## Abstract

By exploiting the accidental degeneracy of the doubly-degenerate dipolar and quadrupolar modes, we show that a two-dimensional dielectric photonic crystal (PC) can exhibit the double Dirac cone dispersion relation at the $\Gamma $ point. Using a perturbation method and group theory, we demonstrate that the double cone is composed of two identical and overlapping Dirac cones with predictable linear slopes, and the linearity of the dispersion is guaranteed by the spatial symmetry of the Bloch eigenstates. Numerical simulations including wave-front shaping, unidirectional transmission and perfect tunneling show that the corresponding PC structure can be characterized by a zero effective refractive index.

© 2015 Optical Society of America

## 1. Introduction

Conical dispersions in periodic systems have attracted ever-increasing attention due to the fantastic properties originating from these unique linear dispersion relations [1–39]. The well-known example is the Dirac cones at the corners of the graphene Brillouin zone (BZ), which lead to many novel phenomena in electronic transports, such as Zitterbewegung oscillation, Klein tunneling, and integer quantum Hall effect [1]. It had been demonstrated that such Dirac cones can also be realized at the BZ boundary in classical wave systems including photonic crystals (PCs) and acoustic crystals, opening up an avenue for the analogous study of properties near the Dirac point in graphene [7–24]. Another type of conical dispersion is referred to as Dirac-like cones which are observed at the BZ center in classical wave systems [25–36]. A Dirac-like cone consists of two linear bands that compose the conical dispersion surfaces and one additional flat band intersecting the conical singularity. Different from the Dirac cone formed at the BZ boundary, the Dirac-like cone at the BZ center is induced by the accidental degeneracy of a doubly-degenerate mode and a single mode. In some PCs and acoustic crystals, Dirac-like cones are related to zero effective refractive index and have been used to realize some intriguing wave propagation behaviors like cloaking [25, 27, 31] and unidirectional transmission [32].

Very recently, a new type of conical dispersion named double Dirac cone, i.e., a pair of two *identical and overlapping* Dirac cones, was found at the BZ center in two-dimensional (2D) acoustic systems with ${C}_{6v}$ symmetry [38,39]. Interestingly, it was reported that when the wavelength in the host medium at the double Dirac point is comparable to the lattice constant, the image of the acoustic crystal was formed in the wave-front of the transmitted waves, a phenomenon called as the Talbot effect [38]. Utilizing the eigenstates of unit cell, the degenerate frequency of the double Dirac cone can be lowered down to a large extent, which makes it possible that the corresponding acoustic system [39] can be mapped onto a zero-index medium (ZIM) [40–45], and many interesting wave manipulation effects such as energy tunneling effect were demonstrated in those context.

In PCs, the existence of double Dirac cone was firstly predicted by Sakoda [35–37], and he pointed that a triangular-lattice electromagnetic metamaterial can be designed to exhibit double Dirac cone dispersion at the BZ center [37], which means that the effective refractive index is zero at the degenerate frequency. However, the metallic components in [37] have intrinsic ohmic losses which may hamper the possible applications and functionalities of the double Dirac point. To minimize/eliminate the influence of metallic losses, using a system comprising purely dielectrics is a good method. On the other hand, it is well known that compared with the Dirac-like cone, the double Dirac cone is usually more difficult to achieve due to the fact that it requires a four-fold accidental degeneracy at the BZ center, instead of a three-fold degeneracy. Besides, in order to map a dielectric PC onto a ZIM and explore the attractive properties inherent to ZIM, the double Dirac point frequency should be relatively low to avoid the diffraction effect, which makes our target indeed non-trivial. Last but not least, only the dielectric constant is an alterable parameter when we search for the double Dirac cone in dielectric PCs, which makes our current task even more difficult as compared with the previous studied acoustic systems where two parameters (mass density plus wave velocity) are alterable.

To resolve the difficulties mentioned above, in this paper we propose a 2D photonic system consisting of a triangular array of dielectric cylindrical shells in an air host to realize the double Dirac cone dispersion at the BZ center. We show that by using the intendedly designed shell structure with specified material and geometry parameters, a double Dirac cone can be realized in a relatively low frequency region by utilizing the accidental degeneracy of a doubly-degenerate dipolar mode and a doubly-degenerate quadrupolar mode. To gain deeper understanding of the origin of the double Dirac cone, we perform a symmetry analysis to study the linear behavior near the degenerate point and resort to a perturbation method to explore the associated physical mechanism, which is shown to be able to accurately predict the linear slopes of the conical dispersions. In the vicinity of the double Dirac point frequency, numerical simulations unambiguously reveal that the proposed PC structure can be effectively described by a zero refractive index, and some interesting wave manipulation phenomena are demonstrated. This paper is organized as follows: In Section 2, the design of a dielectric PC with the double Dirac cone dispersion is presented. The perturbation method is also outlined to explore the origin of the double cone. In Section 3, numerical simulations are presented to demonstrate the equivalence between the PC structure and a ZIM. Conclusions are drawn in Section 4.

## 2. Double Dirac cone in a dielectric photonic crystal

The PC under investigation is a 2D triangular lattice of dielectric cylindrical shells with relative permittivity ${\epsilon}_{r}=12$ and permeability ${\mu}_{r}=1$ immersed in an air host, as illustrated in the center of Fig. 1(a). It is noted that ${\epsilon}_{r}=12$ is close to the parameter of silicon material used in semiconductor electronics. The outer radii and inner radii of the dielectric shells are denoted by ${R}_{1}$ and ${R}_{2}$, respectively. For this 2D system, we consider only the case of transverse magnetic (TM) polarization with electric field along the axis of the cylindrical shells, i.e., $\overrightarrow{E}=(0,0,{E}_{z})$. We use COMSOL Multiphysics, a commercial package based on the finite element method, to calculate the band structure of the PC.

The band structure for the PC with ${R}_{1}=0.45a$ and ${R}_{2}=0.2656a$ is shown in Fig. 1(a), which exhibits that four bands linearly intersect at a four-fold degenerate point, marked as “A”, at frequency ${\omega}_{0}=0.4044(2\pi {c}_{0}/a)$ at the BZ center. Here, $a$ is the lattice constant and ${c}_{0}$ is the speed of light in air. The field distributions of the four degenerate eigenstates at Point “A” are presented in Figs. 1(c)-1(f), with Figs. 1(c) and 1(d) corresponding to dipolar modes, and Figs. 1(e) and 1(f) corresponding to quadrupolar modes, respectively. If the outer/inner radii or material parameters of the dielectric shells are altered, the four-fold degeneracy will break up. When the inner radius ${R}_{2}=0.3a$, for example, the four-fold degeneracy will split into a doubly-degenerate dipolar mode (marked as “${E}_{1}$”), and a doubly-degenerate quadrupolar mode (marked as “${E}_{2}$”), as shown in Fig. 1(b). More importantly, the linear dispersion relations do not exist anymore when ${R}_{2}=0.3a$. Therefore we see that the four-fold degeneracy at Point “A” is *not* a necessary result guaranteed by the lattice symmetry, but instead achieved only when appropriate structural and material parameters are chosen, which means that it is an accidental degeneracy. If we zoom in the band structure at the BZ center (as shown in Fig. 2(a)), we can see that a pair of two identical and overlapping cones are formed near Point “A”. For this reason, Point “A” is called as a double Dirac point, and the associated linear dispersions are called as a double Dirac cone [37].

The linearity of the dispersion relations near Point “A” can be explained from a view point of the spatial symmetry of the four-fold degenerate state. Our previous studies have demonstrated that the linear behavior of band structure around a degenerate point can be predicted by the nonzero, mode-coupling integrals between the degenerate states [33,34], which is called as the “selection rule” in [34]. To be more specific, the criterion for the existence of linear dispersions is that the direct product of the irreducible representations of ${\phi}_{i}$, $\overrightarrow{L}$, and ${\phi}_{j}$ contains the fully symmetrical representation, where ${\phi}_{i}$ represent the *i*-th degenerate eigenstate, and $\overrightarrow{L}$ is a vector differential operator associated with the Maxwell equations [33,34]. Take the PC system shown in Fig. 1 as an example, any Bloch state at the BZ center should correspond to an irreducible representation of the ${C}_{6v}$ point group. From the field distributions of the Bloch states shown in Fig. 1, we can identify that the dipolar modes shown in Figs. 1(c) and 1(d) belong to ${E}_{1}$ representation, and the quadrupolar modes shown in Figs. 1(e) and 1(f) belong to ${E}_{2}$ representation. The operator $\overrightarrow{L}$ corresponds to ${E}_{1}$ representation in ${C}_{6v}$ group. It is easy to prove that the direct product, ${E}_{1}\otimes {E}_{1}\otimes {E}_{2}$, contains ${A}_{1}$, i.e., the fully symmetrical representation, manifesting the existence of linear dispersions. Clearly, the linearity of double Dirac cone is an inevitable result of this particular spatial symmetry of the corresponding Bloch states.

The symmetry analysis based on group theory can only assure that the dispersions near Point “A” should be linear. But it cannot tell whether the double Dirac cone is composed of two identical and overlapping cones. Therefore, a theoretical prediction of the linear slopes of band structure near the double Dirac point needs to be provided. According to Maxwell equations, the propagation of time-harmonic TM waves in a 2D periodic composite structure can be described by the following scalar wave equation:

*n*-th band with Bloch wave vector $\overrightarrow{k}$, and ${\epsilon}_{r}(\overrightarrow{r})=\epsilon (\overrightarrow{r})/{\epsilon}_{0}$ and ${\mu}_{r}(\overrightarrow{r})=\mu (\overrightarrow{r})/{\mu}_{0}$ are the relative permittivity and permeability, respectively. ${\epsilon}_{0}$ and ${\mu}_{0}$ are the permittivity and permeability of air. Due to the periodicity of the 2D PC in the

*xy*-plane, the eigenstates of Eq. (1) are Bloch functions, ${\psi}_{n\overrightarrow{k}}(\overrightarrow{r})={u}_{n\overrightarrow{k}}(\overrightarrow{r}){e}^{i\overrightarrow{k}\cdot \overrightarrow{r}}$, where ${u}_{n\overrightarrow{k}}(\overrightarrow{r})$ is a periodic function. Equation (1) together with the Bloch periodic boundary condition forms a standard Sturm-Liouville eigenvalue problem, with the electric field ${\psi}_{n\overrightarrow{k}}$ being the eigenstate and ${\omega}^{2}/{c}_{\text{0}}^{\text{2}}$ the corresponding eigenvalue. Solving the generalized eigenvalue problem, we can obtain the relation between the eigenfrequency $\omega $ and the Bloch wave vector $\overrightarrow{k}$, i.e., the dispersion relations. Here, we use a perturbation method which is similar to the $\overrightarrow{k}\cdot \overrightarrow{p}$ method in quantum mechanics to study the system, in which the eigenstates around a particular symmetry point in $\overrightarrow{k}$ space are expanded by the eigenstates at that point. Since we are only interested in the linear slopes, it is sufficient to take the four degenerate states at Point “A” as the basis to expand the eigenstates near this point.

The eigenstates at Point “A” can be obtained by employing COMSOL Multiphysics and are denoted by ${\phi}_{j}(j=1,2,3,4)$, whose orthogonality is defined by $\frac{{(2\pi )}^{2}}{\Omega}{\displaystyle \underset{unitcell}{\int}{\phi}_{l}^{*}(\overrightarrow{r}){\epsilon}_{r}(\overrightarrow{r}){\phi}_{j}(\overrightarrow{r})}d\overrightarrow{r}={\delta}_{lj}$, with $\Omega $ and ${\delta}_{lj}$ being the volume of a unit cell and the Kronecker delta, respectively. The Bloch eigenstate ${\psi}_{\overrightarrow{k}}$ for wave vector $\overrightarrow{k}$ near ${\overrightarrow{k}}_{0}$ (${\overrightarrow{k}}_{0}=0$ denotes the Bloch wave vector at Point “A”) is expanded as the linear combination of ${\phi}_{j}$s, i.e., ${\Psi}_{\overrightarrow{k}}(\overrightarrow{r})={\displaystyle \sum _{j=1}^{4}{A}_{j}{e}^{i(\overrightarrow{k}-{\overrightarrow{k}}_{0})\cdot \overrightarrow{r}}{\phi}_{j}(\overrightarrow{r})}$. Substituting this expansion into Eq. (1) and utilizing the orthonormal properties of ${\phi}_{j}(j=1,2,3,4)$, we can get:

*linear*slopes near Point “A”. Solving Eq. (2), we can obtain the dispersions near Point “A”:

Equation (5) reveals the main features of the double Dirac cone. First, there is no flat band in Fig. 2(a), which is significantly different from the case of Dirac-like cone at the BZ center. Second, considering *H* is a $4\times 4$ matrix, there are usually four solutions to Eq. (2). But only two different values of linear slopes are obtained here. This is due to the fact that each root exhibited in Eq. (5) is in fact a double root, which therefore confirms that the corresponding solution is actually a double Dirac cone consisting of two identical and overlapping cones. Third, the slopes of the linear bands are independent of the directions of wave vector $\overrightarrow{k}$ and in consequence the double Dirac cone is isotropic in character. Figure 2(b) shows the three-dimensional dispersion surfaces near the double Dirac point, which distinctly verify the aforementioned characteristics of the double Dirac cone.

## 3. Transmission properties at the double Dirac point

Since the double Dirac cone is located at the BZ center of the PC and stays in a relatively low frequency region, it is applicable to employ an effective refractive index, ${n}_{eff}$, to describe the PC structure near the double Dirac point frequency. Figure 2(a) shows that in the vicinity of double Dirac point, the magnitude of the wave vector, $k$, in the PC is much smaller than that in air, which suggests that the effective refractive index of the PC is very small and close to zero because of the direct proportional relationship between ${n}_{eff}$ and $k$. Therefore, a designed PC structure can mimic a ZIM. Actually, this statement can be validated by the following demonstrations.

Here, we would like to mention that a Dirac cone with double degeneracy of Bloch states can be realized at the $K$ and/or ${K}^{\prime}$ points of a triangular/honeycomb lattice in both quantum and classical wave systems. In such situations, the Dirac point is formed at the BZ corner. To the best of our knowledge, we do not know the realization of an isotropic Dirac cone with double degeneracy at the BZ center. For the Dirac points realized at the BZ corners, since the wave vector $\overrightarrow{k}$ is away from the $\Gamma $ point, we just cannot map the corresponding PC structure onto a zero-refractive-index medium. This difference is one of the advantages for a four-fold degeneracy than a double one.

When the electromagnetic wave propagates through a ZIM, it will accumulate no phase change even over a physically long distance. According to this special property, a given phase distribution of the impinging wave can be shaped to a desired pattern by properly tailoring the exit surface of the ZIM slab [44]. By utilizing a ZIM slab with flat exit surface, for example, the curved phase fronts of the incident wave can be converted into planar ones. In general, an ordinary PC slab cannot serve as such a phase front converter, as illustrated in Fig. 3(a), in which a tightly focused Gaussian wave packet of width $2.5a$ is illuminated to the PC slab at a frequency below the double Dirac point and then transmits through it with curved wave-front. When the incident wave frequency is very close to the double Dirac point, however, the phase front of the transmitted waves would conform to the shape of the right-side exist surface as shown in Fig. 3(b), i.e., the impinging Gaussian wave is converted into a plane wave as what is expected for a ZIM slab. In this sense, the proposed PC structure can be used as a zero-refractive-index system. In fact, the wave-front conversion shown in Fig. 3(b) can be also explained in terms of an angular filtering effect, as will be shown in the explanation for Fig. 4.

Another demonstration on the zero effective refractive index can be given through the simulation displayed in Fig. 4(a), in which a plane wave normally impinges on a prism from the bottom. The prism is made of the designed PC and the incident frequency is equal to the double Dirac point frequency. Figure 4(a) shows that the incident wave passes through the prism and finally exits from the upper surface as a plane wave with little phase distortion, although part of the incident energy is reflected back due to the impedance mismatch. The refractive angle at the upper surface is observed to be zero. From Snell’s law, we can get:

where ${n}_{1}$ is the refractive index of air. As the incident angle $\theta $ equals to ${30}^{\circ}$, the effective refractive index ${n}_{eff}$ should be zero according to Eq. (6).Considering the conservation law of the wave vector component tangential to the PC-air interface, the only allowed propagation modes inside the prism are the modes with zero tangential component of the wave vector. In other words, when the plane wave is obliquely incident on the prism from the top, as shown in Fig. 4(b), almost all the incident energy is reflected at the PC-air interface, which leads to close-to-zero transmittance in total. Therefore, the results shown in Fig. 4 suggest that the unidirectional transmission of electromagnetic wave can be easily realized by using the PC structure. Actually, such kind of manipulation of the wave propagation direction is an inherent property of ZIM: when a plane wave is incident from an ordinary medium (e.g. air) on a ZIM, total reflection is always achieved for any incident angle except for normal incidence [45]. With this example, we show again that the designed PC structure can be characterized by ${n}_{eff}=0$ at the double Dirac point frequency. The phenomenon demonstrated in Fig. 4 also reveals that the zero effective refractive index only allows normal beam to transmit through the PC slab. This is basically an angular filtering effect and it can explain the low field intensity on the right-side of the PC slab in Fig. 3(b).

At the double Dirac point, electromagnetic waves propagate with infinite phase velocity inside the PC, and thus the PC structure can be used to enhance the transmission of the impinging waves through a bend or a narrow junction. As an example, let’s consider a waveguide channel that consists of two rectangular waveguides connected by a narrow junction. Intuitively, one can expect that the abrupt variation of the waveguide’s transverse width would generate strong scattering, thereby destroying the planar wave-front and degrading the transmission efficiency. In Fig. 5(a), the waveguide channel is empty and a plane wave at the double Dirac point frequency is incident from left-side. As a result, substantial reflection can be observed due to the abrupt change of the channel width, and the phase front pattern is significantly disturbed, which is consistent with our expectation. However, when the waveguide channel is filled with the PC structure as shown in Fig. 5(b), an interesting phenomenon happens: the incident plane wave tunnels through the narrow junction and the transmitted wave preserves the planar wave-front. Nevertheless, only part of the incident wave energy can squeeze through the waveguide channel due to the mismatched impedance. To further enhance the transmission efficiency, we modify the surface impedance by truncating the cylindrical shells in the first and last rows of the PC structure with appropriate cutting parameter *L*. Figure 5(c) shows that perfect tunneling with unity transmission can be obtained when $L=1.866{R}_{1}$. Here we would like to remark that this tunneling effect is substantially different from the conventional Fabry-Perot (FP) resonance. For FP resonance, the resonant frequency will be shifted when the horizontal length of the narrow junction is changed, and apparent phase change inside the narrow junction can usually be observed. On the contrary, the tunneling effect achieved by using our PC structure is quite robust to the variation of the junction length. Moreover, no phase change can be observed inside the narrow junction, which is also different from the FP resonance and is actually a result dedicated by the zero effective refractive index.

## 4. Conclusions

To conclude, we propose a 2D photonic crystal composed of a triangular array of dielectric cylindrical shells in an air host to create a double Dirac cone at the center of the Brillouin zone. The double Dirac cone is induced by the accidental degeneracy of a doubly-degenerate dipolar mode and a doubly-degenerate quadrupolar mode. Using a perturbation method together with the group theory, we show that the linear behavior of the dispersion relations near the double Dirac point is determined by the spatial symmetry of the four degenerate Bloch states. Besides, the linear slopes of the bands can be precisely reproduced by our perturbation method, which confirms that the double Dirac cone is composed of two identical and overlapping Dirac cones. Moreover, it is demonstrated that our PC structure can be described by a zero effective refractive index at the double Dirac point frequency, making it an ideal structure to study interesting and attractive functionalities inherent to ZIMs. Accordingly, several ZIM-related numerical simulations are presented at the end of this paper, including wave-front shaping, unidirectional transmission and perfect tunneling. In addition to these demonstrations, potential applications in various possibilities of wave manipulation can also be expected for the double Dirac cone.

## Acknowledgments

The authors wish to thank Prof. Ying Wu for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 11274120), the Fundamental Research Funds for the Central Universities (Grant No. 2014ZG0032).

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