Full Polarization Conical Dispersion and Zero-Refractive-Index in Two-Dimensional Photonic Hypercrystals

Photonic conical dispersion has been found in either transverse magnetic or transverse electric polarization, and the predominant zero-refractive-index behavior in a two-dimensional photonic crystal is polarization-dependent. Here, we show that two-dimensional photonic hypercrystals can be designed that exhibit polarization independent conical dispersion at the Brillouin zone center, as two sets of triply-degenerate point for each polarization are accidentally at the same Dirac frequency. Such photonic hypercrystals consist of periodic dielectric cylinders embedded in elliptic metamaterials, and can be viewed as full-polarized near zero-refractive-index materials around Dirac frequency by using average eigen-field evaluation. Numerical simulations including directional emissions and invisibility cloak are employed to further demonstrate the double-zero-index characteristics for both polarizations in the photonic hypercrystals.

polarizations. By utilizing average eigen-field calculation, we demonstrate the full-polarized zero-refractive-index near the frequency of double Dirac cone. Directional emission and cloaking effect are simulated to illustrate the effective zero-refractive-index characteristics for both polarizations.

Full polarization conical dispersion in photonic hypercrystals
Triangular lattice with elliptic metamaterial background. It is well known that the condition on the accidentally degenerate Dirac cone in either TE or TM polarization is opposite. The TM-polarized Dirac cone prefers to the dielectric cylinders in air host, while the TE polarized Dirac cone prefers to the air rods embedded dielectric host. However, there is an exceptional case when the radii of the air rods are large enough in such way that each rod almost connects with each other. The large air-rod configuration effectively creates nearly "isolated" dielectric region so that it might support the TM-polarized Dirac cone. For example, consider a triangular lattice of air rods as shown in Fig. 1(a). The radius of the air rods is 0.4584a, where a is the lattice constant. The background medium (red) is a kind of elliptic metamaterial with the permittivity value of ε host = (ε // , ε // , ε ⊥ ) = (15.47, 15.47, 12.45). It can be constructed by layer-by-layer natural materials, so as to form the 2D photonic hypercrystal (more details later). Here, the use of anisotropy is to achieve two Dirac points of both polarizations at the same frequencies, as ε // and ε ⊥ can independently manipulate the appearance of accidentally degenerate Dirac cones for each polarization. Figure 1(c) shows the corresponding band structure of the triangular lattice with anisotropic metamaterial background. Blue lines refer to the TM polarization while red lines for the TE case. The conical dispersion and a triply accidental degenerate Dirac point appears at the center of Brillouin zone (Γ point) for both polarizations, with the frequency of ωa/2πc = 0.6965. In other words, there is a six-fold degenerate Dirac point. Detailed eigen-field pattern analysis shows that the eigen-state at the Dirac point consists of two monopolar (a) Schematic diagram of triangular photonic hypercrystal constructed by air holes in elliptic metamaterial of ε host = (ε // , ε // , ε ⊥ ), ε // > 0 and ε ⊥ > 0 (red). (b) Schematic diagram of honeycomb photonic hypercrystal with C 3v symmetry. The host (red) is made by elliptic metamaterial, while the hollow is air hole. Between holes, there are two sets of rods (green and blue) with different isotropic permittivity and same radii. (c) Band structure of the triangular photonic hypercrystal. Blue and red solid lines stand for transverse magnetic (TM) and transverse electric (TE) polarizations, respectively. The Dirac points for both polarizations are at the same frequency of 0.6965 c/a, forming the six-fold accidentally degeneracy at Γ point. (d) Band structure of the honeycomb photonic hypercrystal, showing the six-fold accidentally degenerate Dirac state at 0.6725 c/a and the lifted degeneracy at K point in TE polarization, which will benefit to improve the functionality of the zero-refractiveindex photonic crystals. singlets and two dipolar doublets, implying the feature of polarization independent zero refractive index (will be discussed later).
However, the TE-polarized band suffers from the high-k mode problem. This can be found in the red band of Fig. 1(c), that at the Dirac frequency, there are not only modes in small k near zone center, but also modes in large k near zone boundary due to double degeneracy protected by point group symmetry at K point. As a result, multimodes will be excited when placing a dipolar source inside the photonic hypercrystal, and it may seriously affect the predominant feature of the photonic device such as directional emitter.  . Effective medium parameters in the C 3v honeycomb photonic crystal by using average eigenfield method. All the components of effective permittivity and effective permeability go to zero for both polarizations, when the operating frequency approaches to the Dirac frequency. Note that the TM-polarized eigen-field is used to determine the out-of-plane component of the effective permittivity ε eff ⊥ (blue solid) and the in-plane components of the effective permeability μ eff // (blue dash). On the other hand, the remaining effective parameters are retrieved from the TE-polarized eigen-field patterns, i.e., the in-plane effective permittivity ε eff // (red solid) and the out-of-plane effective permeability μ eff ⊥ (red dash).
Scientific RepoRts | 6:22739 | DOI: 10.1038/srep22739 Honeycomb lattice of elliptic metamaterial background. The solution is to degrade the lattice symmetry in order for breaking the degeneracy at K point. The C 3v symmetry honeycomb structure is a good candidate, e.g., adding a set of small blue rods with different dielectric constant at the corner of the unit cell in Fig. 1(b). Another necessity to annihilate high-k mode issue is to form the frequency isolated point in such a way that the frequency of the flat band near K point are lower than the Dirac frequency at Γ point. This can be accomplished by increasing the severity of the broken degeneracy, e.g. adjusting the radii of small blue rod and big air rod. Note that the "isolated" region of the background should be kept and the dielectric constant of the background needs to be tuned to enable six-fold accidental degeneracy. The TE cone is determined by the in-plane component of dielectric constant ε // , while the TM case by the out-of-plane component ε ⊥ . But it does not guarantee the overlapping of the two Dirac frequencies. This is tricky to be solved when introducing a set of green rods at the other corner of the unit cell in Fig. 1(b). Followed the procedures above, a representative example is that the anisotropic permittivity of the host medium has the value of ε host = (20.27, 20.27, 10.2) [red in Fig. 1(b)], while the isotropic blue and green rods at the corner is 12 and 21, respectively. The radii of the corner and the air rod are set to be 0.119a and 0.456a. Figure 1(d) shows the corresponding band structure. The TM-polarized Dirac cone overlaps with the TE-polarized cone at the frequency of 0.6725 c/a, exhibiting the conical dispersion and six-fold accidentally degeneracy. We also plot the three-dimensional dispersion surface of the band structure near the Dirac point in Fig. 2, in order to illustrate the Dirac cone. One can see that there are two conical cones touching at the Γ point with a flat sheet for both TM and TE polarizations. Note that the polarization-independent Dirac cone is sensitive to the values of permittivity and geometry parameters. The derivation of such parameters may lead to either the lift of double cones, or the frequency shift of the two Dirac points. The very-specific permittivity above is for illustrating the conical dispersion physics in a much more clear way. In realistic system, the specific permittivity is hard to satisfy, but it is possible to control the geometry parameters instead thanks to the nowadays nanoscale technologies.

Proposal of elliptic metamaterial background.
It is worthwhile to note that the anisotropic host in the unit cell of photonic hypercrystal is hard to be constructed by natural material. Composite metamaterial, on the other hand, offers us with a solution. Here, we first consider the multilayer model to build the anisotropic dielectric constant. The multilayer consists of two alternative dielectrics with high and low permittivity, and it is placed horizontally in the xy plane so that the effective permittivity can yield by the formulism 28,29 . The host material [ε host = (15.47, 15.47, 12.45)] of the triangular photonic crystal in Fig. 1(a) can be approximately made by the high dielectric with the permittivity of 15.8 and the air layer when the filling ratio of high dielectric is 0.982. For the host material [ε host = (20.27, 20.27, 10.2)] in the honeycomb photonic crystal in Fig. 1(b), it may be approximately realized when the filling ratio of the high dielectric is 0.947 and the two permittivities are 21.3 and 1.0. Such configuration may be achieved by low loss ceramic and foam in microwave experiment. Alternatively, the host material may be obtained by the structure of metal nanowire embedded dielectric matrix in optics experiment 29 . For example, according to Eqs (18) and (19) in ref. [29], we have the effective parameters of host material in Fig. 1(a), when the permittivity of dielectric and metal is 12.96 and − 19 around λ = 0.62 μm, and the filling ratio of metal nanowire is 0.016. Another example is to set the permittivity of dielectric and metal to be 12.96 and − 26 around λ = 0.72 μm with the metal filling ratio of 0.072, in order to obtain the effective parameters of host material in Fig. 1(b).

Zero-refractive-index characteristics in photonic hypercrystals
Retrieval of the effective zero-refractive-index. In this section, we will prove the existence of the effective zero-refractive-index near the Dirac frequency in the C 3v honeycomb photonic hypercrystal. The method here is the average eigen-field method 30 . Consider the light propagates along Γ M direction of the C 3v honeycomb lattice structure in Fig. 1(b). The effective parameters at a given frequency can be retrieved by   Fig. 1(b). The polarization-independent directional emission is achieved at the Dirac frequency. Blue is for TM case while red for TE case. (b) Same as (a) except for the device of isotropic mu-near-zero materials with the values of ε = 1 and μ = 0.001. Little energy can emit outward due to impedance mismatch. (c) Same as (a) except for the device of the triangular lattice with double Dirac cone and high-k mode issue. The TE-polarized directivity is affected seriously. (d) Same as (a) except for the device of the triangular lattice, with rod radius of 0.184a and rod permittivity of 12.5. There is TM-polarized-only Dirac cone, and thus the TE-polarized direction emission fails. frequency region near the double Dirac cone are almost circle, indicating that n eff y ≈ n eff x for both polarizations. Thus, we reach μ eff x ≈ μ eff y = μ eff // and ε eff x ≈ ε eff y = ε eff // . Using the above method, the effective parameters of the C 3v honeycomb photonic crystal are retrieved from 0.6675 c/a to 0.6755 c/a and plotted in Fig. 3. All the components of the effective parameters exhibit quite a linear relation and have near-zero values in the vicinity of the Dirac frequency of 0.6725 c/a, confirming the polarization-independent effective zero-refractive-index.

Directional emission.
As the effective zero-refractive-index at the Dirac points for both polarizations has been demonstrated in last section, directional emission, as one of the characteristic of zero-refractive-index, is expected in a six-port emitter of the photonic hypercrystal. To see this, we set a E z polarized source at the Dirac frequency of 0.6725 c/a in the center of the finite hexagonal sample with the side length of 15a, whose schematic diagram is illustrated in Fig. 4(a). Propagating wave directionally emits along six directions out of the emitter, as demonstrated in the near field [ Fig. 4(a)] and far field electric pattern [blue in Fig. 5(a)]. The pattern for the TE-polarized case is almost the same as the TM-polarized case when the H z polarized source is employed [ Fig. 4(b) and red in Fig. 5(a)]. In comparison, the emission in another six-port emitter made by single near-zero index material with the parameter of ε = 1, μ = 0.001 is plotted in Fig. 5(b). The far field pattern keeps the directionality but the emitted energy is seriously weaken due to impedance mismatch. To see the high-k mode issue, we employed the triangular photonic hypercrystal of Fig. 1(a) as example. Although the double Dirac cones are present for both polarizations, the TE-polarized wave is seriously affected [Fig. 5(c)]. Figure 5(d) gives another example with photonic crystal with single Dirac cone in TM polarization, reported in ref. [9], with rod radius of 0.184a, and rod permittivity of 12.5. Since TE-polarized Dirac cone does not exist in this structure, and the TE directionality is totally lost in Fig. 5(d). To conclude, the emitter with the honeycomb photonic hypercrystal can overcome various drawbacks above. Cloaking effect. Another interesting characteristic of zero-refractive-index is cloaking effect. Here, we calculated the transmission behaviors through a waveguide made of the honeycomb photonic hypercrystal with the propagating length of 8 3 a and the height of 11a, when a hexagonal obstacle with the side length of 2a is placed inside the photonic hypercrystal waveguide. Note that the perfect magnetic conductor waveguide boundary and obstacle are used for TM polarization while the perfect electric conductor for TE polarization. In addition, two air rectangular area with the length of 3a is placed at both sides of the crystal waveguide to illustrate the profile of the wavefront. The E z field patterns in Fig. 6(a) and the H z field patterns in Fig. 6(b) show at the Dirac frequency, that the waveguide enables to preserve the transmitted wave as the plane wavefront without phase change, even when the obstacle is inserted. We also observe similar cloaking effect within a narrow bandwidth near the Dirac frequency as plotted in Fig. 6(c-f). The phase change is clearly seen inside the crystal as the effective refractive index is around 0.03(− 0.03), but the output plane wavefront is preserved. Note that the validity of the bandwidth allows to relax the sensitivity of the specific values in permittivity and geometry parameters, even though the Dirac frequencies in two polarizations have a little deviation.