Abstract

Accidental degeneracy in a photonic crystal consisting of a square array of elliptical dielectric cylinders leads to both a semi-Dirac point at the center of the Brillouin zone and an electromagnetic topological transition (ETT). A perturbation method is deduced to affirm the peculiar linear-parabolic dispersion near the semi-Dirac point. An effective medium theory is developed to explain the simultaneous semi-Dirac point and ETT and to show that the photonic crystal is either a zero-refractive-index material or an epsilon-near-zero material at the semi-Dirac point. Drastic changes in the wave manipulation properties at the semi-Dirac point, resulting from ETT, are described.

© 2014 Optical Society of America

1. Introduction

Dirac cones in electron systems give rise to many intriguing transport properties, such as Klein tunneling, Zitterbewegung, and anti-localization, because the Ek relation is linear at the corner of the Brillouin zone [13]. Similar dispersion relations, i.e., Dirac or Dirac-like cones, have also been found in classical wave systems [222], which lead to remarkable wave transport behaviors and interesting applications in classical waves. Representative examples are classical analogs of edge states in Quantum-Hall-Effect systems [5], extremal transmission [6,7], classical analogs of Zitterbewegung [8,9], and cloaking of an object [10].

Very recently, a semi-Dirac cone, a type of unique and unprecedented electronic band dispersion, was discovered [23] and studied [24]. It was found that, near a point in the Fermi surface in the two-dimensional (2D) Brillouin zone, the dispersion relation is linear along the symmetry line ((1,1) direction) but quadratic in the perpendicular direction [2326]. This unique feature results in the interesting hybridized property of the associated quasiparticles that are massless along one direction, like those in graphene, but effective-mass-like along the other direction. More interestingly, it is reported that this semi-Dirac point is associated with the topological phase transition between a semi-metallic phase and a band insulator [26]. Although semi-Dirac cones in electronic systems have attracted extensive attention, to the best of my knowledge, there has been no report on their classical analogs. If the special peculiarity of a semi-Dirac cone could be transcribed into a classical system, it is possible to envisage various intriguing consequences that could be attributed to this unique dispersion relation. An obvious one is the super anisotropic wave transport behavior near the semi-Dirac point.

The electromagnetic topological transition (ETT), proposed earlier in the context of optical waves, is the electromagnetic equivalent of the “Liftshiz transition”, in which the iso-frequency surface is transformed from a closed to an open geometry as the frequency changes [27]. This transition leads to drastic changes in the nature of the electromagnetic radiation, such as an enhanced spontaneous emission rate. Recently, a strongly anisotropic metal-dielectric metamaterial was designed to demonstrate the occurrence of the optical topological transition [27]. However, metallic components bring non-negligible losses that might affect the performance of the metamaterial, and, moreover, the size of the metamaterial’s building block is much smaller than (around one twentieth of) the operating wavelength, which means that fabrication of such a material is challenging.

The semi-Dirac point and ETT seem to be two unrelated subjects. In fact, they become connected under certain circumstances. In a 2D dispersive homogeneous anisotropic medium with permittivity ε and uniaxial permeabilityμ=diag(μx,μy), the iso-frequency surface for a transverse-electric (TE) polarized wave, i.e. E=(0,0,Ez), propagating in such a medium follows the relation

kx2μy+ky2μx=ω2ε.

Thus, altering the sign of one component in the permeability tensor, e.g. μy, would result in a transition in the topology of the iso-frequency surface. Furthermore, if the permittivity ε and μy are simultaneously zero at a particular frequency ω0, it can be shown that the dispersion relation at the center of the Brillouin zone exhibits linear-parabolic behavior near ω0, the peculiar yet defining property of a semi-Dirac cone. ETT is therefore intertwined with the semi-Dirac point under the condition that ε=μy=0 and μx0. Interestingly, this condition indicates that the material is hybridized from a zero-index material (ZIM) with zero permittivity and permeability along the x-direction and an epsilon-near-zero material (ENZM) along the y-direction. Both ZIM [10, 2830] and ENZM [3134] exhibit rich physics that gives rise to unconventional wave manipulation properties [35], such as super-coupling [31] and cloaking of an object [10]. A material hybridized from ZIM and ENZM is likely to spawn a large variety of possibilities in wave control. Thus, achieving a semi-Dirac point, an ETT, and hybridized properties in just one simple anisotropic material is fundamentally interesting in both theory and application.

The purpose of this work is to design a material that possesses a semi-Dirac point, supports ETT, and exhibits hybridized properties. This can be accomplished by employing accidental degeneracy in a 2D photonic crystal (PhC) with anisotropic dielectric scatterers. The advantage of using dielectric components is manifested in minimizing the loss that would hamper the functionality of the material. Numerical simulations show that there exists a semi-Dirac point with a finite frequency in the center of the Brillouin zone. At the semi-Dirac point, ETT is observed, where the topology of the iso-frequency surface is changed from an open hyperbola to a closed ellipse, and the PhC is indeed a ZIM along the x-direction and an ENZM along the y-direction. To gain deeper understanding of the semi-Dirac point, I developed a perturbation method [19,20] to investigate the dispersion relations and derived a boundary effective medium theory to study the effective medium properties. The perturbation method accurately predicts the slopes of the dispersion relation, confirming the linear-parabolic behavior of the dispersion relation in the vicinity of the semi-Dirac point. A surprising result unveiled by the method is that the linear slope decreases as k rotates away from the ΓX direction and eventually vanishes when k is in the ΓYdirection. The boundary effective medium theory indeed links the PhC with the special features of the semi-Dirac point and ETT to an anisotropic zero-index material. This theory offers a clear picture of the physical origin of ETT and indicates that the PhC is a hybridized material at the semi-Dirac point.

The remainder of this paper is organized as follows: the design of the PhC and its band structure, which exhibits a semi-Dirac point and ETT, are presented in Section 2. In Section 3, the perturbation method is developed and exploited to study the dispersion relations and the linear-parabolic behavior of the dispersion relation is affirmed. In Section 4, a boundary effective medium theory is deduced and ETT and the hybridized property of the PhC are subsequently discussed in the context of this theory. Two illustrative examples, beam splitting and beam shaping, are also presented in Section 4 to show the drastic change in the wave manipulation behavior induced by ETT. Conclusions are drawn in Section 5.

2. The photonic crystal system and its dispersion relation

The PhC considered in this study is a square array of elliptical cylinders with a dielectric constant of εs=12.5 embedded in air (ε0=1). The inset of Fig. 1(a) illustrates the unit cell of this PhC. The semi-minor axis of each elliptical cylinder is ra=0.188a, where a is the lattice constant, and the semi-major axis is 1.3 times that of the semi-minor axis, i.e. rb=1.3ra. The electromagnetic wave is transverse electric (TE) polarized and its electric field, E=(0,0,Ez), is always perpendicular to the plane of periodicity.

 figure: Fig. 1

Fig. 1 (a) The band structure of the 2D PhC composed of a square array of elliptical dielectric cylinders. The inset shows the unit cell of the PhC. A doubly-degenerate state in the center of the Brillouin zone is found near the dimensionless frequency, 0.540, marked as “A”. In the vicinity of this point, the dispersion relation is linear along the ΓXdirection and quadratic along theΓY direction, which is shown more clearly in Fig. 3(d). Near point “A”, there is another state in the center of the Brillouin zone, marked as “B”. The states at points “A” and “B” are used in the perturbation theory. The branches highlighted by black and blue dots are used to compute the effective medium parameters, which are shown in Fig. 4(a). (b) and (c) Enlarged views of the band structure for smaller and larger elliptical cylinders. The doubly-degenerate state shown in (a) splits into two single states, marked as A1 and A2, where A1 corresponds to a dipolar state and A2 corresponds to a monopolar state.

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Figure 1(a) shows the band structure of this PhC calculated by using COMSOL Multiphysics, a commercial package based on finite element simulations. There exists a doubly-degenerate point, marked as “A”, in the Brillouin zone center at the dimensionless frequency, ω˜=ωa/2πc=0.540, where ω is the angular frequency and c is the wave speed in air. This degenerate point is created by accidental degeneracy [10, 1720] of a monopolar state (see Fig. 3(b)) and a dipolar state (see Fig. 3(c)) at the Γ point, when the frequencies of these two states are deliberately tuned to be identical by adjusting the size or the material of the inclusion. Figures 1(b) and 1(c) show the band structure near the Γ point for smaller and larger dielectric cylinders, whose eccentricity is kept at 1.3 but whose semi-minor axes become 0.180aand 0.195a, respectively. Apparent in both figures are the separated modes at the Γ point marked as “A1” and “A2”, indicating a dipolar and a monopolar mode, respectively. The reversed relative positions of points A1 and A2 for smaller and larger cylinders imply that there must be a case where A1 and A2 coincide, which occurs when ra=0.188a as discussed earlier. This accidental degeneracy produces a very interesting dispersion relation in the vicinity of this degenerate point. Roughly seen in Fig. 1(a) are two linear bands, along the ΓX direction, touching at Point A, and a quadratic band, along the ΓY direction, tangent to a flat band also at Point A. Below the flat band there is a directional gap in the ΓY direction.

Figures 2(a) and 2(b) present simulated three-dimensional dispersion surfaces, from different view angles, in the frequency regime from ω˜=0.45 to 0.70. Two branches are obvious in these figures. The upper branch resembles the semi-Dirac cone discovered in electron systems, and the lower one is shaped like a roof, which is flat in one direction and bends down in the other directions. These two branches touch at a point, which is Point A mentioned earlier. The iso-frequency surface contours for the lower and upper branches are plotted in Figs. 2(c) and 2(d), where respective open hyperbolic and closed elliptical shapes are manifest. The change in the topology of the iso-frequency surface clearly indicates the occurrence of ETT [27] at Point A.

 figure: Fig. 2

Fig. 2 (a) and (b) The three-dimensional band structure of the PhC. The upper surface is a semi-Dirac cone. Near its bottom, it is linear in Δk along all directions except for the ΓY direction, which is quadratic. It touches the lower surface at the Brillouin zone center near the dimensionless frequency, 0.54. The lower surface is flat in one direction and bends down along the other directions. (c) and (d) The iso-frequency surfaces of the lower and higher branches, where hyperbolic and elliptical surfaces are found, respectively.

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3. A perturbation method and the confirmation of the semi-Dirac dispersion

The unusual dispersion relation leads to several interesting questions: does the dispersion relation really behave like a semi-Dirac cone? Are the dispersion relations along the ΓX direction truly linear? If yes, can their slopes be predicted? To answer these questions, I extended a perturbation theory [19,20] that I earlier developed with colleagues to study the dispersion relations, i.e. Δω as a function of Δk, near Point A. The spirit of the method is to take a set of eigenfunctions at a particular symmetry point (k0) of interest as a basis to study the eigenstates in the vicinity of that point. Accurate prediction of the linear slope of the dispersion relation is possible because the eigenfunctions take the multiple-scattering in the PhC into account. Detailed derivations can be found in the Appendix. Here, the purpose is to identify the behavior of the dispersion relation near Point A. In previous studies on Dirac/Dirac-like cones [19,20], my colleagues and I pointed out that only the degenerate states need to be considered to evaluate the linear slopes of the dispersion relations because other far-away states contribute only to the higher order in Δk [36]. Here, it is not sufficient if I consider only the doubly-degenerate states at Point A to compute the linear slopes. The reason is that the eigenstate marked as “B” in Fig. 1(a) is very close to the doubly-degenerate states and its contribution to the linear term of the dispersion relation near Point A is not negligible. Therefore, I need to solve the following 3×3 secular equation:

det|ωk2ωB2c2+P11P12P13P21ωk2ωA2c2+P22P23P31P32ωk2ωA2c2+P33|=0,
where ωA (ωB) is the frequency of the eigenstates at Point A (B), and Pij represents the mode-coupling integrals among the three eigenstates whose profiles are plotted in Figs. 3(a)3(c). Figure 3(a) shows the electric field distribution of the state located at Point B, where there is a dipolar state with its magnetic field parallel to the horizontal axis. This state is denoted as ψ1. Figures 3(b) and 3(c) exhibit the electric field map of the doubly-degenerate states at Point A. Evident are a monopolar state (Fig. 3(b)) labeled as ψ2 and a dipolar state with its magnetic field parallel to the vertical axis (Fig. 3(c)) labeled as ψ3. By substituting ψi(i=1,2,3) into Eq. (8) in the Appendix, the values of Pij are obtained to solve Eq. (2). Here, I wish to point out that the 3×3 matrix can actually be downfolded to a 2×2 one. This is because I am interested in the region close to Point A and I can express ωk as ωk=ωA+Δωk. Thus, ωk2ωA2 is approximated by 2ωAΔωk and ωk2ωB2 becomes a number, ωA2ωB2, which does not depend on Δωk and the dimension of the matrix in Eq. (2) can be reduced. Given that the dimensionless frequency at Point A is ω˜A=0.540, the solution to Eq. (2) is:
Δω˜k=(±0.0459cosβ)Δk+O(Δk2),
where β denotes the angle between k and the ΓX direction, and the dimensionless frequency, Δω˜=Δωa/2πc, is used. This solution clearly points to an important conclusion that the dispersion relation near Point A has non-zero linear dependence in Δk in all directions except for the ΓY direction where cosβ vanishes. The linear slope, Δω˜k/Δk, takes the largest value when k is along the ΓXsymmetry axis, decreases as krotates away from the ΓXdirection, and vanishes when k is along the ΓYsymmetry axis. To validate Eq. (3), I present in Figs. 3(d) and 3(e) close-up views of the dispersion relations near Point A. The dots are the result of numerical simulations and the red lines are obtained by using Eq. (3). Excellent agreement between the dots and the lines is observed in the band structure along the ΓX direction, confirming the validity of the perturbation method. Along the ΓM direction, the lines given by Eq. (3) deviate from the numerical simulations when k is away from the Γ point. The main source of the deviation is the quadratic term in Δk. To verify this, I performed a quadratic fit of the band structure shown in Fig. 3(e). I used ω˜k=ω˜A+α1Δk+α2Δk2 as the fitting function. The coefficients are α1=0.0327 and α2=0.0106 for the upper branch, and α1=0.0327 and α2=0.0116 for the lower branch. The fitting results are plotted in Fig. 3(e) as dashed blue curves. It is worth mentioning that the coefficient of the linear term α1 agrees well with the result of Eq. (3), which gives ±0.0324 when β=π/4 (ΓMdirection), implying that the linear term of the perturbation method is accurate. The dots in Fig. 3(d) indicate that the band structure along the ΓY direction does not have linear component near the doubly-degenerate point, which is predicted by Eq. (3) as well. I again performed a quadratic fit of the band structure along the ΓY direction. For the upper branch, the coefficients are α10 and α2=0.0481, suggesting that the band structure is quadratic near the center of the Brillouin zone. The fitting function is sketched in Fig. 3(d) as blue dashed curves. The green solid curves in Fig. 3(d) are obtained by solving Eq. (2) to the second order in Δk. While the lower green curve overlaps with the flat band, the upper one deviates from the band structure. The discrepancies between the perturbation theory and the band structure in both the ΓMand ΓY directions are due to the fact that the contributions of the far-away states and the second-order perturbations need to be considered to determine an accurate coefficient of Δk2 [36]. Such contributions are beyond the scope of the perturbation method described here and are therefore ignored. Figures 3(d) and 3(e) unambiguously demonstrate the linear-parabolic dispersion relation near Point A, which is further rigorously verified by the perturbation theory and the quadratic fit. In short, Point A is a semi-Dirac point.

 figure: Fig. 3

Fig. 3 (a) The electric field pattern of the eigenstate marked as “B” in Fig. 1(a). Dark red and dark blue indicate the maximum positive and negative values, respectively. This is a dipolar state with a magnetic field parallel to the x-axis, indicated by the arrows. (b) and (c) The electric field patterns of the doubly-degenerate states marked as “A” in Fig. 1(a). A monopolar and a dipolar state with the magnetic field (arrows) perpendicular to the x-axis are evident. (d) An enlarged view of the band structure near the doubly-degenerate point. The dots are calculated by COMSOL. Linear dispersion is seen along the ΓX direction, while a quadratic dispersion relation is manifest along the ΓY direction. Red solid lines and green solid curves are obtained from the perturbation theory. The blue dashed curves represent the results of quadratic fitting. (e) The same as (d) but along the ΓMdirection. A linear dispersion relation is seen again.

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Straightforward linear algebra reveals that the origin of the linear-parabolic dispersion relation resides in the strength of the mode-coupling integral between ψ2 and ψ3, i.e. unitcellψ2*(r)ψ3(r)dr. The linear term disappears as the integral vanishes. Along the ΓY direction, this integration is zero because there is no coupling between ψ2 and ψ3, as is also suggested by the symmetry of these two states. Even though accidental degeneracy is achieved, no linear dispersion is therefore found [20]. In fact, if the eigenstates are examined, it is not difficult to find that the underlying physics lies in ψ3, the dipolar mode of the doubly-degenerate state. As shown in Fig. 3(c), the magnetic field of this dipolar mode is polarized vertically, implying that it is a longitudinal mode along the ΓY direction but a transverse mode along the ΓX direction. In electromagnetic waves, the longitudinal branch is localized and almost does not couple to the incident wave and other branches [10, 37]. Thus, a flat band associated with the longitudinal dipolar mode is found in the ΓY direction. However, since ψ3 is transverse to the ΓXdirection, it easily couples to the incident wave and other branches. The coupling between ψ2 and ψ3 is therefore strong in the ΓX direction and a linear dispersion relation is found.

4. Anisotropic effective medium theory and the electromagnetic topological transition

Earlier discussions show that it is possible to link a semi-Dirac point to ETT by proper effective medium parameters. For the PhC studied here, is the condition of proper effective medium parameters satisfied? From the band structure shown in Fig. 1(a), it is evident that the frequency of the semi-Dirac point is not low because the wavelengths in the scatterer and the background are roughly 0.52 and 1.85 times the lattice constant, respectively. The low-frequency requirements of many effective medium theories that wavelengths in the scatterer and the host are much larger than the size of the unit cell [37] seem to be contradicted. However, as long as the semi-Dirac point is located in the center of the Brillouin zone and is induced by the accidental degeneracy of a monopolar state and a dipolar state, the effective medium description may still be applicable [37]. Here, I adopt a boundary effective medium approach that was originally developed for elastic waves in [38]. The effective medium parameters are computed by using the constitutive relations, i.e.

D¯z=εeffE¯z,and(B¯xB¯y)=(μxeff00μyeff)(H¯xH¯y),
where the average fields are evaluated from the eigenstate fields on the boundaries of the unit cell. For example, for an eigenstate whose Bloch wave vector, k, is along the ΓX direction, the average electric and magnetic fields are respectively:
E¯z=(0aEz(x=0)dy+0aEz(x=a)dy)/2a,
H¯y=(0aHy(x=0)dy+0aHy(x=a)dy)/2a,
and the average electric displacement and magnetic induction fields are respectively:
D¯z=(0aHy(x=a)dy0aHy(x=0)dy)/(iωa2),
B¯y=(0aEz(x=a)dy0aEz(x=0)dy)/(iωa2).
Here, the Maxwell equation, ×B=D/t and ×E=B/t, and Stokes’ theorem are exploited. For the eigenstates with Bloch wave vectors in the ΓY direction, similar expressions can be obtained.

Figure 4(a) shows the results of the effective medium parameters evaluated by this boundary integration method using the eigenstates highlighted by the solid dots shown in Fig. 1(a). Figure 4(a) shows that the effective permittivities calculated from the eigenstates along the ΓXand ΓYdirections are identical, because the electric field is a scalar for TE polarized waves. However, the permeability is anisotropic owing to the vector nature of the in-plane magnetic field. Figure 4(a) shows that μyeff and εeff cross zero simultaneously at the frequency of the semi-Dirac point, whereas μxeffequals zero at a lower frequency ω˜=0.487. I emphasize that the effective medium theory derived here is valid near the center of the Brillouin zone.

 figure: Fig. 4

Fig. 4 (a) Effective medium parameters evaluated with a boundary effective medium theory using the eigenstates highlighted by solid dots in Fig. 1(a). The blue triangles and black squares represent the effective permittivity εeff calculated by using the eigenstates along the ΓYand ΓX directions, respectively. They almost overlap, indicating that εeffis a scalar and does not depend on the direction. The red circles represent μyeff, which crosses zero simultaneously with εeffat the semi-Dirac point. The green triangles represent μxeff, which crosses zero at dimensionless frequency 0.487. Note that both the blue and green triangles are missing from the frequency regime at 0.487 to 0.540, which corresponds to a band gap along the ΓY direction. No eigenstates are thus available to evaluate the related effective medium parameters. (b)-(d) The electric field for a plane wave impinging on a PhC slab in a waveguide whose walls have perfect magnetic conductor boundary conditions at the semi-Dirac frequency 0.540. Dark red and dark blue indicate the maximum positive and negative values, respectively. (b) The real part of the electric field when the incident wave is along the ΓY direction. The transmitted field is very weak. The imaginary part is orders of magnitude smaller than the real part, which is why it is not shown here. (c) and (d) The real and imaginary parts of the electric field when the incident wave is along the ΓXdirection. Both suggest that there is no phase change in the sample, which is a typical property of a ZIM.

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The effective medium results are indeed consistent with the ETT observed earlier. As explained earlier, the iso-frequency surface of an anisotropic material follows the relation described by Eq. (1). For the effective medium of the PhC considered here, in the frequency regime between 0.487 and 0.540, μxeff>0, μyeff<0 and εeff<0. Such a combination of signs in the effective medium parameters not only leads to hyperbolic iso-frequency surfaces, which are close to those in Fig. 2(c), but it also gives rise to a band gap along the ΓY direction and a negative band along the ΓX direction. This is because ky is purely imaginary when kx=0 while nxeff=εeffμyeff<0 when ky=0. Similar analysis can be applied to the higher branch when the frequency is above the semi-Dirac point, where all the material parameters are positive. Elliptical iso-frequency surfaces are therefore expected. The simultaneous zero μyeffand εeffachieved by accidental degeneracy lead to a linear dispersion relation along the ΓX direction in the vicinity of the semi-Dirac point, whereas a single zero in εeffand a positive μxeff make the dispersion relation quadratic along the ΓY direction. All the behaviors predicted by the effective medium parameters are in line with the properties of the simulated band structures.

The ETT results in drastic changes in the wave manipulation properties. In Fig. 5, I demonstrate the radiation properties of a square sample with 16-by-16 rods that are illuminated by a point source located at the center of the sample at two different frequencies. Figures 5(a) and 5(b) show, respectively, the electric field and the flux distributions when the point source radiates at the dimensionless frequency 0.520, which is below the semi-Dirac point. Due to the hyperbolic shape of the dispersion relation at that frequency, the out-going wave splits into four beams, which are consistent with the iso-frequency surface. However, when the frequency is slightly above the semi-Dirac point, i.e. ω˜=0.544, the outgoing beams go mainly along the horizontal direction and the wave front is almost parallel to the vertical surfaces as shown in Fig. 5(c). The field pattern from the same source but inside a homogenous anisotropic medium is plotted in Fig. 5(d), where the effective medium parameters (εeff=0.018, μyeff=0.042, and μxeff=0.495) are obtained from the effective medium theory. Almost the same field pattern is observed in Figs. 5(c) and 5(d), demonstrating the validity of the effective medium theory.

 figure: Fig. 5

Fig. 5 A point source is placed inside the center of a square sample of 16-by-16 rods. (a) and (c) show the electric field patterns when the source frequency is below (0.520) and slightly above (0.544) the semi-Dirac point, respectively. Beam splitting and directional beam shaping are observed. (b) The radial flux as a function of the angle for the case simulated in (a). (d) The same as (c) but the sample is replaced by its effective medium. A similar pattern to that shown in (c) is found. Dark red and dark blue indicate the maximum positive and negative values, respectively.

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The effective medium theory also reveals the important hybridized feature of the semi-Dirac point studied here. Since μyeffand εeffare simultaneously zero while μxeff is positive, the PhC is a ZIM along the ΓX direction and an ENZM along the ΓY direction. This super anisotropic characteristic is different from previously studied anisotropic zero materials, in which one component of the physical parameters is zero [3941]. Figures 4(b)4(d) show the electric field pattern of a TE-polarized plane wave with the frequency of the semi-Dirac point impinging on a PhC slab inside a waveguide. The waveguide has boundary walls that are perfect magnetic conductors. Clearly, when the incident wave is propagating in the ΓX direction, the PhC exhibits the typical transmission property of a ZIM [10], i.e. total transmission is supported without any phase change [35] inside the material as shown in Figs. 4(c) and 4(d) for the real and imaginary parts of the electric field, respectively. However, when the PhC is illuminated by the same incident wave along the ΓY direction, the transmitted field is very weak, which is consistent with the transmission property of an ENZM. This anisotropic transport feature provides evidence that the PhC has hybridized properties of a ZIM and an ENZM.

5. Conclusions

In conclusion, I have demonstrated, by accidental degeneracy, that a 2D PhC composed of a square array of elliptical dielectric cylinders possesses a semi-Dirac point in the center of the Brillouin zone. This semi-Dirac point is associated with an ETT in its iso-frequency surface where the topology of the surface is changed from an open hyperbola to a closed ellipse. The ETT results in a drastic change in the wave manipulation behavior from beam splitting to beam shaping. A perturbation method is developed to affirm that in the vicinity of the semi-Dirac point, the dispersion relation is linear along one symmetry axis (the ΓXdirection) and quadratic along the perpendicular one (the ΓYdirection). Furthermore, this method reveals that the linear slope decays as the Bloch wave vector rotates away from the ΓXdirection. An effective medium theory based on boundary integration is deduced and used to build a link between the semi-Dirac point and ETT and to reveal the hybridized properties of the PhC at the semi-Dirac point. Since the PhC is made of dielectric media, the loss would not be significant compared to if it were made of metal. The working wavelength of the semi-Dirac point is roughly twice that of the lattice constant, which is not very large and would make fabrication of such a dielectric PhC feasible.

Appendix

In this appendix, I give a detailed derivation of the perturbation method that is used in Section 2. In two dimensions, the electric field of a TE polarized wave satisfies the following wave equation:

×(1μ(r)×Ezz^)=ω2c02ε(r)Ezz^,
where ε(r)and μ(r)are the permittivity and permeability, respectively, and c0 is the speed of light in air or vacuum. For periodic systems, the solution of Eq. (5) can be expressed as Bloch wave functions, Ψnk(r)=unk(r)eikr, where unk(r) is a periodic function and k is the Bloch wave vector. The relation between the eigenfrequency (ωnk) and the Bloch wave vector gives the nth branch of the dispersion relations.

In the perturbation theory, the unperturbed Bloch states at k0 are obtained from finite element simulations, which means that Ψnk0(r)=unk0(r)eik0r is known. The Bloch states at k near k0 can be expanded as:

Ψnk(r)=jAnj(k)ei(kk0)rΨjk0(r),
where the unknown periodic function unk(r) is expressed as linear combinations of unk0(r). By substituting Eq. (6) into Eq. (5) and invoking the orthonormal properties of the Bloch wave functions, i.e. (2π)2ΩunitcellΨlk*(r)ε(r)Ψjk(r)dr=δlj with Ω and δlj denoting the volume of a unit cell and the Kronecker delta, respectively,
j[ωj02ωnk2c02δljPlj(k)]Anj(k)=0
is obtained.

Here, Plj(k) represents the mode-coupling integrals between two states Ψlk0(r) and Ψjk0(r), and is expressed as

Plj(k)=(kk0)plj(kk0)2qlj,
whereplj=i(2π)2ΩunitcellΨlk0*(r)[2Ψjk0(r)μ(r)+(1μ(r))Ψjk0(r)]dr and qlj=(2π)2ΩunitcellΨlk0*(r)1μ(r)Ψjk0(r)dr. Since the PhC studied in this work does not involve different magnetic permeability, i.e. μ(r) is 1 everywhere, the expressions for plj and qlj can be greatly simplified.

Acknowledgments

The author is grateful to Prof. Z.Q. Zhang, Prof. C. T. Chan, Prof. J. Li, Prof. J. Mei and Dr. X. Q. Huang for fruitful discussions. Special thanks go to Prof. P. Sheng, Prof. Y. Lai, and Prof. Z. H. Hang for their comments. I also would like to acknowledge insightful comments from anonymous reviewers, which greatly helped to improve the quality of this paper. I thank V. Unkefer for editorial work on this manuscript. This work was supported by KAUST Baseline Research Funds.

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6. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007). [CrossRef]  

7. 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]  

8. 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]  

9. 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]  

10. 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]  

11. 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]  

12. V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011). [CrossRef]  

13. J. Bravo-Abad, J. D. Joannopoulos, and M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012). [CrossRef]   [PubMed]  

14. 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]  

15. 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]  

16. Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013). [CrossRef]  

17. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012). [CrossRef]   [PubMed]  

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

19. 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]  

20. 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]  

21. D. Torrent and J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012). [CrossRef]   [PubMed]  

22. 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]  

23. V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009). [CrossRef]   [PubMed]  

24. S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009). [CrossRef]   [PubMed]  

25. G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009). [CrossRef]  

26. M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011). [CrossRef]  

27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef]   [PubMed]  

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

29. 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]  

30. Y. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013). [CrossRef]  

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

32. A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013). [CrossRef]  

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

34. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef]   [PubMed]  

35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013). [CrossRef]   [PubMed]  

36. B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000). [CrossRef]  

37. Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006). [CrossRef]  

38. Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011). [CrossRef]   [PubMed]  

39. H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012). [CrossRef]  

40. J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012). [CrossRef]  

41. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012). [CrossRef]   [PubMed]  

References

  • View by:

  1. P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947).
    [Crossref]
  2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [Crossref] [PubMed]
  3. 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]
  4. 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]
  5. 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]
  6. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
    [Crossref]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011).
    [Crossref]
  13. J. Bravo-Abad, J. D. Joannopoulos, and M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
    [Crossref] [PubMed]
  14. 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]
  15. 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]
  16. Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
    [Crossref]
  17. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
    [Crossref] [PubMed]
  18. K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012).
    [Crossref] [PubMed]
  19. 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]
  20. 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]
  21. D. Torrent and J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012).
    [Crossref] [PubMed]
  22. 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]
  23. V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
    [Crossref] [PubMed]
  24. S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
    [Crossref] [PubMed]
  25. G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
    [Crossref]
  26. M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011).
    [Crossref]
  27. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
    [Crossref] [PubMed]
  28. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
    [Crossref]
  29. 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]
  30. Y. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
    [Crossref]
  31. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
    [Crossref] [PubMed]
  32. A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
    [Crossref]
  33. A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
    [Crossref]
  34. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
    [Crossref] [PubMed]
  35. N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013).
    [Crossref] [PubMed]
  36. B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000).
    [Crossref]
  37. Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
    [Crossref]
  38. Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
    [Crossref] [PubMed]
  39. H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
    [Crossref]
  40. J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
    [Crossref]
  41. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
    [Crossref] [PubMed]

2013 (7)

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]

Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
[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]

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. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
[Crossref]

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[Crossref]

N. Engheta, “Materials Science. Pursuing Near-Zero Response,” Science 340(6130), 286–287 (2013).
[Crossref] [PubMed]

2012 (10)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

D. Torrent and J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012).
[Crossref] [PubMed]

K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
[Crossref] [PubMed]

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

J. Bravo-Abad, J. D. Joannopoulos, and M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
[Crossref] [PubMed]

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]

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[Crossref]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[Crossref]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
[Crossref] [PubMed]

2011 (4)

Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[Crossref] [PubMed]

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]

V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011).
[Crossref]

M. O. Goerbig, “Electronic properties of graphene in a strong magnetic field,” Rev. Mod. Phys. 83(4), 1193–1243 (2011).
[Crossref]

2010 (3)

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]

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]

2009 (5)

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]

V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
[Crossref] [PubMed]

S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[Crossref] [PubMed]

G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[Crossref]

2008 (5)

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]

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]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

2007 (3)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref] [PubMed]

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

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

2006 (2)

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

Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[Crossref]

2000 (1)

B. A. Foreman, “Theory of the effective Hamiltonian for degenerate bands in an electric field,” J. Phys. Condens. Matter 12(34), R435–R461 (2000).
[Crossref]

1947 (1)

P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947).
[Crossref]

Alu, A.

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

Alù, A.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

Banerjee, S.

S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[Crossref] [PubMed]

Basharin, A. A.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[Crossref]

Bazaliy, Y. B.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Beenakker, C. W. J.

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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).
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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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J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
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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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Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
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Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
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A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
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A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
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M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
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G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
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J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
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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).
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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).
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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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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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J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010).
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J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (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).
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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
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Jiang, W. X.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial Power Combination for Omnidirectional Radiation via Anisotropic Metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012).
[Crossref] [PubMed]

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J. Bravo-Abad, J. D. Joannopoulos, and M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
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A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
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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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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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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
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Lai, Y.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[Crossref]

Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[Crossref] [PubMed]

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]

Li, J.

Y. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
[Crossref]

Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
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Liu, Z.

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[Crossref] [PubMed]

Luo, J.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[Crossref]

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H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[Crossref]

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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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A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
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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).
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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]

Menon, V. M.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Montambaux, G.

G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (2009).
[Crossref]

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]

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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

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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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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]

Nori, F.

Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
[Crossref]

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).
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A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
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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).
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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).
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S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
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V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
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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).
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Pickett, W. E.

V. Pardo and W. E. Pickett, “Half-Metallic Semi-Dirac-Point Generated by Quantum Confinement in TiO2/VO2 Nanostructures,” Phys. Rev. Lett. 102(16), 166803 (2009).
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S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[Crossref] [PubMed]

Piéchon, F.

G. Montambaux, F. Piéchon, J. N. Fuchs, and M. O. Goerbig, “Merging of Dirac points in a two-dimensional crystal,” Phys. Rev. B 80(15), 153412 (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, 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.

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]

Sakoda, K.

Salandrino, A.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(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]

D. Torrent and J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012).
[Crossref] [PubMed]

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, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Sheng, P.

Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[Crossref] [PubMed]

Shi, J. H.

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
[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.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

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

Silveirinha, M. G.

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

Singh, R. R. P.

S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, “Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point,” Phys. Rev. Lett. 103(1), 016402 (2009).
[Crossref] [PubMed]

Soljacic, M.

J. Bravo-Abad, J. D. Joannopoulos, and M. Soljačić, “Enabling single-mode behavior over large areas with photonic Dirac cones,” Proc. Natl. Acad. Sci. U.S.A. 109(25), 9761–9765 (2012).
[Crossref] [PubMed]

Soukoulis, C. M.

A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
[Crossref]

Szameit, A.

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]

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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]

D. Torrent and J. Sánchez-Dehesa, “Acoustic Analogue of Graphene: Observation of Dirac Cones in Acoustic Surface Waves,” Phys. Rev. Lett. 108(17), 174301 (2012).
[Crossref] [PubMed]

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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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P. R. Wallace, “The band theory of Graphite,” Phys. Rev. 71(9), 622–634 (1947).
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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. Wu and J. Li, “Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects,” Appl. Phys. Lett. 102(18), 183105 (2013).
[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]

Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
[Crossref] [PubMed]

Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[Crossref]

Xu, P.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100(22), 221903 (2012).
[Crossref]

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).
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B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
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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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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).
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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).
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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).
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Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
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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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Y. Lai, Y. Wu, P. Sheng, and Z.-Q. Zhang, “Hybrid elastic solids,” Nat. Mater. 10(8), 620–624 (2011).
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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).
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H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14(12), 123010 (2012).
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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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Phys. Rev. B (9)

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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Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: A comparative study,” Phys. Rev. B 87(24), 245134 (2013).
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V. Yannopapas, “Photonic analog of a spin-polarized system with Rashba spin-orbit coupling,” Phys. Rev. B 83(11), 113101 (2011).
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Y. Wu, J. Li, Z.-Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
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A. A. Basharin, C. Mavidis, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Epsilon near zero based phenomena in metamaterials,” Phys. Rev. B 87(15), 155130 (2013).
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Phys. Rev. Lett. (11)

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
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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]

X. Zhang, “Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
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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).
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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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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336(6078), 205–209 (2012).
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Figures (5)

Fig. 1
Fig. 1 (a) The band structure of the 2D PhC composed of a square array of elliptical dielectric cylinders. The inset shows the unit cell of the PhC. A doubly-degenerate state in the center of the Brillouin zone is found near the dimensionless frequency, 0.540, marked as “A”. In the vicinity of this point, the dispersion relation is linear along the ΓX direction and quadratic along the ΓY direction, which is shown more clearly in Fig. 3(d). Near point “A”, there is another state in the center of the Brillouin zone, marked as “B”. The states at points “A” and “B” are used in the perturbation theory. The branches highlighted by black and blue dots are used to compute the effective medium parameters, which are shown in Fig. 4(a). (b) and (c) Enlarged views of the band structure for smaller and larger elliptical cylinders. The doubly-degenerate state shown in (a) splits into two single states, marked as A1 and A2, where A1 corresponds to a dipolar state and A2 corresponds to a monopolar state.
Fig. 2
Fig. 2 (a) and (b) The three-dimensional band structure of the PhC. The upper surface is a semi-Dirac cone. Near its bottom, it is linear in Δk along all directions except for the ΓY direction, which is quadratic. It touches the lower surface at the Brillouin zone center near the dimensionless frequency, 0.54. The lower surface is flat in one direction and bends down along the other directions. (c) and (d) The iso-frequency surfaces of the lower and higher branches, where hyperbolic and elliptical surfaces are found, respectively.
Fig. 3
Fig. 3 (a) The electric field pattern of the eigenstate marked as “B” in Fig. 1(a). Dark red and dark blue indicate the maximum positive and negative values, respectively. This is a dipolar state with a magnetic field parallel to the x-axis, indicated by the arrows. (b) and (c) The electric field patterns of the doubly-degenerate states marked as “A” in Fig. 1(a). A monopolar and a dipolar state with the magnetic field (arrows) perpendicular to the x-axis are evident. (d) An enlarged view of the band structure near the doubly-degenerate point. The dots are calculated by COMSOL. Linear dispersion is seen along the ΓX direction, while a quadratic dispersion relation is manifest along the ΓY direction. Red solid lines and green solid curves are obtained from the perturbation theory. The blue dashed curves represent the results of quadratic fitting. (e) The same as (d) but along the ΓM direction. A linear dispersion relation is seen again.
Fig. 4
Fig. 4 (a) Effective medium parameters evaluated with a boundary effective medium theory using the eigenstates highlighted by solid dots in Fig. 1(a). The blue triangles and black squares represent the effective permittivity ε eff calculated by using the eigenstates along the ΓY and ΓX directions, respectively. They almost overlap, indicating that ε eff is a scalar and does not depend on the direction. The red circles represent μ y eff , which crosses zero simultaneously with ε eff at the semi-Dirac point. The green triangles represent μ x eff , which crosses zero at dimensionless frequency 0.487. Note that both the blue and green triangles are missing from the frequency regime at 0.487 to 0.540, which corresponds to a band gap along the ΓY direction. No eigenstates are thus available to evaluate the related effective medium parameters. (b)-(d) The electric field for a plane wave impinging on a PhC slab in a waveguide whose walls have perfect magnetic conductor boundary conditions at the semi-Dirac frequency 0.540. Dark red and dark blue indicate the maximum positive and negative values, respectively. (b) The real part of the electric field when the incident wave is along the ΓY direction. The transmitted field is very weak. The imaginary part is orders of magnitude smaller than the real part, which is why it is not shown here. (c) and (d) The real and imaginary parts of the electric field when the incident wave is along the ΓX direction. Both suggest that there is no phase change in the sample, which is a typical property of a ZIM.
Fig. 5
Fig. 5 A point source is placed inside the center of a square sample of 16-by-16 rods. (a) and (c) show the electric field patterns when the source frequency is below (0.520) and slightly above (0.544) the semi-Dirac point, respectively. Beam splitting and directional beam shaping are observed. (b) The radial flux as a function of the angle for the case simulated in (a). (d) The same as (c) but the sample is replaced by its effective medium. A similar pattern to that shown in (c) is found. Dark red and dark blue indicate the maximum positive and negative values, respectively.

Equations (12)

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k x 2 μ y + k y 2 μ x = ω 2 ε.
det| ω k 2 ω B 2 c 2 + P 11 P 12 P 13 P 21 ω k 2 ω A 2 c 2 + P 22 P 23 P 31 P 32 ω k 2 ω A 2 c 2 + P 33 |=0,
Δ ω ˜ k =(±0.0459cosβ)Δk+O(Δ k 2 ),
D ¯ z = ε eff E ¯ z , and ( B ¯ x B ¯ y )=( μ x eff 0 0 μ y eff )( H ¯ x H ¯ y ),
E ¯ z =( 0 a E z ( x=0 )dy + 0 a E z ( x=a )dy )/2a,
H ¯ y =( 0 a H y ( x=0 )dy + 0 a H y ( x=a )dy )/2a,
D ¯ z =( 0 a H y ( x=a )dy 0 a H y ( x=0 )dy )/( iω a 2 ),
B ¯ y =( 0 a E z ( x=a )dy 0 a E z ( x=0 )dy )/( iω a 2 ).
×( 1 μ( r ) × E z z ^ )= ω 2 c 0 2 ε( r ) E z z ^ ,
Ψ n k ( r )= j A nj ( k ) e i( k k 0 ) r Ψ j k 0 ( r ),
j [ ω j0 2 ω n k 2 c 0 2 δ lj P lj ( k ) ] A nj ( k ) =0
P lj ( k )=( k k 0 ) p lj ( k k 0 ) 2 q lj ,

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