1. Introduction

Materials with a zero refractive index (ZRI) have unique wave manipulation properties that are of great interest for both fundamental studies of light-matter interactions and numerous technological innovations. ZRI materials hold promise for a transformative impact via applications such as electromagnetic cloaking [1,2], plano-concave focusing lens [1], monomodal and multimodal resonances [3], unidirectional or asymmetric transmission [4,5]. ZRI can be achieved by structuring subwavelength constituents of a metamaterial, i.e. a nano-engineered composite material with properties not found in naturally occurring materials [6]. For applications, it is desirable that such materials be low-loss, e.g. absent resonant metallic nanostructures. One approach to achieve this is to use an all-dielectric nanostructure materials, e.g. a photonic crystal (PC), that provides an effective ZRI in bulk form [7,8]. All-dielectric PCs can be polymer-based. For example, photo-curable polymers such as SU-8 [9,10] provide an excellent platform for the fabrication of PC lattices. The use of such polymers opens up opportunities for the low-cost high-throughput fabrication of ZRI metamaterials. However, achieving ZRI behavior in a polymer-based PC places restrictions on the dielectric constant, a lattice constant and fill fraction. Recent research stipulated a correlation between ZRI and a triply-degenerate state (TDS) at a Dirac-like $\Gamma$-point in the photonic band structure (PBS) [1–8,11]. In diverse periodic systems, such as graphene [12], certain metamaterials [8,13], photonic crystals [11,14] and phononic crystals [11,15], Dirac cones, exhibiting linear dispersion near the center of the Brillouin zone (BZ), have been identified. In a TDS, Dirac cone bands and a flat band intersect at the $\Gamma$-point [1,8,11,16,17]. However, within the framework of the $k.p$ perturbation theory [11,18] the Dirac-like cones on the $\Gamma$ point are formed by accidental degeneracy of two modes with specific combinations of spatial symmetries. Based on $k.p$ theory analytical expressions for the shapes of dispersion curves were obtained in agreement with numerical calculations. A quasi-triply degenerate state (QTDS) is similar to a TDS except that the frequencies of the linear dispersion and flat bands are distinct and minimally spaced at the $\Gamma$-point. We demonstrate the QTDS for a 2D PC that consists of a square lattice of dielectric pillars, with index of refraction $n$ and radius $r$, in air. We use a plane wave expansion (PWE) together with a parametric analysis to tune $n$ and $r$ to achieve a QTDS. We further use finite element method (FEM) analysis to demonstrate that a PC-based metamaterial exhibits ZRI behavior for each of the three QTDS band frequencies at the $\Gamma$-point. For proof-of-concept, we consider a bulk polygon prism PC-based metamaterial. The ability to realize low-loss PC-based ZRI metamaterials as discussed here, should prove useful for a broad range of applications. Furthermore, the computational approach that we follow enables the rational design of such materials to guide fabrication.

2. Accidentally degenerate state

We study a 2D square lattice PC with dielectric pillars in air. The transverse magnetic (TM) wave with the E field polarized along the axis of the pillars is of interest because of the Dirac cone at the $\Gamma$-point. We use a PWE analysis [19] to investigate its PBS and associated Bloch functions (BFs). In our analysis, the refractive index $n$ of the pillars is varied within a range from 1 to 6 to cover most naturally occurring dielectric materials. The radius $r$ of the pillars ranges from $0$ to $0.5a$, where $a$ is the lattice constant. We begin with a PWE analysis of a 2D square lattice all-dielectric PC. The material and the structure parameters are taken from the literature [1], $n^2 = 12.5$ ($\epsilon _r = 12.5$) and $r = 0.2a$, with $\mu _r = 1$. A TDS has been reported for this PC in Ref. [19]. However, the accidental degeneracy of three bands requires precise values for the structure parameters to find the TDS; while the triple degeneracy can be approached as closely as we desire; the probability of choosing the precise degeneracy values is extremely small. From our PWE analysis bands 1, 2 and 3 do not intersect at a TDS, but rather two bands are degenerate with the symmetry of the E mode ($C_{4V}$ group) at the $\Gamma$-point as shown in the top inset of Fig. 1 and the third band of $A_1$ mode symmetry is separated by a miniscule frequency gap [11,18]. For reference, the Brillouin zone with symmetry points ($\Gamma$, M and X) are shown. The highest and lowest frequencies of these bands are labeled $f_{\Gamma H}$ and $f_{\Gamma L}$, respectively. Throughout this paper, we define the frequency spread of the bands $\Delta f_\Gamma$ to be the difference between the highest and lowest frequencies; since two bands are degenerate at the $\Gamma$ point we define the frequency difference as, $\Delta f_\Gamma = f_{\Gamma 1} + f_{\Gamma 3} - 2f_{\Gamma 2}$, where the subscript designates the band. The BFs for these bands are also shown to the right in Fig. 1, i.e. longitudinal and transverse magnetic dipole modes at $f_{\Gamma 3}$ and $f_{\Gamma 2}$, respectively and an electric monopole mode at $f_{\Gamma 1}$. We define a QTDS for the PC to be one in which combinations of the material and structure parameters ($n$ and $r$) produce a minimum frequency spread $\Delta f_\Gamma$ for the three photonic bands at the $\Gamma$-point. A parametric PWE analysis is used to screen PC material and structure parameters to identify combinations of $n$ and $r$ that define a QTDS i.e. that minimize $\Delta f_\Gamma$. We use a set of 361 plane waves symmetric about the $\Gamma$-point and retaining the square lattice symmetry. Figure 2(a) shows the 3 different band normalized frequencies at the $\Gamma$-point as a function of the cylinder refractive index n $\epsilon$ [1,6] with the cylinder radius fixed at $r=0.2a$ as above. Figure 2(c) is a similar plot of the frequencies as a function of r $\epsilon$ [0, $0.5a$] with $n=3.53616$ fixed. Two upper frequencies are degenerate; at the crossing point, the double degeneracy changes from the upper to the lower bands. As the three points approach triple degeneracy, one mode moves so that the degeneracy of bands 2 and 3 crosses over to degeneracy of bands 1 and 2. Figures 2(b) and 2(d) identify a QTDS for the PC configurations, $n = 3.53616$, $r = 0.2a$. This is close to the parameter values ($n = 3.53553$, $r = 0.2a$) reported to define a TDS as reported in Ref. [1]. In Fig. 2(b) the bands 2 and 3 are degenerate as the crossing point is approached from smaller refractive index values. We use FEM analysis to show that a smaller $\Delta f_\Gamma$ (more closely spaced bands) gives rise to a more uniform effective ZRI behavior in the bulk metamaterial. Also, note that the three distinct QTDS Eigen-frequency bands have different symmetries, the degenerate bands have dipole (E symmetry) of the $C_{4v}$ group and the third band has monopole symmetry $A_1$. Due to the modal symmetry difference, the behavior is quite distinct from anti-crossing behavior for modes of common symmetry treated in many fields of optics, physics and chemistry [20–23]. In our case with different symmetries, the modes do not mix and there should exist a set of parameters where the modes are degenerate. Based on our analysis, Figs. 2(b) and 2(d), we conclude that for each fixed value of either $n$ or $r$, we can numerically determine the parameter pair ($r$, $n$), respectively, that identifies a QTDS. It is important to note that each distinct combination of material and structure parameters ($n$, $r$) that we identify defines a distinct PC. Numerical analysis can be used to screen an entire $n-r$ parameter space to identify an infinite family of all dielectric 2D PCs, each with a distinct QTDS. To demonstrate this, we perform a parametric PWE analysis of PBS for each pair of ($n$, $r$) points in an $n-r$ plane that spans the range n $\epsilon$ [1.5,6] and r $\epsilon$ [0.15a,0.35a] as shown in Fig. 2(e). For each r we find $n$ that minimizes $\Delta f_\Gamma$. These QTDS ($n$, $r$) pairs are plotted in the $n-r$ plane and the corresponding frequency intermediate band frequency $\Delta f_{\Gamma 2}$ for each pair ($n$, $r$) is plotted on the vertical axis as shown in Fig. 2(e). Note that as $r$ increases from 0.15a to 0.35a, $f_{\Gamma 2}$ increases from 0.453 to 0.864 and $n$ decreases from 5.62 to 1.2. The lower values of $n$ are consistent with designs using polymer materials. The computationally generated relationship of $r$, $n$ and $f_{\Gamma P}$ at a QTDS can be used for the rational design of low-loss PC based ZRI metamaterials as demonstrated below. Specifically, if a particular refractive index material parameter is desired, then the required scaled radius and frequency are can be extracted from Fig. 2(e). For example, consider a PC made of GaAs pillars with refractive index 3.374 at $\lambda$=1550 nm the curve in Fig. 2(e) yields . From Fig. 2(e), the QTDS scaled frequency near 0.56 yields a lattice parameter $a=868 nm$ and $r=181 nm$. As another example, a typical fluorene-based polymer at visible wavelengths would have an index close to $n=1.6$. If the PC pillars are formed with $r= 0.3a$, then from Fig. 2(e) data the ZRI behavior can be excited near $f_{\Gamma }$ =0.78. For a specific wavelength of 800 nm, the lattice parameter is $a=624 nm$, so that $r=187 nm$. A more in-depth study reveals a correlation between the field excitations, equi-frequency contours (EFCs) and 3D dispersion surfaces near a QTDS. Four EFCs as a function of increasing values of $r/a$ are plotted in Fig. 3 with the values of $n$ and $f_{\Gamma }$ taken from the plot in Fig. 2(e). In Fig. 3, as $r$ increases, the BFs for $f_{\Gamma 2}$ and $f_{\Gamma 3}$ alternate between transverse and longitudinal magnetic dipole BFs, whereas $f_{\Gamma 1}$, which corresponds to the electric monopole BF, remains symmetric at the $\Gamma$-point. Furthermore, as $r$ increases, the apex angle of the cone gradually decreases and the cone peak becomes sharper. Starting from the center circle ring ($\Gamma$ point) the shape of the EFCs change from circular to square and then to concave. This shows that from the top of the apex angle to the bottom, the shape of the cone changes from circular to square and then to concave. The change of the EFCs of band 2 is small due to its nearly flat band character. Although the EFCs are distributed symmetrically, they do not have a regular shape. Therefore, strictly speaking, band 2 is flat in a restricted region around the $\Gamma$-point [1,3,11,14,16,17]. Additional calculations show that for band 3, as $r$ increases and $n$ decreases, EFCs gradually appear in the four corners of the first BZ. This demonstrates that from the bottom of the apex angle to the top, the shape of the upper cone changes from circular to irregular.

 

Fig. 1. PWE analysis of the TM modes of the square lattice 2D PC, $n^2$ = 12.5, $r = 0.2a$, $\mu _r$ = 1 [1]. The modes for Bands 1 to 3 are pictured on the right

Download Full Size | PDF

 

Fig. 2. PWE analysis of bands 1, 2, 3 at the Dirac-like $\Gamma$-point as a function of the index $n$ and the radius $r$ of the pillars: (a) $r = 0.2a$ fixed, n $\epsilon$ [1,6]; (b) an expanded view of the QTDS in (a); (c) $n=3.53616$ fixed, r $\epsilon$ $[0, 0.5a$]; (d) An expanded view of the QTDS in (c); (e) the line follows the ZRI region for values of the cylinder radius in the range r $\epsilon$ $[0.15a, 0.35a$]

Download Full Size | PDF

 

Fig. 3. Three bands for the 2D square lattice PC. Vertical: four values of the scaled rod radius r. Horizontal: EFCs for three bands, axes are $(kx, ky)$ coordinates in the Brillouin zone; BFs with axes $(x, y)$ in the unit cell; and 3D PBSs, axes are $(kx, ky)$ coordinates in the Brillouin zone.

Download Full Size | PDF

3. Zero refractive index

A relationship between the ZRI, $n_{eff}$, of the PC and the normalized frequency $f$ can be written as $n_{eff}=a.k_{f} (\theta )/2 \pi f$ [24]. The value of $k$ is zero at the Dirac-like $\Gamma$-point, i.e., for $f \rightarrow f\Gamma$ , $k_{f\Gamma }(\theta ) \rightarrow 0$ and therefore $n_{eff}=0$ for any of the QTDS frequencies, i.e. bands 1, 2, and 3, independent of the angle of incidence $\theta$. Using FEM simulations [25] we demonstrate this effective ZRI behavior for a metamaterial fabricated using the 2D PC. For the purpose of analysis, we consider a polygon prism structure as shown in Fig. 4(a) and perform an FEM analysis for two different sets of material-structure parameters: $n^2 = 12.5$, $r=0.2a$, and the normalized frequency of the incident wave is 0.53383, which is smaller than the PWE values $f_{\Gamma P}$ = 0.5414030 for bands 2 and 3 and $f_{\Gamma L}$ = 0.5413546. The high-low frequency spread between the three bands is small $\Delta f_{\Gamma }$=$f_{\Gamma H}$-$f_{\Gamma L}$=0.0000484. Our PWE analysis shows that the PBS bands do not intersect at the $\Gamma$-point as reported in [1]. The parameters are close enough to the ZRI value that in Fig. 4(a) the wavelength inside the prism is much larger than the wavelength outside. Interestingly, the light exiting all of prism surfaces is emitted in phase. Our simulations also show strong focusing of light at the output of a PC-based concave lens shaped structure as shown in Fig. 4(b).

 

Fig. 4. FEM analysis of ZRI behavior in a polygon prism metamaterial based on 2D SLAD PC: (a) Prism geometry with $n^2$ = 12.5, $r=0.2a$, $f=0.53383$; (b) PC lens design using the same physical parameters. In all cases $\mu _r=1$

Download Full Size | PDF

In Fig. 5(a), when the normalized frequency of the incident wave on the prism is increased to 0.54065, we find that the phase inside the prism is more uniform and again, the emitted waves at all faces are in phase with one another. The fact that the material and structural parameters give a very small value to $\Delta f_{\Gamma }$ means that the performance closely mimics the expected result for a zero index material. The results are tolerant to differences in the input frequency of the wave. In Fig. 5(b) we introduce a defect in the PC prism by removing three rods and filling the space with gold (refractive index 0.131+4.0263i). The output is largely unchanged from the perfect lattice case. Even in our limited volume structure, an object is cloaked near the zero index frequency.

 

Fig. 5. FEM analysis of PC ZRI behavior: (a) $n^2$ = 12.5, $r=0.2a$, $f=0.54065$; (b) same physical parameters as in (a) cloaking by removing three pillars and filling the space with gold.

Download Full Size | PDF

In summary, the 2D square lattice PC with dielectric rods has a QTDS Dirac-like $\Gamma$-point. Although triple degeneracy of the bands is not observed, the three bands approach one another so close that the observable properties of structures adapted to the QTDS frequency perform as expected of a ZRI material. We demonstrated this concept for a 2D all-dielectric PC and have used parametric PWE analysis to screen PC material and structure parameters $n$ and $r$ to determine combinations that render a QTDS. We have also used FEM analysis to verify that the QTDS parameters and frequencies get mapped to effective ZRI behavior in a bulk metamaterial format. For the pentaprism results in Figs. 4 and 5 we observe equal phase emerging from each face, which means that we can distribute the same phase into several different ports as a common reference phase. Additional demonstrated ZRI effects include cloaking and focusing. Finally, the computational approach demonstrated herein can enable the rational design and optimization of low-loss ZRI materials and devices for numerous photonic applications, including devices designed using materials with refractive indices around 1.5, which lie in the range of polymeric materials.