OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 25181–25194
« Show journal navigation

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

Kazuaki Sakoda  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 25181-25194 (2012)
http://dx.doi.org/10.1364/OE.20.025181


View Full Text Article

Acrobat PDF (1226 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Photonic Dirac cones, or the linear dispersion relation around certain points in the Brillouin zone, have been attracting considerable interest during the last five years [1

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

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

]. Haldane et al. [1

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

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]

] pointed out the presence of photonic Dirac cones on the Brillouin-zone boundary of two-dimensional triangular-lattice photonic crystals due to their structural symmetry and discussed unidirectional propagation of surface modes caused by time-reversal symmetry breaking. Ochiai et al. extended the discussion to honeycomb-lattice photonic crystals [3

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]

]. Zhang proposed optical simulation of 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]

]. A pseudo-diffusive transmission was found by Sepkhanov et al. [5

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]

] and numerically demonstrated by Diem et al. [6

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]

].

Recently, Huang et al. reported that Dirac cones can also be created in the Brillouin-zone center of two-dimensional dielectric photonic crystals by accidental degeneracy of two modes [7

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]

]. They showed for square- and triangular-lattice photonic crystals that combinations of a non-degenerate mode and a doubly degenerate mode yield a Dirac cone together with a quadratic dispersion surface. Because the Dirac point in the Brillouin-zone center is equivalent to a zero effective refractive index [7

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]

], it has much potential for various applications like scatter-free waveguides [13

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]

] and lenses of arbitrary shapes [14

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]

].

On the other hand, we showed by tight-binding approximation and group theory that Dirac cones can also be created in the Brillouin-zone center of metamaterials, which are characterized by well-defined electromagnetic resonant states localized in their unit structures, by accidental degeneracy of two 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]

12

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

]. First, we analyzed one-dimensional regular arrays of metallic unit cells to show that one-dimensional photonic Dirac cones can be created by accidental degeneracy of A1 and B1 modes and that of A2 and B2 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]

]. We also showed that a one-dimensional Dirac cone can be created by the 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]

]. In addition, we proved that the combination of A1 and E modes of square-lattice metamaterials and the combination of A1g and T1u 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]

]. We further proved that the combination of E1 and E2 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]

].

Quite recently, we systematically examined the relation between the symmetry of modes and the shapes of dispersion curves generated by accidental degeneracy for both dielectric photonic crystals and metamaterials with localized electromagnetic resonant states by numerical photonic-band calculation and tight-binding approximation, respectively [12

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

]. We found that the two calculations gave the same results, which strongly suggested the presence of universality of mode symmetries that enabled the creation of photonic Dirac cones irrespective of the details of the sample structure, irrespective of the presence or absence of localized resonant states, and irrespective of the approximation used in the calculation.

In this paper, anticipating that the presence or absence of photonic Dirac cones is solely determined by the combination of mode symmetries, we develop a new theory that does not depend on the details of the sample structure, but depends only on the spatial symmetry of the accidentally degenerate modes.

This paper is organized as follows. In Section 2, a degenerate perturbation theory is formulated for the vector electromagnetic field of periodic systems and it is applied to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. A necessary condition is derived by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. In Section 3 to Section 6, this method is applied to four periodic systems, that is, the one-dimensional lattice of C2v symmetry, the square lattice of C4v symmetry, the triangular lattice of C6v symmetry, and the simple-cubic lattice of Oh 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.

2. Theory

The eigen equation for the magnetic field of periodic systems is given by
Hkn×[1ε(r)×Hkn]=λknHkn,
(1)
where 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
λkn=ωkn2c2,
(2)
where c is the speed of light in free space and ωkn denotes the eigen angular frequency. ε should be real for ωkn to be real. According to Bloch’s theorem, the eigen function, Hkn, is a product of an exponential factor and a vector field with the lattice-translation symmetry:
Hkn(r)=eikrukn(r),
(3)
ukn(r+a)=ukn(r),
(4)
where a is the elementary translation vector.

When we assume that 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
H1|H21VVdrH1*(r)H2(r),
(5)
where V is the volume on which the periodic boundary condition is imposed [15

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

]. We normalize the eigen functions as
Hkn|Hkn=δkkδnn.
(6)
Then, ukn is an eigen function of operator ℒk defined by
keikreikr=(+ik)×[1ε(r)(+ik)×],
(7)
which is a Hermitian operator in the Hilbert space of complex vector fields with the lattice-translation symmetry. So, {ukn| n = 1, 2,…} is a complete set. We normalize it as
ukn|ukn01V0V0drukn*(r)ukn(r)=δnn,
(8)
where V0 denotes the volume of the unit cell. Thus, in particular, for k = 0,
{u0n|n=1,2,}
(9)
is an orthonormal complete set. Therefore, we can express any eigen function ukl of operator ℒk by a linear combination of eigen functions {u0n} of operator ℒ0. Thus, for small k in the vicinity of the Γ point, we can calculate λkl perturbatively using {u0n} 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:
Δkk0Δk(1)+Δk(2),
(10)
where
Δk(1)=ik×[1ε(r)×],
(11)
Δk(2)=×[1ε(r)ik×].
(12)

We assume according to the situation of our problem that {u0l| l = 1, 2, … ,M} are degenerate and denote their eigenvalue by λ0(=ω02/c2). By the degenerate perturbation theory, the first-order solution for ukl (l = 1, 2, … ,M) is obtained by diagonalizing the matrix whose ij (1 ≤ i, jM) element is given by
Cij(k)=u0i|Δk|u0j0,
(13)
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 Ck=(Cij(k)) are non-zero.

For this purpose, we examine Ck using the spatial symmetry of {u0l}. 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
Rt=R1anddetR=±1,
(14)
since R is an orthogonal matrix. Second, from the nature of inner and outer products, we have
u0i|Δk(1)|u0j0=ikPij,
(15)
where Pij is defined as
Pij=1V0V0dru0i*(r)×[1ε(r)×u0j(r)]u0i|Δ|u0j0.
(16)
We also have
V0u0i|Δk(2)|u0j0Vdr{×[1ε(r)ik×u0j(r)]}u0i*(r)=S0dS{[1ε(r)ik×u0j(r)]×u0i*(r)}n+V0dr[1ε(r)ik×u0j(r)][×u0i*(r)],
(17)
where S0 denotes the surface of V0 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
u0i|Δk(2)|u0j0=ikPji*.
(18)
So, we have
Cij(k)=ik(Pij+Pji*).
(19)
Thirdly, we can prove [15

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

]
(×)1=detR(×).
(20)
By a similar calculation, we can also prove
(×)1=detR(×).
(21)

Next, by introducing five pairs of ℛ−1ℛ (= identity operator) to the definition of Pij, we obtain
kPij=kV0V0dr1u0i*(r)1×1[1ε(r)1×1u0j(r)]=(detR)2V0kV0dr1[u0i*](r)×{1ε(r)×[u0j](r)}=(Rk)V0V0dr[u0i*](r)×{1ε(r)×[u0j](r)}=(Rk)u0i|Δ|u0j0,
(22)
where the prime denotes r′ = Rr and by definition,
[u0i](r)Ru0i(R1r).
(23)
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.

So, unless the product of k, u0i, and u0j contains a term invariant for all ℛ ∈ 𝒢, k · Pij, and consequently, Cij(k) vanish. In this case, the linear term is absent in the dispersion curve, so Dirac cones do not exist. Therefore, the presence of an invariant term in the above-mentioned product is a distinct necessary condition for the Dirac cone.

3. One-dimensional lattice of C2v symmetry

We start with the one-dimensional lattice of C2v 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]

]. There are four one-dimensional representations (A1, A2, B1, B2) for the C2v point group [16

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

]. As can be verified easily, vector k, which should be regarded as a one-dimensional vector (or scalar), has the B1 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]

] whether the product of k, u0i, and u0j contains an invariant term, or the totally symmetric A1 representation. As is summarized in Table 1, { Cij(k)} is non-zero only for the combinations of (A1, B1) modes and (A2, B2) modes.

Table 1. Types of dispersion curves generated by accidental degeneracy of two modes (Mode 1 and Mode 2) for one-dimensional lattices of the C2v symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { Cij(k)} vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D and Q denote Dirac cone and quadratic dispersion surface, respectively.

table-icon
View This Table
| View All Tables

By examining all symmetry transformations ℛ ∈ C2v, we can prove that matrix Ck has the following form for the combination of the A1 (u01) and B1 (u02) modes.
Ck=(0,bkb*k,0).
(24)
In Eq. (24),
b=ie[u01|Δ|u020+u02|Δ|u010*],
(25)
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, Δλ, to the eigenvalue of Eq. (1) is given by
|Δλ,bkb*k,Δλ|=0.
(26)
Thus, its solutions are given by
Δλ=±|b|k,
(27)
which leads to the following dispersion relation in the vicinity of the Γ point:
ωkω0±|b|c2k2ω0.
(28)
So, there is a one-dimensional Dirac cone. This result is consistent with 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]

], in which the creation of the linear dispersion was verified for the accidental degeneracy of an A1 mode and a B1 mode by tight-binding approximation based on the resonant states of unit cells of periodic metamaterials. For the combination of the A2 and B2 modes, we have the same relations as Eq. (24), so there is also a Dirac cone.

Let us make a remark here. If two modes have the same symmetry, they are mixed and their dispersion curves repel each other when their eigen frequencies are close. So, accidental degeneracy does not take place in this case. Nevertheless, this case is included in Table 1 and other tables in the following sections to show that their dispersion curves are quadratic for small k.

4. Square lattice of C4v symmetry

Next, we examine the two-dimensional square lattice of C4v 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]

]. There are four one-dimensional representations (A1, A2, B1, B2) 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]

]. We can verify easily that vector k, which should be regarded as a two-dimensional vector for this case, has the E symmetry. The nature of Cij(k) is summarized in Table 2. It is non-zero for the combinations of an A1, A2, B1, or B2 mode and an E mode.

Table 2. Types of dispersion curves generated by accidental degeneracy of two modes (Mode 1 and Mode 2) for square lattices of the C4v symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { Cij(k)} vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D and Q denote Dirac cone and quadratic dispersion surface, respectively.

table-icon
View This Table
| View All Tables

Let us first examine the case of the combination of an E mode and an A1 mode. From group theory [16

16. 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 one of the two eigen functions of the E mode is transformed like the x coordinate and the other like the y coordinate by the symmetry operation ℛ ∈ C4v. We denote the first and second functions by u01 and u02, respectively. The eigen function of the A1 mode is denoted by u03. By examining all transformation ℛ ∈ C4v, we can prove that matrix Ck has the following form:
Ck=(00bkx00bkyb*kxb*ky0).
(29)
The secular equation for the first-order correction is given by
|Δλ,0,bkx0,Δλ,bkyb*kx,b*ky,Δλ,|=0,
(30)
whose solutions are
Δλ=0,±|b|k,
(31)
where k=kx2+ky2. Therefore, two of the three dispersion curves compose an isotropic Dirac cone. For the third mode, the linear term is absent, so it gives a quadratic dispersion. This result agrees with Ref. [10

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

], in which the presence of the Dirac cone was shown by the tight-binding approximation. When we compare the calculation given in Ref. [10

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

] with the present one, the matrix that we have to diagonalize is much simpler in the present study due to the perturbative nature of the calculation, so the diagonalization is much easier.

For the combinations of (E, A2), (E, B1), and (E, B2) modes, we can prove that Ck has the following forms, respectively:
(E,A2):Ck=(00bky00bkxb*kyb*kx0),(E,B1):Ck=(00bkx00bkyb*kxb*ky0),(E,B2):Ck=(00bky00bkxb*kyb*kx0).
(32)
For all these cases, the solutions are given by the same equation as Eq. (31). So, there is an isotropic Dirac cone and a quadratic dispersion surface. These conclusions are consistent with Ref. [12

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

] in which the shapes of dispersion curves were analyzed by tight-binding approximation for metamaterials and by numerical photonic-band calculation for photonic crystals.

5. Triangular lattice of C6v symmetry

In this section, we examine the two-dimensional triangular lattice of C6v symmetry. There are four one-dimensional representations (A1, A2, B1, B2) and two two-dimensional representations (E1, E2) [16

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

]. We can verify easily that vector k has the E1 symmetry. The nature of Cij(k) is summarized in Table 3.

Table 3. Types of dispersion curves generated by accidental degeneracy of two modes for the triangular lattice of the C6v symmetry. Shapes of the dispersion curves are given in the right column, where D, DD and Q denote Dirac cone, double Dirac cones, and quadratic dispersion surface, respectively.

table-icon
View This Table
| View All Tables

Let us examine the shapes of the dispersion curves for the cases of non-zero { Cij(k)}. For the combination of an E1 mode and an E2 mode, we can prove by examining all transformations ℛ ∈ C6v that matrix Ck has the following form:
Ck=(00bkybkx00bkxbkyb*kyb*kx00b*kxb*ky00),
(33)
where we assumed without loss of generality that the two eigen functions of the E1 mode (u01, u02) are transformed like x and y and those of the E2 mode (u03, u04) are transformed like 2xy and x2y2 [16

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

]. The secular equation for the first-order correction can be solved easily and its solutions are given by
Δλ=±|b|k(doubleroots).
(34)
So, we have isotropic double Dirac cones with the same slope. This conclusion agrees with the result of the tight-binding calculation given in Ref. [11

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

]. The matrix that has to be diagonalized is again much simpler in the present study due to the perturbative nature of the calculation, so the diagonalization is much easier compared with that in Ref. [11

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

].

For the (E1, A1) and (E2, B2) combinations, matrix Ck has the following form:
Ck=(00bkx00bkyb*kxb*ky0).
(35)
On the other hand, for the (E1, A2) and (E2, B1) combinations,
Ck=(00bky00bkxb*kyb*kx0).
(36)
For these cases, the solutions of the secular equation are the same and given by
Δλ=0,±|b|k.
(37)
So, there are an isotropic Dirac cone and a quadratic dispersion surface. These conclusions are consistent with Ref. [12

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

] in which dispersion curves were analyzed by tight-binding approximation and numerical photonic-band calculations.

6. Simple-cubic lattice of Oh symmetry

Finally, we examine the simple-cubic lattice of the Oh symmetry. There are four one-dimensional representations (A1g, A1u, A2g, A2u), two two-dimensional representations (Eg, Eu), and four three-dimensional representations (T1g, T1u, T2g, T2u) [16

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

]. We can verify that vector k has the T1u symmetry. The nature of Cij(k) is summarized in Table 4.

Table 4. Types of dispersion curves generated by accidental degeneracy of two modes for the simple-cubic lattice of the Oh symmetry. Symmetries of the magnetic field of the two modes on the Γ point are given in the left column. The middle column shows whether { Cij(k)} vanishes by symmetry (×) or it can be non-zero (○). Shapes of the dispersion curves are given in the right column, where D, DD, and Q denote Dirac cone, double Dirac cones, and quadratic dispersion surface, respectively.

table-icon
View This Table
| View All Tables

Next, we examine the combination of a T1u mode (u01, u02, u03) and an Eg mode (u04, u05). We can assume that two eigen functions of the Eg mode are transformed like 2z2x2y2 and 3(x2y2) [16

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

]. We can prove easily that matrix Ck has the following form:
Ck=(000bkx3bkx000bky3bky0002bkz0b*kxb*ky2b*kz003b*kx3b*ky000)
(40)
Then, the secular equation is
Δλ[(Δλ)44|b|2k2(Δλ)2+12(kx2ky2+ky2kz2+kz2kx2)|b|4]=0.
(41)
When we express the solutions by the spherical coordinates, (k, θ, ϕ),
Δλ={0,±2|b|k1±13F(θ,ϕ),
(42)
where
F(θ,ϕ)=sin4θsin22ϕ+sin22θ4.
(43)
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 (T1g,Eu), we can prove that Ck has the same structure as Eq. (40), so it also yields two anisotropic Dirac cones and a quadratic dispersion surface.

For the combinations of (T2u,Eg) and (T2g,Eu), Ck has the following form:
Ck=(0003bkxbkx0003bkybky00002bkz3b*kx3b*ky000b*kxb*ky2b*kz00)
(44)
This Ck leads to exactly the same secular equation as Eq. (41). So, the combinations of (T2u,Eg) and (T2g,Eu) also yield two anisotropic Dirac cones and a quadratic dispersion surface.

Finally, let us examine the case of two triply degenerate modes. For the combination of a T1g mode (u01, u02, u03) and a T1u mode (u04, u05, u06), Ck has the following structure:
Ck=(0000bkzbky000bkz0bkx000bkybkx00b*kzb*ky000b*kz0b*kx000b*kyb*kx0000)
(45)
which is sufficiently sparse so that we can easily obtain its analytical solutions:
Δλ={0(doubleroots),±|b|k(doubleroots).
(46)
So, there are isotropic double Dirac cones with the same slope and two quadratic dispersion surfaces. We can prove that the combination of T2g and T2u modes has Ck of the same structure, so this combination also yields double Dirac cones and two quadratic dispersion surfaces.

For the combinations of (T1g, T2u) and (T2g, T1u), Ck has the following form:
Ck=(0000bkzbky000bkz0bkx000bkybkx00b*kzb*ky000b*kz0b*kx000b*kyb*kx0000),
(47)
which leads to the following secular equation for the first-order correction.
(Δλ)3|b|2k2Δλ±2|b|3kxkykz=0.
(48)
In general, three solutions of
ξ3+pξ+q=0
(49)
are given by
ξl=e2πil/3q2+(q2)2+(p3)33+e2πil/3q2(q2)2+(p3)33,
(50)
where l = 1, 2, 3. So, six solutions of Eq. (48) are
Δλl±=|b|k(e2πil/3F1±+e2πil/3F2±),
(51)
where
F1±=[±sinθsin2θsin2ϕ4+sin2θsin22θsin22ϕ16127]13,
(52)
F2±=[±sinθsin2θsin2ϕ4sin2θsin22θsin22ϕ16127]13.
(53)
So, there are three anisotropic Dirac cones.

To check these analytical results, we performed photonic-band calculations by the plane-wave expansion method [15

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

]. We assumed a regular simple-cubic lattice of dielectric spheres with a dielectric constant of 12.6 (GaAs). By changing the radius of the spheres, we found several cases of accidental degeneracy among the lowest 20 dispersion curves, three of which are shown in Fig. 1, which agree with the results of analytical calculations listed in Table 4. The rest of the cases also agreed with Table 4.

Fig. 1 Dispersion curves of a photonic crystal composed of the simple-cubic lattice of dielectric spheres with a dielectric constant of 12.6. The vertical axis is the normalized frequency (ωa/2πc), and the horizontal axis is the wave vector in the R (= (π/a, π/a, π/a)), and X (= (π/a, 0, 0)) directions, where R/6, for example, means that the horizontal axis is magnified by six times. The radius of the spheres is 0.160a for (a), 0.210a for (b), and 0.416a for (c). The number associated with each dispersion curve is the multiplicity of the mode.

Let us make six remarks. First, in this paper, we formulated a degenerate perturbation theory for the vector electromagnetic field of periodic structures and applied it to the problem of the creation of Dirac cones by accidental degeneracy of two modes. We obtained an efficient necessary condition (non-zero Ck) 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 Ck by symmetry operations for four periodic systems of different spatial symmetries, we could determine the structure of Ck 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.

Second, the present theory can of course be applied to periodic structures of other spatial symmetries. We can examine Ck 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.

Third, the secular equations obtained by the present theory are generally much simpler than those obtained by tight-binding approximation in Refs. [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]

], [10

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

], [11

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

], and [12

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

] due to the perturbative nature of the calculations. As a result, we could easily solve equations up to the sixth degree to obtain the first-order corrections to the eigen values.

Fourth, we assumed that the dielectric constant of the periodic structure does not depend on frequency in order to make our problem well-defined. When we have to deal with dispersive materials with frequency-dependent dielectric constants, we can use the value at the degenerate frequency. Thus, we can obtain qualitatively correct solutions in the vicinity of the degenerate frequency.

Fifth, although the numerical confirmation was performed for photonic crystals because of the ease of calculation, the present theory is equally applicable to periodic metamaterials with well-defined resonant states localized in unit cells.

Finally, the present theory may be regarded as an extension of the 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]

] to the vector electromagnetic field. In the usual 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

We formulated a degenerate perturbation theory for the vector electromagnetic field of periodic structures like photonic crystals and metamaterials and applied it to the problem of the creation of Dirac cones in the center of the Brillouin zone by accidental degeneracy of two modes. We derived an efficient necessary condition for the Dirac cone by which we can easily select candidates of mode combinations that enable the creation of the Dirac cones. We can analyze the structure of a matrix (Ck) 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.

All these findings clearly show that the presence or absence of the Dirac cone in the zone center due to accidental degeneracy of two modes does not depend on the details of the periodic structure that we analyze but is solely determined by the spatial symmetry of the two modes. Thus, we succeeded in proving the universality of mode symmetries in creating photonic Dirac cones, which was anticipated in our recent study [12

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

].

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.

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]

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]

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]

9.

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

10.

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

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]

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]

15.

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

16.

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

17.

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

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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. A78, 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. B80, 155103 (2009). [CrossRef]
  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. A75, 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 B405, 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]
  8. K. Sakoda and H-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express18, 27371–27386 (2010). [CrossRef]
  9. K. Sakoda and H.-F. Zhou, “Analytical study of two-dimensional degenerate metamaterial antennas,” Opt. Express19, 13899–13921 (2011). [CrossRef] [PubMed]
  10. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express20, 3898–3917 (2012). [CrossRef] [PubMed]
  11. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express20, 9925–9939 (2012). [CrossRef] [PubMed]
  12. K. Sakoda, “Universality of mode symmetries in creating photonic Dirac cones,” J. Opt. Soc. Am. B29, 2770–2778 (2012). [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. B75, 155410 (2007). [CrossRef]
  15. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).
  16. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1990). [CrossRef]
  17. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition (World Scientific, Singapore, 2009). [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
 

« Previous Article

OSA is a member of CrossRef.

CrossCheck Deposited