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 Γ 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 [139]. 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 [724]. Another type of conical dispersion is referred to as Dirac-like cones which are observed at the BZ center in classical wave systems [2536]. 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 C6v 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) [4045], 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 [3537], 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 εr=12 and permeability μr=1 immersed in an air host, as illustrated in the center of Fig. 1(a). It is noted that εr=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 R1 and R2, 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., E=(0,0,Ez). We use COMSOL Multiphysics, a commercial package based on the finite element method, to calculate the band structure of the PC.

 figure: Fig. 1

Fig. 1 (a) Band structure of a two-dimensional PC consisting of a triangular array of dielectric cylindrical shells (εr=12and μr=1) embedded in an air host, whose structure is schematically shown in the center of (a). The outer and inner radii of dielectric shells are R1=0.45a and R2=0.2656a, respectively. Here, a is the lattice constant. A four-fold degenerate state is created at Point “A”. (b) Band structure for R1=0.45a and R2=0.3a. The four-fold degenerate state splits into two doubly-degenerate states. (c)-(f) Electric field distributions of the four degenerate eigenstates at Point “A”. The color patterns represent the electric field, whose positive and negative maxima are represented by dark red and dark blue, respectively. Arrows show the in-plane magnetic field, whose magnitude is proportional to the length of the arrows. The field patterns in (c) and (d) are dipolar modes, and those in (e) and (f) are quadrupolar modes.

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The band structure for the PC with R1=0.45a and R2=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 ω0=0.4044(2πc0/a) at the BZ center. Here, a is the lattice constant and c0 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 R2=0.3a, for example, the four-fold degeneracy will split into a doubly-degenerate dipolar mode (marked as “E1”), and a doubly-degenerate quadrupolar mode (marked as “E2”), as shown in Fig. 1(b). More importantly, the linear dispersion relations do not exist anymore when R2=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].

 figure: Fig. 2

Fig. 2 (a) Enlarged view of band structure around Point “A”. Blue dots and red curves represent the results obtained by using COMSOL Multiphysics and predicted by the kp perturbation method, respectively. (b) Three-dimensional dispersion surfaces near Point “A”. Clearly, two identical and overlapping Dirac cones compose a double Dirac cone.

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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 φi, L, and φj contains the fully symmetrical representation, where φi represent the i-th degenerate eigenstate, and 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 C6v 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 E1 representation, and the quadrupolar modes shown in Figs. 1(e) and 1(f) belong to E2 representation. The operator L corresponds to E1 representation in C6v group. It is easy to prove that the direct product, E1E1E2, contains A1, 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:

(1μr(r)ψnk(r))=ω2c02εr(r)ψnk(r),
where ψnk(r) represents the electric field Ez on the n-th band with Bloch wave vector k, and εr(r)=ε(r)/ε0 and μr(r)=μ(r)/μ0 are the relative permittivity and permeability, respectively. ε0 and μ0 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, ψnk(r)=unk(r)eikr, where unk(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 ψnk being the eigenstate and ω2/c02 the corresponding eigenvalue. Solving the generalized eigenvalue problem, we can obtain the relation between the eigenfrequency ω and the Bloch wave vector k, i.e., the dispersion relations. Here, we use a perturbation method which is similar to the kp method in quantum mechanics to study the system, in which the eigenstates around a particular symmetry point in 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 φj(j=1,2,3,4), whose orthogonality is defined by (2π)2Ωunitcellφl*(r)εr(r)φj(r)dr=δlj, with Ω and δlj being the volume of a unit cell and the Kronecker delta, respectively. The Bloch eigenstate ψk for wave vector k near k0 (k0=0 denotes the Bloch wave vector at Point “A”) is expanded as the linear combination of φjs, i.e., Ψk(r)=j=14Ajei(kk0)rφj(r). Substituting this expansion into Eq. (1) and utilizing the orthonormal properties of φj(j=1,2,3,4), we can get:

det|Hωk2ω02c02I|=0,
where ωk (ω0) is the frequency of k point (k0 point), and H is the reduced Hamiltonian with matrix elements:
Hlj=(kk0)plj,
with
plj=i(2π)2Ωunitcellφl*(r)[2φj(r)μr(r)+(1μr(r))φj(r)]dr,
It is noted that plj=iφl|L|φj represents the mode-coupling integrals between the degenerate states, and L is the aforementioned vector operator. The second-order term of Δk=|kk0| in the definition of Hlj can be omitted if we are only interested in the linear slopes near Point “A”. Solving Eq. (2), we can obtain the dispersions near Point “A”:
Δωk=ωkω0=γβΔk,
in which the slopes of the linear dispersions are given by γβ=±0.1594c0. The red curves in Fig. 2(a) represent the dispersions predicted by Eq. (5), which agree very well with the rigorous band structure calculations by using COMSOL Multiphysics that are shown by blue dots.

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 4×4 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 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, neff, 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 neff 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 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 k is away from the Γ 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.

 figure: Fig. 3

Fig. 3 Electric field distributions produced by a Gaussian beam driven (a) at a frequency below the double Dirac point ω=0.3336(2πc0/a), and (b) exactly at the double Dirac point frequency ω0=0.4044(2πc0/a). Perfect-matched-layer boundary condition is used on the upper and lower walls of the waveguide channel. White arrows represent the direction of the incident waves. Panel (a) shows that the phase front is still curved at the exit port, and panel (b) shows that the incident phase front is re-shaped to a planar one.

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 figure: Fig. 4

Fig. 4 Electric field distributions under the excitation of a plane wave at the double Dirac point frequency ω0=0.4044(2πc0/a). (a) The plane wave is normally incident on the PC prism from the bottom. The impinging wave transmits through the prism and exits from the upper surface along the normal direction. (b) The plane wave is incident on the PC prism from the top. Obviously, the incident wave is not allowed to pass through the prism. Perfect-matched-layer boundary condition is used on the left-side and right-side walls. White arrows in (a) & (b) represent the direction of the incident plane waves. Black arrows in (a) denote the incident and refractive directions at the upper surface.

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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:

neffsinθ=n1sin0=0.
where n1 is the refractive index of air. As the incident angle θ equals to 30, the effective refractive index neff 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 neff=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=1.866R1. 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.

 figure: Fig. 5

Fig. 5 (a) Electric field distributions when a plane wave passes through an empty waveguide channel (incident from the left-side). Perfect-magnetic-conductor boundary condition is used on the upper and lower walls of the waveguide channel. H2/H1=0.2, and the working frequency is at the double Dirac point ω0=0.4044(2πc0/a). Most of the incident wave energy is reflected back and the planar wave-front is significantly disturbed. (b) The same as (a), except that the waveguide is filled with the PC structure. The plane wave squeezes through the narrow junction with little distortion and no phase variation is observed inside the narrow junction. Nevertheless, only 18% of the incident wave energy can pass through the narrow junction. (c) The same as (b), except that the cylindrical shells in the first/last rows are truncated with an appropriate cutting parameterL=1.866R1, where perfect tunneling effect can be seen. The inset of (c) shows a schematic view of the cylindrical shell in the first row after truncation.

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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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36. K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic-crystal slabs,” Phys. Rev. A 90(1), 013835 (2014). [CrossRef]  

37. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012). [CrossRef]   [PubMed]  

38. Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014). [PubMed]  

39. Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014). [CrossRef]   [PubMed]  

40. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef]   [PubMed]  

41. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010). [CrossRef]  

42. V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010). [CrossRef]   [PubMed]  

43. Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009). [CrossRef]  

44. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]  

45. F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012). [CrossRef]  

References

  • View by:

  1. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
    [Crossref]
  2. R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
    [Crossref]
  3. M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
    [Crossref]
  4. Y. Wu, “A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal,” Opt. Express 22(2), 1906–1917 (2014).
    [Crossref] [PubMed]
  5. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
    [Crossref]
  6. S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
    [Crossref] [PubMed]
  7. J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
    [PubMed]
  8. F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
    [Crossref] [PubMed]
  9. R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
    [Crossref]
  10. S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
    [Crossref] [PubMed]
  11. S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
    [Crossref]
  12. 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(1), 013904 (2008).
    [Crossref] [PubMed]
  13. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
    [Crossref]
  14. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
    [Crossref]
  15. X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
    [Crossref] [PubMed]
  16. X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
    [Crossref] [PubMed]
  17. Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36(13), 2513–2515 (2011).
    [Crossref] [PubMed]
  18. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
    [Crossref] [PubMed]
  19. A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
    [Crossref] [PubMed]
  20. M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
    [Crossref] [PubMed]
  21. G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
    [Crossref] [PubMed]
  22. D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
    [Crossref] [PubMed]
  23. D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
    [Crossref]
  24. J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
    [Crossref]
  25. 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(8), 582–586 (2011).
    [Crossref] [PubMed]
  26. F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
    [Crossref]
  27. F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
    [Crossref]
  28. P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
    [Crossref]
  29. N. Mattiucci, M. J. Bloemer, and G. D’Aguanno, “Phase-matched second harmonic generation at the Dirac point of a 2-D photonic crystal,” Opt. Express 22(6), 6381–6390 (2014).
    [Crossref] [PubMed]
  30. X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
    [Crossref]
  31. Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
    [Crossref]
  32. Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
    [Crossref]
  33. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
    [Crossref]
  34. Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
    [Crossref] [PubMed]
  35. K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
    [Crossref] [PubMed]
  36. K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic-crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
    [Crossref]
  37. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012).
    [Crossref] [PubMed]
  38. Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
    [PubMed]
  39. Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
    [Crossref] [PubMed]
  40. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
    [Crossref] [PubMed]
  41. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
    [Crossref]
  42. V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
    [Crossref] [PubMed]
  43. Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
    [Crossref]
  44. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
    [Crossref]
  45. F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
    [Crossref]

2014 (14)

R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
[Crossref]

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

Y. Wu, “A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal,” Opt. Express 22(2), 1906–1917 (2014).
[Crossref] [PubMed]

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

N. Mattiucci, M. J. Bloemer, and G. D’Aguanno, “Phase-matched second harmonic generation at the Dirac point of a 2-D photonic crystal,” Opt. Express 22(6), 6381–6390 (2014).
[Crossref] [PubMed]

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic-crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
[Crossref]

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

2013 (6)

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[Crossref]

2012 (7)

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
[Crossref] [PubMed]

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012).
[Crossref] [PubMed]

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[Crossref]

2011 (3)

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

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(8), 582–586 (2011).
[Crossref] [PubMed]

Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36(13), 2513–2515 (2011).
[Crossref] [PubMed]

2010 (4)

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[Crossref] [PubMed]

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

2009 (4)

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

2008 (5)

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

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(1), 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

2007 (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

2006 (1)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

Alù, A.

R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Barnes, W. L.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Beenakker, C. W. J.

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

Bellec, M.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

Bittner, S.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Bloemer, M. J.

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Chan, C. T.

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[Crossref]

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(8), 582–586 (2011).
[Crossref] [PubMed]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Chang, K.

Chen, H.

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Chen, L.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

Chen, X.

Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[Crossref] [PubMed]

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

Chen, Y.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Chen, Y.-F.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Chen, Z.-G.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

D’Aguanno, G.

de Dood, M. J. A.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
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Deng, F.

Deng, F. S.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Dietz, B.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Dong, J.

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Engheta, N.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
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R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
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Fu, L.

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
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Fu, Y.

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
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A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
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Haldane, F. D. M.

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(1), 013904 (2008).
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S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
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V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
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D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Hang, Z. H.

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

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(8), 582–586 (2011).
[Crossref] [PubMed]

Hao, J.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

He, C.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Hess, O.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Hu, N.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Huang, X.

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

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(8), 582–586 (2011).
[Crossref] [PubMed]

Jiang, H.

Jiang, H. T.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Joannopoulos, J. D.

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[Crossref]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Ke, M.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Kuhl, U.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

Lai, Y.

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(8), 582–586 (2011).
[Crossref] [PubMed]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Li, Y.

Li, Y. H.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Liang, Q.

Liu, F.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

Liu, Z.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

Lu, J.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Lu, L.

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[Crossref]

Lu, M.-H.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Ma, Y. G.

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Mariani, E.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Mattiucci, N.

Mayou, D.

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

McPhedran, R. C.

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

Mei, J.

Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
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J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
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P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

Meylan, M. H.

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

Min, R.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Miski-Oglu, M.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Montambaux, G.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

Mortessagne, F.

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Nguyen, V. C.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

Ni, X.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Nilsson, J.

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

Nolte, S.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Novoselov, K. S.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

Ong, C. K.

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

Oria Iriarte, P.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Pei, L.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Peng, P.

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

Peres, N. M. R.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Qiu, C.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Qiu, M.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

Raghu, S.

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(1), 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
[Crossref]

Rechtsman, M. C.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Richter, A.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Sakoda, K.

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Sánchez-Dehesa, J.

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

Schäfer, F.

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Sepkhanov, R. A.

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

Shi, Y.

Shi, Y. L.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Silveirinha, M.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Skirlo, S. A.

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

Smith, M. J. A.

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

Soljacic, M.

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[Crossref]

Sun, X.-C.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Sun, Y.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

Szameit, A.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
[PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Torrent, D.

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

Tse, W.-K.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Wang, F.

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[Crossref]

Wang, P.

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

Wang, X.

F. Deng, Y. Sun, X. Wang, R. Xue, Y. Li, H. Jiang, Y. Shi, K. Chang, and H. Chen, “Observation of valley-dependent beams in photonic graphene,” Opt. Express 22(19), 23605–23613 (2014).
[Crossref] [PubMed]

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

Wang, Z.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Wei, W.

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Weick, G.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Woollacott, C.

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

Wu, Y.

Y. Wu, “A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal,” Opt. Express 22(2), 1906–1917 (2014).
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Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Y. Li, Y. Wu, X. Chen, and J. Mei, “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[Crossref] [PubMed]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

Xu, L.

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

Xu, S.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

Xue, R.

Yan, C.

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
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Yan, W.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

Yan, Y.

Ye, Y.

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
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Zandbergen, S. R.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
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Zeuner, J. M.

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states in disordered photonic graphene,” Opt. Lett. 39(3), 602–605 (2014).
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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
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Zhang, X.

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

Zhang, Z.-Q.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Zheng, H.

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(8), 582–586 (2011).
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Zheng, L.-Y.

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Zi, J.

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
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Appl. Phys. Lett. (5)

F. Liu, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Appl. Phys. Lett. 100(7), 071911 (2012).
[Crossref]

Y. Fu, L. Xu, Z. H. Hang, and H. Chen, “Unidirectional transmission using array of zero-refractive-index metamaterials,” Appl. Phys. Lett. 104(19), 193509 (2014).
[Crossref]

Y. Li, Y. Wu, and J. Mei, “Double Dirac cones in phononic crystals,” Appl. Phys. Lett. 105(1), 014107 (2014).
[Crossref] [PubMed]

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
[Crossref]

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009).
[Crossref]

Europhys. Lett. (2)

F. Wang and C. T. Chan, “On the transition between complementary medium and zero-refractive-index medium,” Europhys. Lett. 99(6), 67002 (2012).
[Crossref]

X. Wang, H. T. Jiang, C. Yan, F. S. Deng, Y. Sun, Y. H. Li, Y. L. Shi, and H. Chen, “Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals,” Europhys. Lett. 108(1), 14002 (2014).
[Crossref]

J. Appl. Phys. (1)

Z. Wang, W. Wei, N. Hu, R. Min, L. Pei, Y. Chen, F. Liu, and Z. Liu, “Manipulation of elastic waves by zero index metamaterials,” J. Appl. Phys. 116(20), 204501 (2014).
[Crossref]

Nat. Mater. (2)

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
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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(8), 582–586 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[Crossref]

Nature (1)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. A (2)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
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K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic-crystal slabs,” Phys. Rev. A 90(1), 013835 (2014).
[Crossref]

Phys. Rev. B (10)

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

F. Liu, Y. Lai, X. Huang, and C. T. Chan, “Dirac cones atk→=0, ” Phys. Rev. B 84(22), 224113 (2011).
[Crossref]

P. Peng, J. Mei, and Y. Wu, “Lumped model for rotational modes in phononic crystals,” Phys. Rev. B 86(13), 134304 (2012).
[Crossref]

D. Torrent, D. Mayou, and J. Sánchez-Dehesa, “Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,” Phys. Rev. B 87(11), 115143 (2013).
[Crossref]

J. Lu, C. Qiu, S. Xu, Y. Ye, M. Ke, and Z. Liu, “Dirac cones in two-dimensional artificial crystals for classical waves,” Phys. Rev. B 89(13), 134302 (2014).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

S. Bittner, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, “Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene,” Phys. Rev. B 82(1), 014301 (2010).
[Crossref]

R. Fleury and A. Alù, “Manipulation of electron flow using near-zero index semiconductor metamaterials,” Phys. Rev. B 90(3), 035138 (2014).
[Crossref]

R. A. Sepkhanov, J. Nilsson, and C. W. J. Beenakker, “Proposed method for detection of the pseudospin-1/2,” Phys. Rev. B 78(4), 045122 (2008).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Phys. Rev. Lett. (10)

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105(23), 233908 (2010).
[Crossref] [PubMed]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[Crossref] [PubMed]

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
[Crossref] [PubMed]

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(1), 013904 (2008).
[Crossref] [PubMed]

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals,” Phys. Rev. Lett. 101(26), 264303 (2008).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Topological transition of Dirac points in a microwave experiment,” Phys. Rev. Lett. 110(3), 033902 (2013).
[Crossref] [PubMed]

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110(10), 106801 (2013).
[Crossref] [PubMed]

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102(12), 123904 (2009).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

Sci Rep (1)

Z.-G. Chen, X. Ni, Y. Wu, C. He, X.-C. Sun, L.-Y. Zheng, M.-H. Lu, and Y.-F. Chen, “Accidental degeneracy of double Dirac cones in a phononic crystal,” Sci Rep 4, 4613 (2014).
[PubMed]

Waves Random Complex Media (1)

M. J. A. Smith, R. C. McPhedran, and M. H. Meylan, “Double Dirac cones at k=0 in pinned platonic crystals,” Waves Random Complex Media 24(1), 35–54 (2014).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Band structure of a two-dimensional PC consisting of a triangular array of dielectric cylindrical shells ( ε r =12 and μ r =1 ) embedded in an air host, whose structure is schematically shown in the center of (a). The outer and inner radii of dielectric shells are R 1 =0.45a and R 2 =0.2656a , respectively. Here, a is the lattice constant. A four-fold degenerate state is created at Point “A”. (b) Band structure for R 1 =0.45a and R 2 =0.3a . The four-fold degenerate state splits into two doubly-degenerate states. (c)-(f) Electric field distributions of the four degenerate eigenstates at Point “A”. The color patterns represent the electric field, whose positive and negative maxima are represented by dark red and dark blue, respectively. Arrows show the in-plane magnetic field, whose magnitude is proportional to the length of the arrows. The field patterns in (c) and (d) are dipolar modes, and those in (e) and (f) are quadrupolar modes.
Fig. 2
Fig. 2 (a) Enlarged view of band structure around Point “A”. Blue dots and red curves represent the results obtained by using COMSOL Multiphysics and predicted by the k p perturbation method, respectively. (b) Three-dimensional dispersion surfaces near Point “A”. Clearly, two identical and overlapping Dirac cones compose a double Dirac cone.
Fig. 3
Fig. 3 Electric field distributions produced by a Gaussian beam driven (a) at a frequency below the double Dirac point ω=0.3336( 2π c 0 /a ) , and (b) exactly at the double Dirac point frequency ω 0 =0.4044( 2π c 0 /a ) . Perfect-matched-layer boundary condition is used on the upper and lower walls of the waveguide channel. White arrows represent the direction of the incident waves. Panel (a) shows that the phase front is still curved at the exit port, and panel (b) shows that the incident phase front is re-shaped to a planar one.
Fig. 4
Fig. 4 Electric field distributions under the excitation of a plane wave at the double Dirac point frequency ω 0 =0.4044( 2π c 0 /a ) . (a) The plane wave is normally incident on the PC prism from the bottom. The impinging wave transmits through the prism and exits from the upper surface along the normal direction. (b) The plane wave is incident on the PC prism from the top. Obviously, the incident wave is not allowed to pass through the prism. Perfect-matched-layer boundary condition is used on the left-side and right-side walls. White arrows in (a) & (b) represent the direction of the incident plane waves. Black arrows in (a) denote the incident and refractive directions at the upper surface.
Fig. 5
Fig. 5 (a) Electric field distributions when a plane wave passes through an empty waveguide channel (incident from the left-side). Perfect-magnetic-conductor boundary condition is used on the upper and lower walls of the waveguide channel. H 2 / H 1 =0.2 , and the working frequency is at the double Dirac point ω 0 =0.4044( 2π c 0 /a ) . Most of the incident wave energy is reflected back and the planar wave-front is significantly disturbed. (b) The same as (a), except that the waveguide is filled with the PC structure. The plane wave squeezes through the narrow junction with little distortion and no phase variation is observed inside the narrow junction. Nevertheless, only 18% of the incident wave energy can pass through the narrow junction. (c) The same as (b), except that the cylindrical shells in the first/last rows are truncated with an appropriate cutting parameter L=1.866 R 1 , where perfect tunneling effect can be seen. The inset of (c) shows a schematic view of the cylindrical shell in the first row after truncation.

Equations (6)

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( 1 μ r ( r ) ψ n k ( r ))= ω 2 c 0 2 ε r ( r ) ψ n k ( r ),
det| H ω k 2 ω 0 2 c 0 2 I |=0,
H lj =( k k 0 ) p lj ,
p lj =i (2π) 2 Ω unit cell φ l * ( r )[ 2 φ j ( r ) μ r ( r ) +( 1 μ r ( r ) ) φ j ( r ) ] d r ,
Δ ω k = ω k ω 0 = γ β Δk,
n eff sinθ= n 1 sin0=0.

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