Top-down, decoupled control of constitutive parameters in electromagnetic metamaterials with dielectric resonators of internal anisotropy

A meta-atom platform providing decoupled tuning for the constitutive wave parameters remains as a challenging problem, since the proposition of Pendry. Here we propose an electromagnetic meta-atom design of internal anisotropy (εr ≠ εθ), as a pathway for decoupling of the effective- permittivity εeff and permeability μeff. Deriving effective parameters for anisotropic meta-atom from the first principles, and then subsequent inverse-solving the obtained decoupled solution for a target set of εeff and μeff, we also achieve an analytic, top-down determination for the internal structure of a meta-atom. To realize the anisotropy from isotropic materials, a particle of spatial permittivity modulation in r or θ direction is proposed. As an application example, a matched zero index dielectric meta-atom is demonstrated, to enable the super-funneling of a 50λ-wide flux through a sub-λ slit; unharnessing the flux collection limit dictated by the λ-zone.

Here, we propose a new platform for electromagnetic metamaterial: a meta-atom of decoupled constitutive parameters, enabling top-down design -where the target ε eff and μ eff are first specified and the design parameters are subsequently determined, all the while using readily available lossless dielectrics. We first analytically solve the problem of decoupled effective-permittivity ε eff and permeability μ eff , by introducing a hypothetical meta-atom of internal anisotropic susceptibility χ r ≠ χ θ ; for their axis are set in conform to the electric-and magnetic-characteristic movements of the electron, or equivalently to their corresponding dipole moments p r (χ r ) and m z (χ θ ) ( Fig. 1). We then realize top-down design for the internal structure of a meta-atom, by inversely-solving the decoupled equation to get the required χ r and χ θ (or ε r and ε θ ) for targeted ε eff and μ eff . Finally, a meta-atom implementation based on isotropic materials of spatial (r, θ) permittivity modulation is proposed to realize the required (ε r , ε θ ) for a matched zero index, along with the demonstration of the super-funneling for a 50λ-wide flux through a sub-λ slit.

Results
Structure of the meta-atom for decoupled permittivity and permeability. We consider the two dimensional problem shown in Fig. 2(a); where a transverse electric (TE) plane wave is incident onto a cylindrical particle of radius R with split ε r , ε θ anisotropy. To derive ε eff and μ eff , we start from the zeroth order expressions for the electric and magnetic polarizabilities (α e and α m ) 11 of the isolated particle, where ε y and ε θ being the permittivity of the particle along the y and θ direction respectively, and the integration is taken over the particle cross section C. It is noted that in Eq. (1) we treat only the dipole polarizability terms, with an implicit assumption of long-wavelength approximation. Solving the wave equation in polar coordinate for the general solution of E and H (details in Supplementary Information), and then keeping only the lowest order terms (within good approximation under ε r , ε θ ≫ 1), we achieve analytical solutions for α e and α m ,  Scientific RepoRts | 7:42447 | DOI: 10.1038/srep42447 From α e and α m of the isolated particle (2), it is then straightforward to calculate ε eff and μ eff for a square lattice of meta-atoms. Using the mixing formula 12 we arrive, Inspection of Eqs (2) and (3) clearly shows the dependence of μ eff only on ε θ , or complete decoupling of μ eff (or α m ) from ε r , which supports separate tunability of ε eff and μ eff from the adjustment of ε r and ε θ ; confirming the proposed ansatzs based on anisotropic susceptibility in relation to the respective current patterns exhibited by the electric and magnetic modes [Figs 1 and 2(b,c)]. Because the obtained decoupling condition α e (ε r , ε θ ) and α m (ε θ ) are based on (2) -which are expanded from (1), the condition of decoupling becomes to follow the generic constraint of long-wavelength approximation in metamaterial applications; lattice period (a)/wavelength (λ/n eff ) ~ 1/10. For example, with a = 5λ, the validity of our approximation would hold till |n eff | < 0.5. We also note, Eq. (3) works well in the low index regime (n eff ≪ λ/2a 12 ), while ε eff and μ eff of high values also always can be determined from S-matrix parameters 13 .
Focusing here on the low index case, we now proceed to inverse-solve the problem of (3), in order to determine required ε r (α e ) and ε θ (α e , α m ) by using (2), from target ε eff and μ eff . Meanwhile the complete solution with Bessel-Fourier series (Supplementary Eq. (S3)) can also be used, here we show the simpler form of first-order approximated solution, near the first zeros of the Bessel functions 11 (see Supplementary Information for details). For example, by specifying target ε eff and μ eff equal to 0, we arrive to a set of simple and intuitive relations which are used to calculate the required values of ε r and ε θ for the matched zero index; and Supplementary Fig. S1), and α 0 (~2.405), α 1 (~3.831) are the first zeros of the zeroth and first order Bessel functions. Again, for a given particle radius and frequency φ k = k 0 R, the achievement of μ eff = 0 from the single parameter ε θ is evident from B(μ eff = 0; φ k ) in Eq. (4). Subsequent realization of ε eff = 0 from the determination of ε r is made then using A(ε eff = 0; φ k ) in (4). In Fig. 2(d) we show the solution obtained with Supplementary Eq. (S5), giving the values of (ε r , ε θ ) that support a matched zero index at different target frequencies; for the particles of normalized radius R = 0.4a (solid lines) and 0.45a (dashed lines), of periodicity (a). The required (ε r , ε θ ) value set depends on the frequency and particle size, and get smaller as either f or R increase. We also demonstrate in Fig. 2(e) the tunability of ε eff and μ eff by addressing matched index properties at n eff = ± 0.1, again, with the use of (3). It is worth to note that in all cases, the required ε r is greater than ε θ , red-shifting the usually higher energy electric dipole resonance (ε r , ε θ ) closer toward the lower energy magnetic dipole resonance (ε θ ).

Dielectric implementation of the anisotropic meta-atom.
To realize the set of required ε r and ε θ from isotropic materials, we spatially modulate the permittivity inside the particle along a given axis (r or θ). A proposed structure of nano-pizza cross-section is shown in Fig. 3(a). Extending the concept of average permittivity 33 from Gauss' law in polar coordinates we obtain, where ε 1 , ε 2 (ε 1 < ε 2 ) are the permittivities of constituent dielectrics shown in Fig. 3(a), and p is the fill factor of slices containing ε 2 . While it is also possible to design the meta-atom for fixed ε 1 (1, for example) by changing p and ε 2 , we here focus on the case of fill factor p = 0.5, without any loss of generality (design example with p = 0.83, for ε 1 = 1 (air) and ε 2 = 12.25 (silicon) is shown in the Supplementary Information). On the other hand, as the arithmetic mean is always larger than the harmonic mean in (5), the condition of ε r ≥ ε θ for matched zero index realization [ Fig. 2(d)] is only met with nano-pizza cross-section geometry [we note, ε θ ≥ ε r for the nano-donut cross-section -inset of Fig. 3(a)]. Using (5)  complexity, a structure with reduced number of slices has also been tested [ Fig. 3(b)]. Even though the calculation of (ε r , ε θ ) from (p; ε 1 , ε 2 ) started to deviate from Eq. (5) when the size of slices was increased, it was still possible to determine (ε 1 , ε 2 ) = (15.12, 171.9) providing a matched zero index for the 8-slice structure at f = 0.212 c/a [marked with '+ ' symbol in Fig. 3(c)], by using few Newton iterations for the zero-index frequency deviation. It is noted that this value determined from the mixing formula (3) is in excellent agreement with exact values of (ε 1 , ε 2 ) = (14.53, 179.2) extracted from S-matrix parameters 13 .
It is emphasized that, experimentally available, smaller permittivity values using Si and SiO 2 for example [40 slices: (ε 1 , ε 2 ) = (2.43, 15.13), 8 slices: (2.22, 12.96)], can be readily accessed by increasing the radius R of the particle to 0.45a, to give matched zero index at f = 0.546 c/a (e. g., λ = 1100 nm for a = 600 nm) [ Fig. 3(d)]. The designs with 2D-slab structure (of height = 2λ, at GHz operation frequency) and a void at the particle center region are also discussed in Supplementary Information and Figures.

Realization of the zero index super funneling through a subwavelength slit.
Using the matched zero index, we now investigate the problem of extraordinary optical transmission (EOT) [34][35][36][37][38] , for which the maximum field enhancement is limited by the λ-zone 34 . Applications of zero index tunneling have been demonstrated in the past 11,14-16 , yet the possibility of EOT beyond the λ-zone has not been investigated. A perfect electric conductor (PEC) having sub-wavelength (0.21λ) slit, of flux reception width far larger (17λ) than the λ-zone has been tested, with the application of single-layer matched zero index meta-atoms (ε eff = μ eff = 0, f = 0.212 c/a) covering the input/output regions of PEC. It is important to note that the tuning of meta-atom near the slit gap is necessary since the effective medium theory starts to deviates with the introduction of the metal slit in the meta-atom array, breaking the periodicity of the lattice. The detailed tuning procedure is described in the Supplementary Information. The transmittance of the matched zero-index meta-atom nanoslit shows almost perfect transmittance of 0.97 [ Fig. 4(a,d)], a ~50 times increase compared to the slit without zero-index meta-atom coating [ Fig. 4(a,e)]; demonstrating the super-funneling of flux which is 17 times greater than the λ-zone. A low-index 8-slice structure [(ε 1 , ε 2 ) = (2.22, 12.96) at R = 0.45a providing matched zero index at f = 0.546 c/a, see Fig. 3(d)] over much larger flux reception area (50λ) also has been tested, to compensate for a factor of ~90 channel width variation (50λ to 0.55λ). A transmittance of 0.85 was achieved, showing the super-funneling of 42λ-flux (50λ · 0.85) through the meta-atom coated slit [ Fig. 4(a,f)].

Discussion
To summarize, a hypothetical meta-atom of internal (r, θ) anisotropy has been proposed. Introducing the splitsymmetry of susceptibility χ r ≠ χ θ conforming to the orthogonal axes of current pathways of the respective electric-and magnetic-dipoles, we show analytically the decoupling and separate tunability of ε eff and μ eff. The desired target optical response ε eff and μ eff are provided by top-down, analytically determined ε r and ε θ values, which are readily achieved with conventional isotropic materials in radial-or angular-anisotropic spatial arrangements. We note that, our approach widens the scope of metamaterial design; offering a top-down optical response (including both matched zero and negative index) from lossless dielectrics, meanwhile lifting the stringent restrictions of accidental degeneracy 6 which itself was limited to matched zero index at fixed frequency. In an application to EOT, utilizing a single layer of matched zero index meta-atoms, we demonstrated for the first time a super-funneling of electromagnetic flux, overcoming the usual λ-zone limit by two orders. Our proposal of coordinate-conforming anisotropy for decoupling the electric and magnetic responses and thus the separate control of ε eff and μ eff should be applicable to elementary resonators in other exotic coordinate systems compliant to current pathways of chosen electric/magnetic resonances. We expect future development of other anisotropic meta-atom families based on our approach.