From group theory [21

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

], we can assume without loss of generality that two eigen functions of the E1 mode transform like x and y when they are subjected to any symmetry operation of point group C6v. Then, we can derive various relations among Lm(ij) (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]

]:
L0(11)=L0(22)ω12c2+M11,
(31)
L0(12)=L0(21)=0,
(32)
L1(11)=L4(11)M12,
(33)
L1(22)=L4(22)M13,
(34)
L1(12)=L4(12)=L1(21)=L4(21)=0,
(35)
L2(11)=L3(11)=L5(11)=L6(11)M14,
(36)
L2(22)=L3(22)=L5(22)=L6(22)M15,
(37)
L2(12)=L3(12)=L5(12)=L6(12)=L2(21)=L3(21)=L5(21)=L6(21)M16,
(38)
where ω1 is the resonance frequency of the E1 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]

]:
M14=M12+3M134,
(39)
M15=3M12+M134,
(40)
M16=3(M12M13)4.
(41)
Note that notations that are slightly different from those 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]

] are used here to distinguish the E1 and E2 modes.

Similarly we can derive relations among the transfer integrals for the E2 mode whose two eigen functions can be assumed without loss of generality to transform like 2xy and x2y2 when they are subjected to any symmetry operation of C6v [21

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

]. 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]

], it was shown that the relations for the E2 mode have the same structure as the E1 mode:
L0(33)=L0(44)ω22c2+M21,
(42)
L0(34)=L0(43)=0,
(43)
L1(33)=L4(33)M22,
(44)
L1(44)=L4(44)M23,
(45)
L1(34)=L4(34)=L1(43)=L4(43)=0,
(46)
L2(33)=L3(33)=L5(33)=L6(33)M24,
(47)
L2(44)=L3(44)=L5(44)=L6(44)M25,
(48)
L2(34)=L3(34)=L5(34)=L6(34)=L2(43)=L3(43)=L5(43)=L6(43)M26,
(49)
M24=M22+3M234,
(50)
M25=3M22+M234,
(51)
M26=3(M22M23)4,
(52)
where ω2 is the resonance frequency of the E2 mode.

In addition to these relations, we have to newly derive relations among such transfer integrals that contain one E1-mode eigen function and one E2-mode eigen function like L1(13). For this purpose, we examine the results of transformation of the variable of integration by symmetry operation RC6v [19

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

, 20

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

]. In general, the transformation of the variable of integration from r to r′ = Rr leads to the following equation:
Lm(ij)=1VVdrH(i)*(R1r)[𝒧H(j)(R1rrm)],
(53)
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
Lm(ij)=1VVdr[R1RH(i)*(R1r)][R1R𝒧R1RH(j)(R1rrm)]=1VVdr[RH(i)*(R1r)][𝒧RH(j)(R1rrm)],
(54)
where we introduced
𝒧R𝒧R1,
(55)
which is an operator in the r′ coordinate system equivalent to 𝒧 in the r coordinate system. We can prove 𝒧′ = 𝒧 [2

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

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

]. We generally denote the transformation by
RH(i)(R1r)[RH(i)](r).
(56)
Due to the general properties of irreducible representations (like E1 and E2) of point groups (like C6v), RH(i) can be expressed by a linear combination of eigen functions of each irreducible representation. Then
Lm(ij)=1VVdr[RH(i)*(R1r)][𝒧RH(j)(R1{rRrm})]=1VVdr[RH(i)*](r)𝒧[RH(j)](rRrm)
(57)
must be a linear combination of { Lm(ij)}, so we obtain their mutual relations.

By examining all combinations of m, i, j, and R for transfer integrals that contain one E1-mode eigen function and one E2-mode eigen function, we obtain
L0(13)=L0(14)=L0(23)=L0(24)=L0(31)=L0(41)=L0(32)=L0(42)=0,
(58)
L1(13)=L1(24)=L1(31)=L1(42)=L4(13)=L4(24)=L4(31)=L4(42)=0,
(59)
L1(14)=L4(14)=L1(41)*=L4(41)*M1,
(60)
L1(23)=L4(23)=L1(32)*=L4(32)*M2,
(61)
L2(13)=L3(13)=L5(13)=L6(13)=L2(31)*=L3(31)*=L5(31)*=L6(31)*=L2(24)
(62)
=L3(24)=L5(24)=L6(24)=L2(42)*=L3(42)*=L5(42)*=L6(42)*M3,
(63)
L2(14)=L3(14)=L5(14)=L6(14)=L2(41)*=L3(41)*=L5(41)*=L6(41)*M4,
(64)
L2(23)=L3(23)=L5(23)=L6(23)=L2(32)*=L3(32)*=L5(32)*=L6(32)*M5.
(65)
We can further prove
M3=3(M1+M2)4,
(66)
M4=M1+3M24,
(67)
M5=3M1M24.
(68)
To derive these equations, we used the following property:
Lm(ij)*=Lm(ji),
(69)
where −m denotes the lattice vector given by −rm.

B. Solutions of secular equations

We express the Bloch eigen function of the electromagnetic field in a triangular-lattice metamaterial by a linear combination of four basis functions over all unit structures in volume V with k-dependent phase factors:
Hk(r)=1Vmeikrmi=14AiH(i)(rrm),
(70)
where {H(1), H(2)} are resonance states of the E1 symmetry and {H(3), H(4)} are those of the E2 symmetry. Then, the eigenvalue equation,
𝒧Hk(r)=ωk2c2Hk(r),
(71)
leads to the secular equation
|Bωk2c2I|=0,
(72)
where I is the rank-4 unit matrix. Elements of matrix B are defined by
Bij=meikrmLm(ij).
(73)
When we keep the nearest neighbor terms, we obtain Eqs. (6)(14).

We analytically solve the secular equation, Eq. (72). In the following, we denote
ξ=ωk2c2,ξ1=ω12c2,ξ2=ω22c2.
(74)
Then, the secular equation reduces to
ξ4+b3ξ3+b2ξ2+b1ξ+b0=0,
(75)
where
b3=(B11+B22+B33+B44),
(76)
b2=B11B22+B11B33+B11B44+B22B33+B22B44+B33B44(|B12|2+2|B13|2+|B14|2+|B23|2+|B34|2),
(77)
b1=(B22B33B44+B11B33B44+B11B22B44+B11B22B33)+|B12|2(B33+B44)+|B13|2(B11+B22+B33+B44)+|B14|2(B22+B33)+|B23|2(B11+B44)+|B34|2(B11+B22)+B12B13B14*+B12*B13*B14B12B13*B23B12*B13B23*B13B14*B34B13*B14B34*+B13B23*B34*+B13*B23B34,
(78)
b0=B11B22B33B44+|B13|4+|B14|2|B23|2|B12|2B33B44|B13|2(B11B33+B22B44)|B14|2B22B33|B23|2B11B44|B34|2B11B22B11(B13B23*B34*+B13*B23B34)+B22(B13B14*B34+B13*B14B34*)B33(B12B13B14*+B12*B13*B14)+B44(B12B13*B23+B12*B13B23*)+B132B14*B23*+B13*2B14B23.
(79)
In Eqs. (76)(79), terms higher than the fourth order of kx and ky were omitted, since they are not necessary in the following analysis.

For the Γ point (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
ξ={ξ1+32(M12+M13)ξΓ(1)(doubleroot),ξ2+32(M22+M23)ξΓ(2)(doubleroot),
(80)
where we use Eqs. (39) and (50). To solve Eq. (75) for general k, we change the unknown variable from ξ to
η=ξ+b34.
(81)
Then, Eq. (75) is reduced to
η4+pη2+qη+r=0,
(82)
where
p=b23b328,
(83)
q=b1b2b32+b338,
(84)
r=b0b1b34+b2b32163b34256.
(85)
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 and ky, respectively. The results are
p12(ξΓ(1)ξΓ(2))2+2(ξΓ(1)ξΓ(0))(M12+M13M22M23)(sin2kxa2+2sin2kxa4+2sin23kya4)4(|M1|2+|M2|2)sin2kxa16(|M4|2+|M5|2)sin2kxa232|M3|2sin23kya28(M1M4*+M1*M4+M2M5*+M2*M5)sinkxasinkxa2.
(86)
q0
(87)
r116(ξΓ(1)ξΓ(2))2+12(ξΓ(1)ξΓ(2))3(M22+M23M12M13)×(sin2kxa2+2sin2kxa4+2sin23kya44sin2kxa4sin23kya4)+(ξΓ(1)ξΓ(2))2×[8|M3|2(14sin2kxa4)sin23kya2+12{M122+M132+M222+M232+10(M12M13+M22M23)6(M12+M13)(M22+M23)}sin4kxa2+2{M142+M152+M242+M252+10(M14M15+M24M25)6(M14+M15)(M24+M25)}(sin2kxa4+sin23kya4)+2{5(M12+M13+M22+M23)216(M12+M13)(M22+M23)4(M12M14+M13M15+M22M24+M23M25)}×sin2kxa2(sin2kxa4+sin23kya4)4(|M16|2+|M26|2)sin2kxa2sin23kya216(|M4|2+|M5|2)sin2kxa2sin23kya4(M1M4*+M1*M4+M2M5*+M2*M5)sinkxasinkxa2sin23kya4+|M1sinkxa+2M4sinkxa2|2+|M2sinkxa+2M5sinkxa2|2]+(ξΓ(1)ξΓ(2))×[16{M3*(M16+M26)(M2sinkxa+2M5sinkxa2)+M3(M16*+M26*)(M2*sinkxa+2M5*sinkxa2)M3(M16+M26)(M1*sinkxa+2M4*sinkxa2)M3*(M16*+M26*)(M1sinkxa+2M4sinkxa2)}sinkxa2sin23kya2+32|M3|2(M22+M23M12M13)sin23kya2×(sin2kxa2+2sin2kxa4+2sin23kya4)+8{(M22M13)sin2kxa2+2(M24M15)(sin2kxa4+sin23kya4)}×|M1sinkxa+2M4sinkxa2|2+8{(M23M12)sin2kxa2+2(M25M14)(sin2kxa4+sin23kya4)}×|M2sinkxa+2M5sinkxa2|2]+256|M3|4sin43kya2+64M32sin23kya2(M2*sinkxa+2M5*sinkxa2)(M1*sinkxa+2M4*sinkxa2)+64M3*2sin23kya2(M2sinkxa+2M5sinkxa2)(M2sinkxa+2M4sinkxa2)+16|M1sinkxa+2M4sinkxa2|2|M2sinkxa+2M5sinkxa2|2.
(88)

We consider the case of accidental degeneracy of the E1 and E2 modes at the Γ point in the following. So, we assume
ξΓ(1)=ξΓ(2)ξΓ.
(89)
Then, by expanding sine functions in the limit of small kx and ky, we obtain from Eqs. (86)(88) and (66)(68),
p24|M3|2k2a2,
(90)
q0,
(91)
r144|M3|4k4a4,
(92)
where k=kx2+ky2. So,
η±23|M3|ka(doubleroots).
(93)
Since
b34ξΓ+32Mk2a2,
(94)
where M = M12 + M13 + M22 + M23, the solutions of the secular equation, Eq. (75), are
ξξΓ±23|M3|ka38Mk2a2(doubleroots).
(95)
So, in the vicinity of the Γ point where ckωΓcξΓ, we obtain
ωkωΓ±3|M3|ac2kωΓ3Ma2c2k216ωΓ(doubleroots).
(96)

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.

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

2.

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

3.

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

4.

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

5.

C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Antennas and Propagation Society International Symposium 2, 412–415 (2002).

6.

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

7.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14, 68–70 (2004). [CrossRef]

8.

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

10.

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

11.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009). [CrossRef]

12.

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

13.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett. 86, 47008 (2009). [CrossRef]

14.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007). [CrossRef]

15.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010). [CrossRef]

16.

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009). [CrossRef] [PubMed]

17.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011). [CrossRef]

18.

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

19.

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

21.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 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: 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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, Berlin, 2004).
  3. K. Sakoda and J. W. Haus, “Science and engineering of photonic crystals,” Prog. Opt.54, 271–317 (2010). [CrossRef]
  4. C. Caloz and T. ItohElectromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (Wiley, 2006).
  5. C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE Antennas and Propagation Society International Symposium2, 412–415 (2002).
  6. A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag. 5, 34–50 (2004). [CrossRef]
  7. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett.14, 68–70 (2004). [CrossRef]
  8. M. Plihal and A. A. Maradudin, “Photonic band structure of a two-dimensional system: The triangular lattice,” Phys. Rev. B44, 8565–8571 (1991). [CrossRef]
  9. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett.100, 013904 (2008). [CrossRef] [PubMed]
  10. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78, 033834 (2008). [CrossRef]
  11. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B80, 155103 (2009). [CrossRef]
  12. X. Zhang, “Observing zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett.100, 113903 (2008). [CrossRef] [PubMed]
  13. L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” Europhys. Lett.86, 47008 (2009). [CrossRef]
  14. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75, 063813 (2007). [CrossRef]
  15. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405, 2990–2995 (2010). [CrossRef]
  16. L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett.34, 1510–1512 (2009). [CrossRef] [PubMed]
  17. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater.10, 582–586 (2011). [CrossRef]
  18. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18, 27371–27386 (2010). [CrossRef]
  19. K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express19, 13899–13921 (2011). [CrossRef] [PubMed]
  20. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20, 3898–3917 (2012). [CrossRef] [PubMed]
  21. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, Berlin, 1990). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited