Abstract
Materials with anisotropic material parameters can be utilized to fabricate many fascinating devices, such as hyperlenses, metasolids and oneway waveguides. In this study, we analyze the effects of geometric anisotropy on a twodimensional metamaterial composed of a rectangular array of elliptic cylinders and derive an effective medium theory for such a metamaterial. We find that it is possible to obtain a closedform analytical solution for the anisotropic effective medium parameters, provided the aspect ratio of the lattice and the eccentricity of the elliptic cylinder satisfy certain conditions. The derived effective medium theory not only recovers the wellknown MaxwellGarnett results in the quasistatic regime, but is also valid beyond the longwavelength limit, where the wavelength in the host medium is comparable to the size of the lattice so that previous anisotropic effective medium theories fail. Such an advance greatly broadens the applicable realm of the effective medium theory and introduces many possibilities in the design of structures with desired anisotropic material characteristics. A real sample of a recently theoretically proposed anisotropic medium, with a nearzero index to control the flux, is achieved using the derived effective medium theory and control of the electromagnetic waves in the sample is clearly demonstrated.
Introduction
Metamaterials, that is, artificial materials that possess unconventional material parameters, have been employed to achieve unprecedented functionality in the control of electromagnetic and acoustic waves, such as negative refraction^{1,2,3} and superlensing^{4,5}. One prominent class of metamaterials is anisotropic metamaterials^{6}, the material parameters of which are not scalars but tensors, with their principle components taking different values. This property causes the dispersion relations to display elliptic or hyperbolic shapes^{7}. Such anisotropic metamaterials exhibit distinctive properties, including negative refraction^{8,9}, superresolution in the farfield through image magnification^{10} and enhanced spontaneous emission^{11}. When one principle component in the material parameter tensor changes sign, a topological transition occurs^{12,13}. Earlier this year, Luo et al. proposed a method to arbitrarily control electromagnetic flux using a type of anisotropic medium. In this method, only one principle component is near zero and the other components take positive values^{14}. However, a real sample of such a medium is yet to be reported.
The unconventional material parameters of a metamaterial are based on the following two premises: (1) the structure has a subwavelength nature and (2) the metamaterial has local resonances in its building blocks. The subwavelength scale allows the heterogeneous material to be considered as a homogenized effective medium, whereas local resonances lead to exotic values of the effective medium parameters that are rarely or never observed in nature. The existence of resonances poses a considerable challenge to conventional effective medium theories (EMTs), such as the wellknown MaxwellGarnett theory and the Bruggeman theory^{15}. This is because the basic principle of a conventional EMT is to minimize the scattering at the quasistatic limit, while the local resonances usually occur in or even beyond the longwavelength regime. In the longwavelength regime, the wavelength in the host medium (λ_{0}) is large compared to the size of the unit cell, but the wavelength in the scatterer (λ_{s}) can be very small^{16}. In contrast, both λ_{0} and λ_{s} should be much larger than the size of the unit cell in the quasistatic limit.
Efforts have been made to extend conventional EMTs to higher frequency (or short wavelength) regimes. For example, a coherent potential approximation (CPA) method has been applied to both electromagnetic and elastic waves to enlarge the applicability range of the EMTs^{17,18}. Equivalent results were also obtained by taking full account of the interactions among the scatterers using the multiplescattering formalism^{19,20}. In addition, a rigorous approach based on the Floquet representation was proposed to homogenize metamaterials with periodic arrays of dielectric inclusions^{21,22,23}. Later, this approach was generalized to incorporate both dielectric and magnetic materials and a firstprinciples homogenization scheme was developed from dyadic Green's functions and polarizability coefficients. An analytical solution was obtained for periodic systems with isotropic unit cells^{24,25}. Very recently, a method based on reproducing the lowest orders of scattering amplitudes from a finite volume of metamaterials was proposed. This can give accurate predictions of the effective medium parameters over almost the entire Brillouin zone^{26}. These schemes work well for isotropic media in which both the scatterers and the lattice structures are isotropic. For anisotropic media, however, the homogenization scheme is more challenging, as it involves more degrees of freedom than in the case of isotropic media. Many conventional anisotropic EMTs are extensions of the MaxwellGarnett theory^{7,27,28,29} and are consequently limited in application to the quasistatic regime. A multiplescatteringbased scheme has been introduced to study the effective medium properties of metamaterials with anisotropic lattices and isotropic scatterers^{30,31}, yielding a scalar bulk modulus and tensorial mass density at finite frequencies in the longwavelength regime. There exist other schemes that are also applicable to anisotropic scatterers, including the fieldaveraging^{32,33}, boundaryintegration^{13,34} and parameterretrieving methods^{35,36,37,38}. The fieldaveraging and boundaryintegration methods require prior knowledge of field distributions, while the parameterretrieving method requires information about the transmission and reflection coefficients and may give nonunique solutions. More importantly, none of these three methods offer a closedform analytical solution that can directly predict reliable effective medium parameters from the material and geometric information of the system.
In this work, we consider a rectangular array of elliptic cylinders embedded in air and study its scattering properties. We discover that the special properties of elliptic coordinates and Mathieu functions (solutions to Helmholtz equations in elliptic coordinates) enable us to derive a closedform analytical solution for the anisotropic effective medium parameters, provided the aspect ratio of the lattice and the eccentricity of the elliptic cylinder satisfy certain conditions. We verify the derived EMT by comparing its predictions with fullwave bandstructure simulations and excellent agreements are found at finite frequencies beyond the longwavelength limit. This new EMT suggests promising opportunities to expand the design of anisotropic metamaterials. We show that a recently theoretically proposed anisotropic nearzero material, which can manipulate electromagnetic flux, can be achieved from the predictions of the derived EMT. The metamaterial is composed of common dielectric materials with simple structures, which makes the fabrication process feasible and would therefore greatly benefit the practical realization of the material.
Results
Modeling and the analytical solution
The system considered in our study is a twodimensional (2D) metamaterial consisting of a periodic rectangular array of elliptic cylinders with permittivity, ε_{s} and magnetic permeability, μ_{s}, embedded in a background material with permittivity, ε_{0} and magnetic permeability, μ_{0}. A unit cell of the metamaterial is illustrated in Fig. 1(a). The elliptic cylinder's semimajor and semiminor axes are a_{s} and b_{s}, respectively and its filling ratio, i.e., the ratio of the area of the elliptic cylinder to the area of the unit cell, is f. Given a_{s}, b_{s} and f, the length, a and the width, b, of the unit cell are determined by and abf = πa_{s}b_{s}. For the dispersion microstructure^{15}, in which the scatterers are always dispersed in the matrix, the CPA scheme considers the scattering of a coated cylinder in an effective medium (as shown in Fig. 1(b)). The inner elliptic cylinder represents the scatterer in the metamaterial and the coating layer is the background medium, while the semimajor and semiminor axes of the outer elliptic cylinder are a_{0} and b_{0}, respectively. Such a coated elliptic cylinder represents the microstructure of the metamaterial or the average cell^{16} in the CPA scheme, as the outside environment has been averaged as an effective medium. It is generalized from the circular (or spherical) average cells of isotropic lattices with isotropic scatterers^{17,39}. The crosssectional area of the average cell should be identical to that of a unit cell, i.e., πa_{0}b_{0} = ab, so that the filling fraction of the scatterer in the average cell is fixed to that of the metamaterial. The aspect ratio of the average cell should also equal that of the rectangular unit cell, i.e., a_{0}/b_{0} = a/b, in order to preserve the symmetry properties. With this average cell, the anisotropic property of the metamaterial is maintained and the scattering property in the effective medium is correctly produced. The effective medium parameters () are obtained when the total scattering of the average cell vanishes in the limit . In the Method section, we present detailed steps to derive the EMT of the metamaterial for a transverseelectric (TE)polarized wave, in which the electric field is parallel to the elliptic cylinders (). Here, we refer only to the final solutions, which are expressed as
and
where the effective permittivity, ε_{eff}, is a scalar related to the monopolar mode and the effective permeability is a tensor whose principle components μ_{eff,x} and μ_{eff,y} are associated with the ypolarized and xpolarized dipolar modes, respectively. The x and yaxes are set in the directions of the semimajor and semiminor axes of the elliptic cylinders, respectively. Note that all of the notations and subscripts appearing in Eq. (1) are defined or introduced in the Method section.
Verification of the EMT
In Fig. 2(a), we plot the band structure of a metamaterial obtained from a fullwave simulation using black dots. The metamaterial is composed of elliptic cylinders in a rectangular lattice embedded in air and the geometric sizes of the scatterer and the lattice are a_{s} = 0.26r, b_{s} = 0.2r, a = 1.16r and b = 1.12r, where r is a normalized length unit. The material parameters are chosen as ε_{s} = 12, μ_{s} = 1 for the scatterer and ε_{0} = 1, μ_{0} = 1 for air. Also plotted in Fig. 2(a) (in red solid curves) are the band structures predicted by the EMT, i.e., Eq. (1). The corresponding effective permittivity and permeability are shown in Figs. 2(b) and 2(c), respectively. They provide us with a clear picture and understanding of the dispersion relations. We label three points on the band structure at the Brillouin zone center as “A”, “B” and “C” (the blue dots in Fig. 2(a)). The eigenfrequencies of these points are , and , respectively and the dimensionless frequency, , is used (c_{0} is the wave velocity in air). Comparing Fig. 2(a) with Figs. 2(b) and 2(c), we find that , and correspond exactly to the frequencies at which μ_{eff,y}, ε_{eff} and μ_{eff,x} become zero. Because the dispersion relations of such an anisotropic medium are determined by^{7}
it is easy to obtain the dispersion relations in different directions. For example, in the ΓX (ΓY) direction, i.e., k_{eff,y} = 0 (k_{eff,x} = 0), we have (). If both ε_{eff} and μ_{eff,y} are positive (negative) over a frequency range, then there is a positive (negative) band in the ΓX direction. If these two quantities have different signs, then there is a gap in the ΓX direction rather than a pass band. The same rules apply to the dispersion relations along the ΓY direction if we replace μ_{eff,y} with μ_{eff,x}. With these rules, all the dispersion relations near points “A”, “B” and “C” can be easily interpreted. For example, for frequencies between and , both ε_{eff} and μ_{eff,x} are negative and μ_{eff,y} is positive. Thus, there is a negative band in the ΓY direction, but a gap in the ΓX direction. When the frequency is slightly higher and located between and , both ε_{eff} and μ_{eff,y} are positive and μ_{eff,x} is negative, explaining the positive band in the ΓX direction and the gap in the ΓY direction. The flat bands near points “A” and “C” in the ΓY and ΓX directions are in fact the longitudinal bands induced by μ_{eff,y} and μ_{eff,x} equal to zero^{17}, respectively.
Figure 2(a) illustrates the excellent agreements between the numerical simulations and the derived EMT in the center of the Brillouin zone. We also notice that the red curve deviates from the black dots when the Bloch wave vector is far removed from the Γ point. This is reasonable because we used the condition in deriving Eq. (1), which limits the range of applicability of the EMT. When the Bloch wave vector is sufficiently large that this condition no longer holds, the EMT is deemed to be inaccurate. Nevertheless, the derived EMT still yields accurate predictions for the effective medium parameters near the Γ point. Note that the red curves coincide with the black dots in the frequency regimes and (0.48, 0.61) in the ΓX direction and (0, 0.11) and (0.53, 0.66) in the ΓY direction. We also computed the transmission spectrum of a plane wave normally incident on a 9layer metamaterial sample embedded in air in these frequency regimes and the results are plotted in Figs. 2(d) (xdirection) and 2(e) (ydirection) using black dots. For comparison, the transmission spectrum of the same sample but with the metamaterial replaced by a slab of effective medium is represented by the red curves, which are calculated from the standard formula of the transmission coefficient of a layered medium^{40}. Good agreements between the numerical simulation and the effective medium prediction are again observed. Since the band structure and transmission coefficients can be used to determine the effective velocity and the impedance of the sample, respectively, Figs. 2(a), 2(d) and 2(e) offer us clear evidence that the EMT is valid.
A systematic study of the applicability of the EMT is presented in the Discussion section. Here we emphasize that, for this case, Eq. (1) is valid even when the dimensionless frequency is as high as 0.66, at which the wavelength in the background medium is 1.52a (or 1.57b), far beyond the quasistatic limit. Figures 2(f)–2(h) illustrate the field distributions of the eigenstates at points “A”, “B” and “C”, which clearly show an xpolarized dipolar mode, a monopolar mode and a ypolarized dipolar mode, respectively. These figures again support the results given by Eq. (1) that μ_{eff,y}, ε_{eff} and μ_{eff,x} are determined by the scattering coefficients of the xpolarized m = 1 mode, the m = 0 mode and the ypolarized m = 1 mode, respectively.
An anisotropic zeroindex metamaterial
As shown in Fig. 2, when the frequency takes values of , and , the system can be regarded as an anisotropic zeroindex material, because one of the effective material parameters is near zero. Zeroindex materials have unprecedented abilities to manipulate electromagnetic waves^{13,14,41,42,43,44,45}. Here, we would like to focus particularly on , where μ_{eff,x} = 0.002→0^{+}, μ_{eff,y} = 0.5637≫μ_{eff,x} and ε_{eff} = 0.1175. This indicates that the system is an anisotropic zeroindex material with only one component of the permeability tensor near zero. Very recently, such a medium was theoretically proposed and found to be capable of cloaking an arbitrarily shaped defect and of exciting evanescent waves near the defect boundaries, which therefore offers a new method of controlling the electromagnetic flux^{14}. Below, we provide simulated results of the wave transmission through such a metamaterial loaded with defects. Figure 3(a) illustrates a schematic picture of the sample, which is a waveguide filled with a metamaterial slab (composed of 12 × 10 previously mentioned unit cells). Three defects labeled “1”, “2” and “3” are distributed within the slab, as shown in Fig. 3(a), with respective sizes of 2a × 2b, 2a × b and 3a × 2b and permeability μ = 1.5, 0.4 and 2.1, respectively. The permittivity of the defects is set to 1. A TEpolarized plane wave with frequency is incident from the left.
As a comparison, we plot in Fig. 3(b) the electric field for the same sample shown in Fig. 3(a), but without the metamaterial. Strong scattered waves are excited by the defects, which significantly distort the incident wave fronts. However, the results are significantly altered in the presence of the metamaterial. Figures 3(c), 3(e) and 3(g) show, respectively, the electric field and the magnetic fields in the x and ydirections. The field patterns at the outlet of the waveguide are almost the same as those of the incident wave, indicating the good cloaking effect of the metamaterial. From Fig. 3(c), we clearly observe an almost uniform field distribution in the ydirection (vertical direction) and an apparent phase change in the xdirection (horizontal direction) in the metamaterial, implying that the metamaterial is highly anisotropic. The wavelength is nearly infinite along the ydirection, but finite along the xdirection. The corresponding field distribution patterns for the same case, but with the metamaterial replaced by the effective medium, are plotted in Figs. 3(d), 3(f) and 3(h). Similar patterns to those shown in Figs. 3(c), 3(e) and 3(g) at the inlet and outlet of the waveguide are seen, suggesting that the EMT indeed describes the physical properties of the metamaterial. From Fig. 3(f), we find that evanescent waves around the defects are induced, which are essential for high transmittance^{14}.
Figure 3 demonstrates the functionality of the anisotropic zeroindex metamaterial. Noting that the building blocks of the metamaterial are dielectric elliptic cylinders, which are easily attainable and that there are no complex structures involved, we believe that the fabrication of such a metamaterial is feasible.
Discussion
We support the validity and application of our anisotropic EMT by illustrating a simulated example, in which a set of values of a_{s}/b_{s}, ε_{s}, μ_{s} and a filling ratio of f = πa_{s}b_{s}/ab are chosen and good agreements between the numerical simulations and the EMT predictions are observed. In the following, we conduct a systematic study of the manners in which the material and geometric parameters influence the accuracy of the EMT. In Fig. 4, we plot the frequencies at which zero effective medium parameters are obtained as functions of various parameters. The curves are obtained from Eq. (1) and the dots correspond to the frequencies of the lowest monopolar and dipolar states at the Γ point, which are results of the band structure calculations. In Figs. 4(a)–4(c), we fix the permeability of the scatterers to 1 and change the aspect ratio, permittivity and the filling ratio of the scatterers, respectively. In the lower panel of Fig. 4, we study similar cases to those in the upper panel but with the permittivity of the scatterers fixed at 1. Figure 4 demonstrates that the predictions of our EMT in general coincide with the band structure simulations. When the aspect ratio and filling ratio increase, the predictions deviate from the numerical results. This is reasonable as higher angular momentum terms, i.e., m ≥ 2, contribute to the eigenmodes at low frequencies when the elliptic cylinder becomes flatter or larger. This effect leads to inaccurate predictions, because our derived effective medium scheme does not consider higher angular momentum terms.
In summary, we have derived an anisotropic EMT for a 2D electromagnetic metamaterial. This theory can provide closedform analytical solutions for anisotropic effective medium parameters and reveal the link between the effective medium parameters and the resonant modes. It is found that the effective permittivity is related to the monopolar mode and the effective permeability tensor is associated with the dipolar modes. The validity of the theory is verified by band structure and transmission spectra calculations and we find that the theory is valid even when the wavelength in the background medium is comparable to the size of the lattice, which is beyond the longwavelength limit. At the quasistatic limit, our EMT recovers the MaxwellGarnett formula. We expect that the EMT developed here will facilitate the design of new metamaterials and we show that a recently proposed anisotropic zeroindex material can indeed be fabricated from a periodic structure. Additional anisotropic metamaterials with various desired properties may also be devised based on the predictions of our EMT. Although this theory is derived for electromagnetic metamaterials, it can be generalized to its acoustic counterpart because of the mathematical mapping between these two systems in two dimensions.
Methods
Solution of the Helmholtz equation in elliptic coordinates
Considering the microstructure shown in Fig. 1(b) for a TEpolarized wave, the electric field in the effective medium can be expressed as^{46}
and, similarly, the electric field in the background medium of the coating layer is
Here, η and ξ, where 0 ≤ η < 2π and 0 ≤ ξ < ∞, represent elliptic coordinates that can be transformed into Cartesian coordinates according to x = c cos(η)cosh(ξ)and y = c sin(η)sinh(ξ), where represents the focal length of the elliptic coordinate system. In Eqs. (3) and (4), S_{m}(q; η) denote the angular Mathieu functions of the first kind, while J_{m}(q; ξ) and are the radial Mathieu functions of the first and third kinds, respectively. The subscript m is an integer denoting the order of the Mathieu functions. The angular and radial Mathieu functions form solutions to the Helmholtz equation in elliptic coordinates, which split into decoupled even (denoted by subscript e) and odd modes (denoted by subscript o) with respect to the xaxis for nonzero m. Here, the general notation γ = e, or o, is used. The variable q_{0} (q_{eff}) is a dimensionless quantity and is equal to (), where () is the wave vector in the background (effective) medium.
Boundary Conditions
The expansion coefficients in Eqs. (3) and (4), i.e. α_{γm}(σ)and β_{γm}(σ)with σ = 0, or eff, are related through the boundary conditions, which are the continuities of the tangential components of both the electric and magnetic fields on the interface between the background and effective medium. The boundary conditions can be expressed as E_{z}(eff) = E_{z}(0) and H_{η}(eff) = ∂_{ξ}E_{z}(0)/μ_{0} at ξ = ξ_{0}, where ξ_{0} = cosh^{−1}(a_{0}/c) = sinh^{−1}(b_{0}/c) is the outer boundary of the coated cylinder and H_{η}(eff) is expressed in the anisotropic effective medium as
with Δ = cosh^{2}(ξ_{0})sin^{2}(η) + sinh^{2}(ξ_{0})cos^{2}(η). Substituting Eqs. (3) and (4) into the boundary conditions, we obtain
where F_{γ} is
and
where
and
Analytical solution for the effective medium
The effective medium condition requires that the scattering of the coated cylinder vanishes. Since the scattered field of the coated cylinder is represented by , a vanishing scattered wave in the effective medium implies that β_{γm}(eff) = 0. According to Eq. (5), such a condition leads to
where D_{γm}(0) represent the Mie scattering coefficients of a scatterer of the metamaterial. These coefficients can be obtained by solving the Helmholtz equation and matching the boundary conditions between the scatterer and the background medium. They have the form
in which the subscript “s” means that the quantities take the corresponding values of the scatterer, while ξ_{s} indicates the boundary of the scatterer.
When the wavelength in the effective medium is much larger compared to the size of the coated cylinder, i.e., , the scattering of the coated cylinder is dominated by monopolar (m = 0) and dipolar (m = 1) terms. Under this condition, we substitute Eqs. (6c) and (6d) into Eq. (7) and approximate the zero and firstorder Mathieu functions associated with the effective medium by S_{e}_{0}(q_{eff}; η) = 1, , S_{e}_{1}(q_{eff}; η) = cos(η), , S_{o}_{1}(q_{eff}; η) = sin(η), , J_{e}_{0}(q_{eff}; ξ_{0}) = 1, , , , and , with and . We obtain Eq. (1), i.e.,
and
where Y_{γm}(q_{0}; ξ_{0}) are the Mathieu Neumann functions. Similar to the results for the isotropic media^{17}, the effective permittivity and permeability are determined by monopolar (m = 0) and dipolar (m = 1) modes, respectively. For the anisotropic case discussed here, however, the effective permeability is no longer a scalar, but a diagonalized tensor. It is interesting to note that the elements of the tensor, μ_{eff,x} and μ_{eff,y}, correspond exactly to the scattering coefficients of the firstorder ypolarized and xpolarized dipolar modes.
Results in the quasistatic limit
In the quasistatic limit, i.e., and , the Mathieu functions in Eq. (1) can be approximated in the same manner as that used to obtain Eq. (1) and the Mathieu Neumann functions. Y_{γm}(q_{χ}; ξ_{χ}) and its derivatives, , can be approximated as , , , , and , with and , where χ = 0, or s. Notice that is also satisfied in the quasistatic limit. We can treat the corresponding Mathieu Bessel and Neumann functions, as well as their derivatives, J_{γm}(q_{0}; ξ_{s}), , Y_{γm}(q_{0}; ξ_{s}) and , in a similar manner as previously. Eq. (1) can be reduced to
and
where f = a_{s}b_{s}/a_{0}b_{0} is the filling ratio of the elliptic cylinder. It is worth mentioning that Eq. (9) is exactly the MaxwellGarnett (MG) version EMT^{47}, in which the effective parameters are functions of the filling ratio and do not depend on the frequency.
Results in the limit of vanishing eccentricity
In the limit of vanishing eccentricity, i.e., a_{s}/b_{s} → 1 (or c → 0), the scatterer becomes an isotropic cylinder and the rectangular lattice correspondingly becomes a square lattice, according to the relation . In this limit, the angular and radial Mathieu functions transform into the trigonometric and Bessel functions, respectively^{46}. As a result, Eq. (1) is reduced to Eq. (7) in Ref. 17.
Numerical simulations
All the numerical simulations presented here are performed using COMSOL Multiphysics, a commercial package based on the finiteelement method. Figures 2(a) and 2(f)–2(h) are computed using the eigenfrequency study in the RF module. The Bloch boundary conditions are imposed on the boundaries of the unit cells. The black dots in Figs. 2(d) and 2(e) are calculated using the frequency domain study in the RF module. The same module is used in Figs. 3(b) –3(h). A radiation boundary condition is placed at the waveguide outlet so that there is no reflected wave, while periodic boundary conditions are set on the upper and lower boundaries of the waveguide. The TEpolarized plane wave with frequency is incident from the left.
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Acknowledgements
The work described here is supported by King Abdullah University of Science and Technology. The authors would like to thank Prof. P. Sheng, Prof. Z. Q. Zhang, Prof. J. Mei and Dr. M. Yang for stimulating discussions.
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X.J.Z. derived the analytic formulae, conducted numerical simulations, drew figures and prepared the manuscript. Y.W. proposed the research direction, supervised the work and revised the manuscript.
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Zhang, X., Wu, Y. Effective medium theory for anisotropic metamaterials. Sci Rep 5, 7892 (2015). https://doi.org/10.1038/srep07892
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DOI: https://doi.org/10.1038/srep07892
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