1. INTRODUCTION

Photonic Dirac cones, or the isotropic linear dispersion relation around certain points in the Brillouin zone, have been attracting considerable interest during the last five years, since they offer various novel possibilities for physics and engineering. Haldane et al. [1,2] 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 due to the Faraday effect. Ochiai and Onoda extended the discussion to honeycomb-lattice photonic crystals [3]. Zhang proposed optical simulation of Zitterbewegung, or trembling motion, in particle physics by propagating an optical pulse of the Dirac point frequency [4]. A pseudodiffusive transmission whose intensity was inversely proportional to the thickness of the photonic crystal was found by Sepkhanov et al. [5] and numerically demonstrated by Diem et al. [6].

Recently, Nam et al. analyzed the wave propagation in a periodic metal-dielectric waveguide array, which may be regarded as a one-dimensional photonic crystal, to show the presence of the linear dispersion relation, or a one-dimensional Dirac cone, in the Brillouin-zone center by adjusting the coupling constants between the waveguides [7]. 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 bands [8]. They showed for square- and triangular-lattice photonic crystals that combinations of a nondegenerate band and a doubly degenerate band yield a Dirac cone together with an auxiliary quadratic dispersion surface. Because the Dirac point in the Brillouin-zone center is equivalent to a zero effective refractive index [8], it has much potential for various applications such as scatter-free waveguides [9] and lenses of arbitrary shapes [10].

On the other hand, we showed by tight-binding approximation and group theory that we can also create Dirac cones 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 [11,12]. We proved that the combination of $A_1$ and $E$ modes of square-lattice metamaterials and the combination of $A_{1g}$ and $T_{1u}$ modes of simple-cubic-lattice metamaterials create a Dirac cone with auxiliary quadratic dispersion surfaces [11]. We further proved that the combination of $E_1$ and $E_2$ modes of triangular-lattice metamaterials yields double Dirac cones, or a pair of identical Dirac cones, without an auxiliary quadratic dispersion surface [12].

Because the presence of Dirac cones was found in two extreme systems, i.e., dielectric photonic crystals for which the extended-wave picture is appropriate and metamaterials with localized electromagnetic resonant states for which the tight-binding picture is appropriate, it is highly likely that there is a common physical origin in this phenomenon irrespective of the details of the sample structure.

In this paper, based on this anticipation, we systematically examine 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 calculation and tight-binding approximation, respectively. We show that the two calculations give the same results, which strongly suggests the presence of universality of mode symmetries that enable 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.

This paper is organized as follows. In Section 2, the methods for calculating the dispersion curves of metamaterials and photonic crystals are briefly described. In Section 3, the shapes of the dispersion curves thus calculated are presented in the case of accidental degeneracy for various combinations of mode symmetries. A summary of the present study is given in Section 4. In Appendix A, we show the formulation of the tight-binding approximation with the magnetic fields of the resonant states of a single unit structure of metamaterials as basis functions. The method for deriving mutual relations among different electromagnetic transfer integrals is also described. The shapes of the dispersion curves thus calculated for metamaterials of the $C_{6v}$ and $C_{4v}$ symmetries are presented in Appendices B and C, respectively.

2. THEORY

For the analytical description of dispersion curves of periodic metamaterials by tight-binding approximation, electromagnetic transfer integrals defined by the following equation play the main role [11,12]:

We analytically solve the secular equation of the electromagnetic field for wave vector $\{\mathbf{k}\}$ in the vicinity of the Brillouin-zone center by taking into account $L_m^{(ij)}$’s for the nearest-neighbor lattice points [11,12]. The spatial symmetry of the resonant states results in many relations among $L_m^{(ij)}$’s and simplifies the secular equation (see Appendix A). In the case of accidental degeneracy of $E_1$ and $E_2$ modes of triangular-lattice metamaterials with $C_{6v}$ (regular triangle) symmetry, for example, we can thus analytically solve the secular equation that is quartic in ${\omega }^2$ [12]. For numerical calculations of the dispersion relation of photonic crystals, a conventional plane-wave method is used [13].

3. RESULTS AND DISCUSSION

First, we examine the triangular lattice of $C_{6v}$ symmetry. In Table 1, the shapes of dispersion curves generated by accidental degeneracy on the $\mathrm{\Gamma }$ point of the Brillouin zone are summarized for metamaterials. As we mentioned previously, we obtain double Dirac cones by combining $E_1$ and $E_2$ modes [12]. On the other hand, we found in the present study that ($A_1$, $E_1$), ($A_2$, $E_1$), and ($B_2$, $E_2$) combinations yield a Dirac cone with an auxiliary quadratic dispersion surface. For the rest of the mode combinations listed in Table 1, we obtain two or three different quadratic dispersion curves (see Appendix B for details).

Table 1. Types of Dispersion Curves Generated by Accidental Degeneracy of Two Modes (Mode 1 and Mode 2) for Triangular-Lattice Metamaterials of $C_{6v}$ Symmetry^a

View Table | View all tables in this article

For the photonic crystal, we assume a periodic triangular array of circular air cylinders in a uniform material with a dielectric constant of 12.6 (GaAs). First, we examine the structural parameters for materializing accidental degeneracy. Figure 1 shows the second to fifth lowest eigenfrequencies on the $\mathrm{\Gamma }$ point for transverse electric (TE) polarization with its electric field perpendicular to the cylinder axis. In this figure, the vertical axis denotes the eigen angular frequency $\omega$ normalized by the lattice constant of the photonic crystal $a$ and the light velocity in free space, while the horizontal axis denotes the radius of the air cylinder $\rho$ normalized by the lattice constant. The spatial symmetry of the magnetic field is also given to each mode.

 

Fig. 1. Normalized eigenfrequency ($\omega a/2\pi c$) for the TE polarization on the $\mathrm{\Gamma }$ point of the photonic crystal composed of a regular triangular array of circular air cylinders in a medium with a dielectric constant of 12.6 (GaAs). The horizontal axis is the normalized radius of the air cylinders ($\rho /a$). The degeneracy points denoted by $a$ to $e$ correspond to the five panels in Fig. 2.

Download Full Size | PDF

Figure 1 shows that there are six values of the normalized radius that materialize accidental degeneracy of eigenfrequencies. Dispersion curves for five of the six cases are given in Figs. 2(a)–2(e). They are calculated for $\mathbf{k}$ in the $\mathrm{\Gamma }$-to-$M$ and $\mathrm{\Gamma }$-to-$K$ directions in the Brillouin zone [see Fig. 2(f)]. The sixth case (normalized radius of 0.494) is the degeneracy of the $E_1$ and $E_2$ modes and gives a feature similar to Fig. 2(a).

 

Fig. 2. Dispersion curves of the triangular-lattice photonic crystal for TE polarization with a normalized radius of (a) 0.407, (b) 0.433, (c) 0.459, (d) 0.474, and (e) 0.492. (f) Brillouin zone of the triangular lattice. The frequencies of accidental degeneracy are circled in (a)–(e). $M/10$ and $K/10$ in (e) mean that the horizontal axis is magnified by 10 times.

Download Full Size | PDF

Accidental degeneracy of the $E_1$ and $E_2$ modes results in double Dirac cones. As shown in Fig. 2(a), their dispersion in the vicinity of the $\mathrm{\Gamma }$ point is linear and isotropic and the slopes of the two dispersion lines are the same on the $\mathrm{\Gamma }$ point. For the combination of $A_1$ and $E_1$ modes [Fig. 2(b)] and that of $B_2$ and $E_2$ modes [Fig. 2(e)], accidental degeneracy results in the formation of a Dirac cone with an auxiliary quadratic dispersion surface. Two of the three dispersion curves are linear and isotropic in the vicinity of the $\mathrm{\Gamma }$ point, while the third curve is quadratic. Finally, for the combination of $B_2$ and $E_1$ modes [Fig. 2(c)] and that of $A_1$ and $E_2$ modes [Fig. 2(d)], accidental degeneracy results in three quadratic dispersion curves. All these features are the same as the case of triangular-lattice metamaterials listed in Table 1, which was analytically proved assuming the presence of localized resonant states in each unit structure.

In Fig. 3, structural parameters for realizing accidental degeneracy are examined for transverse magnetic (TM) polarization with its magnetic field perpendicular to the cylinder axis. The spatial symmetry of the magnetic field is also given to each mode.

 

Fig. 3. The second to fifth lowest normalized eigenfrequency ($\omega a/2\pi c$) for the TM polarization on the $\mathrm{\Gamma }$ point of the photonic crystal composed of a regular triangular array of circular air cylinders in a dielectric medium with $\epsilon =12.6$ (GaAs). The horizontal axis is the normalized radius of the air cylinders ($\rho /a$). The degeneracy points denoted by $a$ and $b$ correspond to the two panels in Fig. 4.

Download Full Size | PDF

Let us make one remark here. The TM polarization is often referred to as $E$ polarization, and its electric field, which is parallel to the cylinder axis, is discussed traditionally. Because the electric field is a genuine vector and the magnetic field is an axial vector, their transformation by symmetry operations is generally different. When we denote the symmetry operation and its matrix representation by $\mathcal{R}$ and $R$ and the character of transformation of the electric (magnetic) field by ${\chi }^{(E)}$ (${\chi }^{(H)}$), we have the following relation [13]:

There are two values of the normalized radius that materialize accidental degeneracy for the TM polarization as shown in Fig. 3. The combination of $A_2$ and $E_1$ modes [Fig. 4(a)] results in the creation of a Dirac cone with an auxiliary quadratic dispersion surface, whereas the combination of $A_2$ and $E_2$ modes [Fig. 4(b)] results in the formation of three quadratic dispersion curves. Both features agree with the case of metamaterials listed in Table 1.

 

Fig. 4. Dispersion curves of the triangular-lattice photonic crystal for TM polarization with a normalized radius of (a) 0.445, (b) 0.483. The frequencies of accidental degeneracy are circled.

Download Full Size | PDF

Now we proceed to the case of square lattices of $C_{4v}$ (regular square) symmetry. Table 2 summarizes the shapes of dispersion curves of periodic metamaterials generated by accidental degeneracy on the $\mathrm{\Gamma }$ point. As we mentioned previously, we obtain a Dirac cone with an auxiliary quadratic dispersion surface by combining $A_1$ and $E$ modes [11], whereas ($A_1$, $B_1$) and ($A_1$, $B_2$) combinations yield two quadratic curves [12]. We further found in the present study that ($B_1$, $E$) and ($B_2$, $E$) combinations give a Dirac cone and an auxiliary quadratic dispersion surface and that ($A_1$, $A_2$) and ($A_2$, $B_2$) combinations yield two quadratic dispersion curves (see Appendix C for details).

Table 2. Types of Dispersion Curves Generated by Accidental Degeneracy of Two Modes (Mode 1 and Mode 2) for Square-Lattice Metamaterials of $C_{4v}$ Symmetry^a

View Table | View all tables in this article

On the other hand, for square-lattice photonic crystals of the $C_{4v}$ symmetry composed of circular air cylinders in a uniform material with a dielectric constant of 12.6, we found three values of the normalized radius that materialize accidental degeneracy on the $\mathrm{\Gamma }$ point. Dispersion curves for these three cases are shown in Fig. 5. Their shapes agree with the metamaterials listed in Table 2.

 

Fig. 5. (a) Brillouin zone of the square lattice of $C_{4v}$ symmetry. Dispersion curves of the square-lattice photonic crystal (b) for TE polarization with a normalized radius of 0.299 and for TM polarization with a normalized radius of (c) 0.421 and (d) 0.464. The frequencies of accidental degeneracy are circled. $M/2$ and $X/2$ in (c) mean that the horizontal axis is magnified by two times.

Download Full Size | PDF

Thus, in all cases that we could examine numerically, dielectric photonic crystals and metamaterials with localized electromagnetic resonant states have the same relation between the mode symmetries and the shapes of the dispersion curves generated by accidental degeneracy in the vicinity of the $\mathrm{\Gamma }$ point. This fact strongly suggests the presence of universality in combinations of mode symmetries that enable the creation of photonic Dirac cones and double Dirac cones irrespective of the details of the sample structure. This knowledge is quite useful for designing specimens with Dirac cones to realize many applications described in Section 1. However, we should note that the universality has not been proved rigorously yet, since the present study is partly based on numerical calculation for a limited number of examples. Three-dimensional systems in particular were not analyzed, although we expect a similar universality of mode symmetries. So, the rigorous proof of the universality remains a future problem.

4. CONCLUSION

We analytically solved secular equations of periodic metamaterials of the $C_{6v}$ and $C_{4v}$ symmetries with localized electromagnetic resonant states by tight-binding approximation and obtained the shapes of dispersion curves in the vicinity of the Brillouin-zone center. Thus we clarified the combinations of mode symmetries that materialize Dirac cones and double Dirac cones by accidental degeneracy. We also examined dispersion curves of dielectric photonic crystals numerically by the plane-wave expansion method. By changing a structural parameter of assumed samples, we found 11 cases of accidental degeneracy on the $\mathrm{\Gamma }$ point and compared the shapes of dispersion curves with the metamaterials.

In all 11 cases, the relation between the mode symmetries and the shape of the dispersion curves was common to metamaterials with localized electromagnetic resonant states and dielectric photonic crystals without well-defined resonant states, which strongly suggests the presence of universality of mode symmetries that enable 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.

APPENDIX A: FORMULATION

Maxwell’s wave equation for the magnetic field $\mathbf{H}(\mathbf{r},t)$ is given by

(A1)$$\mathcal{L}\mathbf{H}\equiv \nabla \times [\frac{1}{\epsilon (\mathbf{r})}\nabla \times \mathbf{H}]=-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\mathbf{H},$$

where $\epsilon (\mathbf{r})$ is the periodic dielectric constant of the system. The magnetic permeability is assumed to be unity, since we do not deal with magnetic materials. We impose the periodic boundary condition on $\mathbf{H}$ and assume that $\epsilon (\mathbf{r})$ is real to make our problem well-defined [13]. Thus, Eq. (A1) leads to an eigenvalue problem.

For a single metallic unit structure described by dielectric constant ${\epsilon }_s(\mathbf{r})$, we assume the presence of localized resonant states, which characterize the optical response of the metamaterial. Thus, their magnetic fields satisfy the following eigenvalue equation:

(A2)$${\mathcal{L}}_s{\mathbf{H}}^{(i)}(\mathbf{r})\equiv \nabla \times [\frac{1}{{\epsilon }_s}\nabla \times {\mathbf{H}}^{(i)}(\mathbf{r})]=\frac{{\omega }_i^2}{c^2}{\mathbf{H}}^{(i)}(\mathbf{r}),$$

where operator ${\mathcal{L}}_s$ is defined by the first equality of this equation, $i$ is the suffix to distinguish the resonant states, and ${\omega }_i$ is the resonance angular frequency. Without loss of generality, we can assume that the eigenfunctions are normalized as follows:

(A3)$${\int }_V\mathrm{d}\mathbf{r}{\mathbf{H}}^{(i)*}(\mathbf{r})·{\mathbf{H}}^{(j)}(\mathbf{r})=V{\delta }_{ij},$$

where $V$ is the volume on which we impose the periodic boundary condition.

We assume that the unit cell of the periodic metamaterial has a certain spatial symmetry and denote the group of its symmetry operations by $\mathcal{G}$. In Appendices B and C, we deal with two cases of $\mathcal{G}=C_{6v}$ and $\mathcal{G}=C_{4v}$. By definition, the transformation of the magnetic field by symmetry operation $\mathcal{R}\in \mathcal{G}$ is given by

(A4)$$[\mathcal{R}\mathbf{H}](\mathbf{r})\equiv R\mathbf{H}(R^{-1}\mathbf{r}),$$

where $R$ is the matrix representation of $\mathcal{R}$. For all $\mathcal{R}\in \mathcal{G}$, we can prove the following equation [12].

(A5)$$\begin{gathered} L_m^{(ij)}=\frac{1}{V}{\int }_V\mathrm{d}\mathbf{r}[\mathcal{R}{\mathbf{H}}^{(i)*}](\mathbf{r}) \\ ·\mathcal{L}[\mathcal{R}{\mathbf{H}}^{(j)}](\mathbf{r}-R{\mathbf{r}}_m), \end{gathered}$$

where $L_m^{(ij)}$ is defined by Eq. (1). Because ${\mathbf{H}}^{(i)}$ is an eigenfunction of ${\mathcal{L}}_s$, it is a basis of an irreducible representation of $\mathcal{G}$, so $\mathcal{R}{\mathbf{H}}^{(i)}$ can be expressed by a linear combination of eigenfunctions of the same irreducible representation [14]. Then, the right side of Eq. (A5) must be a linear combination of $\{L_m^{(ij)}\}$, so we obtain their mutual relations. We can thus derive many relations among $\{L_m^{(ij)}\}$ for $C_{6v}$ and $C_{4v}$, as will be shown in Appendices B and C.

Now, we solve the eigenvalue problem given by

(A6)$$\mathcal{L}{\mathbf{H}}_{\mathbf{k}}(\mathbf{r})=\frac{{\omega }_{\mathbf{k}}^2}{c^2}{\mathbf{H}}_{\mathbf{k}}(\mathbf{r}),$$

by the tight-binding approximation. In Eq. (A6), $\mathbf{k}$ is a wave vector in the Brillouin zone. The secular equation has the following form [12]:

(A7)$$|B-\frac{{\omega }_{\mathbf{k}}^2}{c^2}I|=0,$$

where $I$ is a unit tensor and elements of matrix $B$ are given by

(A8)$$B_{ij}=\sum _m{e}^{i\mathbf{k}·{\mathbf{r}}_m}{L}_m^{(ij)}.$$

When we analyze the dispersion curves for small $\mathbf{k}$ around the Brillouin-zone center, the summation for $m$ can be limited to the nearest-neighbor lattice points, since other terms give contributions of higher orders of $\mathbf{k}$.

APPENDIX B: TRIANGULAR LATTICE

1. $A_1+B_1$

In this appendix, we examine the dispersion relation in the vicinity of the Brillouin-zone center for the triangular lattice of unit cells of the $C_{6v}$ symmetry. We denote the origin and the nearest-neighbor lattice points by 0 to 6, as shown in Fig. 6, where symmetry operations of the $C_{6v}$ point group are also shown.

 

Fig. 6. Numbering of the lattice points of the triangular lattice and symmetry operations of the $C_{6v}$ point group.

Download Full Size | PDF

We start with the case of accidental degeneracy of an $A_1$ mode and a $B_1$ mode on the $\mathrm{\Gamma }$ point of the Brillouin zone. We denote the magnetic fields of the $A_1$ and $B_1$ modes by ${\mathbf{H}}^{(1)}$ and ${\mathbf{H}}^{(2)}$, respectively. By using all $\mathcal{R}\in C_{6v}$ in Eq. (A5), we can derive the following relations:

(B1)$$L_0^{(11)}\equiv M_{10},L_0^{(22)}\equiv M_{20},$$

(B2)$$L_0^{(12)}=L_0^{(21)}=0,$$

(B3)$$\begin{gathered} L_1^{(11)}=L_2^{(11)}=L_3^{(11)}=L_4^{(11)}=L_5^{(11)}=L_6^{(11)}, \\ \equiv M_{11}, \end{gathered}$$

(B4)$$\begin{gathered} L_1^{(22)}=L_2^{(22)}=L_3^{(22)}=L_4^{(22)}=L_5^{(22)}=L_6^{(22)}, \\ \equiv M_{21}, \end{gathered}$$

(B5)$$\begin{gathered} L_1^{(12)}=L_3^{(12)}=L_5^{(12)}=-L_2^{(12)}=-L_4^{(12)}=-L_6^{(12)} \\ \equiv M_1. \end{gathered}$$

In addition to these relations, we can generally prove the following equation.

(B6)$$L_m^{(ij)}=L_{-m}^{(ji)*},$$

where $-m$ denotes the lattice point given by $-{\mathbf{r}}_m$.

From Eq. (A8), we obtain

(B7)$$\begin{gathered} B_{11}={\xi }_{\mathrm{\Gamma }}^{(1)}+M_{11}[2(\cos k_{x}a-1) \\ +4(\cos \frac{k_{x}a}{2}\cos \frac{\sqrt{3}{k}_{y}a}{2}-1)], \end{gathered}$$

(B8)$$\begin{gathered} B_{22}={\xi }_{\mathrm{\Gamma }}^{(2)}+M_{21}[2(\cos k_{x}a-1) \\ +4(\cos \frac{k_{x}a}{2}\cos \frac{\sqrt{3}{k}_{y}a}{2}-1)], \end{gathered}$$

(B9)$$B_{12}=2i{M}_1(\sin k_{x}a-2\sin \frac{k_{x}a}{2}\cos \frac{\sqrt{3}{k}_{y}a}{2}),$$

where

(B10)$${\xi }_{\mathrm{\Gamma }}^{(1)}\equiv M_{10}+6M_{11},$$

(B11)$${\xi }_{\mathrm{\Gamma }}^{(2)}\equiv M_{20}+6M_{21},$$

which are the square of the eigen angular frequencies at the $\mathrm{\Gamma }$ point divided by $c^2$ for the $A_1$ and $B_1$ modes, respectively. Because the secular equation in the present case is a quadratic equation of ${\omega }^2$, it can easily be solved. When we assume the accidental degeneracy of the two modes at the $\mathrm{\Gamma }$ point—that is, ${\xi }_{\mathrm{\Gamma }}^{(1)}={\xi }_{\mathrm{\Gamma }}^{(2)}\equiv {\xi }_{\mathrm{\Gamma }}$—the dispersion relation for small $\mathbf{k}$ is

(B12)$$\omega \approx \{\begin{array}{l}{\omega }_{\mathrm{\Gamma }}-\frac{3a^2{c}^2{k}^2{M}_{11}}{2{\omega }_{\mathrm{\Gamma }}},\\ {\omega }_{\mathrm{\Gamma }}-\frac{3a^2{c}^2{k}^2{M}_{21}}{2{\omega }_{\mathrm{\Gamma }}},\end{array}$$

where $k=\sqrt{k_x^2+k_y^2}$ and ${\omega }_{\mathrm{\Gamma }}=c\sqrt{{\xi }_{\mathrm{\Gamma }}}$. $M_1$ gives higher-order corrections. So, the linear term is absent and the dispersion relations are quadratic in $k$ in the vicinity of the $\mathrm{\Gamma }$ point.

2. $A_1+B_2$

For the combination of an $A_1$ mode (mode 1) and a $B_2$ mode (mode2), we can derive

(B13)$$L_m^{(12)}=0(m=0,1,\dots ,6)$$

by examining Eq. (A5) for all $\mathcal{R}\in C_{6v}$. So, there is no interaction between the two modes and the dispersion relations are quadratic for small $\mathbf{k}$.

3. $A_1+E_1$

$E_1$ modes are doubly degenerate. From group theory [14], we can assume without loss of generality that two eigenfunctions of the $E_1$ mode transform like $x$ and $y$. We denote the magnetic field of the $A_1$ mode by ${\mathbf{H}}^{(0)}$ for this time and those of the $E_1$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $x$, and ${\mathbf{H}}^{(2)}$, which transforms like $y$. Then, we can derive the following relations from Eq. (A5):

(B14)$$L_0^{(00)}\equiv M_{01},L_0^{(11)}\equiv L_0^{(22)}=M_{11},$$

(B15)$$\begin{gathered} L_1^{(00)}=L_2^{(00)}=L_3^{(00)}=L_4^{(00)}=L_5^{(00)}=L_6^{(00)} \\ \equiv M_{02}, \end{gathered}$$

(B16)$$L_1^{(11)}=L_4^{(11)}\equiv M_{12},L_1^{(22)}=L_4^{(22)}\equiv M_{13},$$

(B17)$$L_2^{(11)}=L_3^{(11)}=L_5^{(11)}=L_6^{(11)}\equiv M_{14},$$

(B18)$$L_2^{(22)}=L_3^{(22)}=L_5^{(22)}=L_6^{(22)}\equiv M_{15},$$

(B19)$$L_0^{(12)}=L_1^{(12)}=L_4^{(12)}=0,$$

(B20)$$L_2^{(12)}=-L_3^{(12)}=L_5^{(12)}=-L_6^{(12)}\equiv M_{16},$$

(B21)$$L_0^{(01)}=L_0^{(02)}=L_1^{(02)}=L_4^{(02)}=0,$$

(B22)$$L_1^{(01)}=-L_4^{(01)}\equiv M_1,$$

(B23)$$L_2^{(01)}=-L_3^{(01)}=-L_5^{(01)}=L_6^{(01)}\equiv M_2,$$

(B24)$$L_2^{(02)}=L_3^{(02)}=-L_5^{(02)}=-L_6^{(02)}\equiv M_3.$$

We can further prove that

(B25)$$M_{14}=\frac{M_{12}+3M_{13}}{4},$$

(B26)$$M_{15}=\frac{3M_{12}+M_{13}}{4},$$

(B27)$$M_{16}=\frac{\sqrt{3}(M_{12}-M_{13})}{4},$$

(B28)$$M_1=2M_2,M_3=\sqrt{3}{M}_2.$$

Then, we can obtain the elements of matrix $B$ according to Eq. (A8). The secular equation is a cubic equation of ${\omega }^2$. To obtain its analytical solutions, we need a lengthy calculation similar to that presented in [11] The solutions are, however, rather simple when we assume the accidental degeneracy of the $A_1$ and $E_1$ modes at the $\mathrm{\Gamma }$ point. The result is

(B29)$$\omega \approx \{\begin{array}{l}{\omega }_{\mathrm{\Gamma }}\pm \frac{3ak{c}^2|M_2|}{{\omega }_{\mathrm{\Gamma }}}-\frac{a^2{k}^{2}M}{4{\omega }_{\mathrm{\Gamma }}},\\ {\omega }_{\mathrm{\Gamma }}-\frac{a^2{k}^{2}M}{4{\omega }_{\mathrm{\Gamma }}},\end{array}$$

where $M=M_{02}+M_{12}+M_{13}$. Two of the three dispersion curves are linear and isotropic with respect to $\mathbf{k}$ in the vicinity of the $\mathrm{\Gamma }$ point, and the third one is quadratic. So, we have a Dirac cone with an auxiliary quadratic dispersion surface.

4. $B_2+E_2$

$E_2$ modes are doubly degenerate. From group theory [14], we can assume without loss of generality that two eigenfunctions of the $E_2$ mode transform like $2xy$ and $x^2-y^2$. We denote the magnetic field of the $B_2$ mode by ${\mathbf{H}}^{(0)}$ and those of the $E_2$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $2xy$, and ${\mathbf{H}}^{(2)}$, which transforms like $x^2-y^2$. Then, we can derive the relations among $L_m^{(ij)}$’s by using Eq. (A5). The results are completely the same as the case of $A_1+E_1$. So we have a Dirac cone with an auxiliary quadratic dispersion surface in the vicinity of the $\mathrm{\Gamma }$ point.

5. $A_2+E_1$

We denote the magnetic field of the $A_2$ mode by ${\mathbf{H}}^{(0)}$ and those of the $E_1$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $y$, and ${\mathbf{H}}^{(2)}$, which transforms like $x$, for this time. Then, we can derive the relations among $L_m^{(ij)}$’s according to Eq. (A5). We obtain the same results as the case of $A_1+E_1$, except

(B30)$$M_3=-\sqrt{3}{M}_2.$$

Since the dispersion relation, Eq. (B29), does not depend on the sign of $M_3$, we obtain the same dispersion, i.e., a Dirac cone with an auxiliary quadratic dispersion surface.

6. $B_2+E_1$

We denote the magnetic field of the $B_2$ mode by ${\mathbf{H}}^{(0)}$ and those of the $E_1$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $x$, and ${\mathbf{H}}^{(2)}$, which transforms like $y$. Then, the relations among $L_m^{(ij)}$’s with $1\le i$, $j\le 2$ are the same as in Appendix B.3. For the rest of $L_m^{(ij)}$’s, we can derive the following relations:

(B31)$$L_0^{(00)}\equiv M_{01},$$

(B32)$$\begin{gathered} L_1^{(00)}=L_2^{(00)}=L_3^{(00)}=L_4^{(00)}=L_5^{(00)}=L_6^{(00)} \\ \equiv M_{02}, \end{gathered}$$

(B33)$$L_0^{(01)}=L_0^{(02)}=L_1^{(01)}=L_4^{(01)}=0,$$

(B34)$$L_2^{(01)}=-L_3^{(01)}=L_5^{(01)}=-L_6^{(01)}\equiv M_1,$$

(B35)$$L_1^{(02)}=L_4^{(02)}\equiv M_2,$$

(B36)$$L_2^{(02)}=L_3^{(02)}=L_5^{(02)}=L_6^{(02)}\equiv M_3.$$

We can further prove

(B37)$$M_1=-\sqrt{3}{M}_3,M_2=-2M_3.$$

Then, we obtain the elements of matrix $B$ in Eq. (A8) and solve the secular equation, Eq. (A7). After a simple but lengthy calculation, we obtain the dispersion relation as follows.

(B38)$$\omega \approx \{\begin{array}{l}{\omega }_{\mathrm{\Gamma }}-\frac{[M\pm \sqrt{3({|M_1|}^2+{|M_{16}|}^2{\sin }^{2}2\varphi )}]a^2{c}^2{k}^2}{4{\omega }_{\mathrm{\Gamma }}},\\ {\omega }_{\mathrm{\Gamma }}-\frac{a^2{c}^2{k}^{2}M}{4{\omega }_{\mathrm{\Gamma }}},\end{array}$$

where $M=M_{02}+M_{12}+M_{13}$ and $\varphi$ is the angle between vector $\mathbf{k}$ and the $x$ axis. So, there are three quadratic dispersion surfaces.

7. $A_2+E_2$

We denote the magnetic field of the $A_2$ mode by ${\mathbf{H}}^{(0)}$ and those of the $E_2$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $x^2-y^2$, and ${\mathbf{H}}^{(2)}$, which transforms like $2xy$, for this time. Then, we can derive the relations among $L_m^{(ij)}$’s according to Eq. (A5). We obtain the same results as the case of $B_2+E_1$, except

(B39)$$M_1=\sqrt{3}{M}_3.$$

Because the dispersion relation, Eq. (B38), does not depend on the sign of $M_1$, we obtain the same dispersion, i.e., three quadratic dispersion surfaces.

8. $A_1+E_2$

We denote the magnetic field of the $A_1$ mode by ${\mathbf{H}}^{(0)}$ and those of the $E_2$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $2xy$, and ${\mathbf{H}}^{(2)}$, which transforms like $x^2-y^2$. Then, the relations derived for $L_m^{(ij)}$’s are the same as the case of $B_2+E_1$. So, there are three quadratic dispersion surfaces in the vicinity of the $\mathrm{\Gamma }$ point.

APPENDIX C: SQUARE LATTICE

1. $A_1+B_1$

Next, we examine the square lattice of unit cells of the $C_{4v}$ symmetry. We denote the origin and the nearest-neighbor lattice points by 0 to 4, as shown in Fig. 7, where symmetry operations of the $C_{4v}$ point group are also shown.

 

Fig. 7. Numbering of the lattice points of the square lattice and symmetry operations of the $C_{4v}$ point group.

Download Full Size | PDF

We start with the case of accidental degeneracy of an $A_1$ mode and a $B_1$ mode in the Brillouin-zone center. We denote the magnetic fields of the $A_1$ and $B_1$ modes by ${\mathbf{H}}^{(1)}$ and ${\mathbf{H}}^{(2)}$, respectively. By using all $\mathcal{R}\in C_{4v}$ in Eq. (A5), we can derive the following relations:

(C1)$$L_0^{(11)}\equiv M_{10},L_0^{(22)}\equiv M_{20},L_0^{(12)}=0,$$

(C2)$$L_1^{(11)}=L_2^{(11)}=L_3^{(11)}=L_4^{(11)}\equiv M_{11},$$

(C3)$$L_1^{(22)}=L_2^{(22)}=L_3^{(22)}=L_4^{(22)}\equiv M_{22},$$

(C4)$$L_1^{(12)}=-L_2^{(12)}=L_3^{(12)}=-L_4^{(12)}\equiv M_1\mathrm{.}$$

Then, we obtain the elements of matrix $B$ and solve the secular equation. The dispersion relation thus obtained for small $\mathbf{k}$ in the vicinity of the $\mathrm{\Gamma }$ point is

(C5)$$\begin{gathered} \omega \approx {\omega }_{\mathrm{\Gamma }}-\frac{a^2{c}^2{k}^2}{4{\omega }_{\mathrm{\Gamma }}}[M_{11}+M_{21} \\ \pm \sqrt{{(M_{11}-M_{21})}^2+4{|M_1|}^2\cos \varphi }], \end{gathered}$$

where $\varphi$ is the angle between vector $\mathbf{k}$ and the $x$ axis. So, there are two quadratic dispersion surfaces.

2. $A_1+A_2$

For this combination,

(C6)$$L_0^{(12)}=L_1^{(12)}=L_2^{(12)}=L_3^{(12)}=L_4^{(12)}=0.$$

So, there is no interaction between the two modes and the dispersion relations are quadratic.

3. $A_1+B_2$

Equation (C6) holds for this combination, too. So, the dispersion relations are quadratic with respect to $\mathbf{k}$ in the vicinity of the $\mathrm{\Gamma }$ point.

4. $A_2+B_2$

We have completely the same relations for $L_m^{(ij)}$’s as $A_1+B_1$. So, there are two quadratic dispersion surfaces.

5. $B_1+E$

$E$ modes are doubly degenerate. From group theory [14], we can assume without loss of generality that two eigenfunctions of the $E$ mode transform like $x$ and $y$. We denote the magnetic field of the $B_1$ mode by ${\mathbf{H}}^{(0)}$ and those of the $E$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $x$, and ${\mathbf{H}}^{(2)}$, which transforms like $y$. Then, we can derive the following relations by examining Eq. (A5) for all $\mathcal{R}\in C_{4v}$:

(C7)$$L_0^{(00)}\equiv M_{01},L_0^{(11)}\equiv L_0^{(22)}=M_{11},$$

(C8)$$L_1^{(00)}=L_2^{(00)}=L_3^{(00)}=L_4^{(00)}\equiv M_{02},$$

(C9)$$L_1^{(11)}=L_3^{(11)}=L_2^{(22)}=L_4^{(22)}\equiv M_{12},$$

(C10)$$L_2^{(11)}=L_4^{(11)}=L_1^{(22)}=L_3^{(22)}\equiv M_{13},$$

(C11)$$L_0^{(12)}=L_1^{(12)}=L_2^{(12)}=L_3^{(12)}=L_4^{(12)}=0,$$

(C12)$$L_0^{(01)}=L_0^{(02)}=0,$$

(C13)$$L_1^{(01)}=-L_3^{(01)}=-L_2^{(02)}=L_4^{(02)}\equiv M_1,$$

(C14)$$L_1^{(02)}=L_3^{(02)}=L_2^{(01)}=L_4^{(01)}=0.$$

Then, from Eq. (A8), elements of matrix $B$ are

(C15)$$B_{00}=M_{01}+2M_{02}(\cos k_{x}a+\cos k_{y}a),$$

(C16)$$B_{11}=M_{11}+2M_{12}\cos k_{x}a+2M_{13}\cos k_{y}a,$$

(C17)$$B_{22}=M_{11}+2M_{13}\cos k_{x}a+2M_{12}\cos k_{y}a,$$

(C18)$$B_{01}=2i{M}_1\sin k_{x}a,$$

(C19)$$B_{02}=-2i{M}_1\sin k_{y}a,$$

(C20)$$B_{12}=0.$$

When we compare the structure of matrix $B$ with that for the combination of $A_1$ and $E$ modes, Eqs. (9)–(14) of [11], they are the same with each other except the sign of $B_{02}$, which is equivalent to changing the sign of $k_y$. So, the dispersion relation of the $B_1+E$ case is obtained by changing the sign of $k_y$ in the dispersion relation of $A_1+E$. Because the structure of the metamaterial that we assume in this paper is symmetric about the $x$ axis due to the $C_{4v}$ symmetry, the dispersion relation is invariant for the change of the sign of $k_y$. In conclusion, the shape of the dispersion curves is the same for the $B_1+E$ case and the $A_1+E$ case, which is a Dirac cone with an auxiliary quadratice dispersion surface.

6. $B_2+E$

Finally, we examine the combination of a $B_2$ mode and an $E$ mode. We denote the magnetic field of the $B_2$ mode by ${\mathbf{H}}^{(0)}$ and those of the $E$ mode by ${\mathbf{H}}^{(1)}$, which transforms like $y$, and ${\mathbf{H}}^{(2)}$, which transforms like $x$, for this time. Then, the relations among $L_m^{(ij)}$ are the same as the $B_1+E$ case except

(C21)$$L_1^{(01)}=-L_3^{(01)}=L_2^{(02)}=-L_4^{(02)}\equiv M_1.$$

The structure of these relations are completely the same as the $A_1+E$ case, Eqs. (27)–(32) of [11]. So, the accidental degeneracy of the $B_2$ and $E$ modes at the $\mathrm{\Gamma }$ point results in a Dirac cone with an auxiliary quadratice dispersion surface.

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 No. 22109007).

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]  

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]  

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. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18, 10120–10126 (2010). [CrossRef]  

8. 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]  

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

10. 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]  

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

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

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

14. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).