Metamaterial Homogenization Research Articles

Metamaterial eigenmodes beyond homogenization

Metamaterial homogenization theories usually start with crude approximations that are valid in certain limits in zero order, such as small frequencies, wave vectors and material fill fractions. In some cases they remain surprisingly robust exceeding their initial assumptions, such as the well-established Maxwell-Garnett theory for elliptical inclusions that can produce reliable results for fill fractions far above its theoretical limitations. We here present a rigorous solution of Maxwell’s equations in binary periodic materials employing a combined Greens-Galerkin procedure to obtain a low-dimensional eigenproblem for the evanescent Floquet eigenmodes of the material. In its general form, our method provides an accurate solution of the multi-valued complex Floquet bandstructure, which currently cannot be obtained with established solvers. It is thus shown to be valid in regimes where homogenization theories naturally break down. For small frequencies and wave numbers in lowest order, our method simplifies to the Maxwell-Garnett result for 2D cylinder and 3D sphere packings. It therefore provides the missing explanation why Maxwell-Garnett works well up to extremely high fill fractions of approximately 50% depending on the constituent materials, provided the inclusions are arranged on an isotropic lattice.

Effective acoustic metamaterial homogenization based on Hamilton's principle with a multiple scales approximation.

This paper derives and demonstrates a one-dimensional acoustic metamaterial homogenization method. The homogenization method uses a multiple-scales approximation with Hamilton's principle, a weak-form representation of the dynamic equation. While the multiple-scales approximation makes the predicted effective material properties of this method inexact, the method is shown to be highly versatile. Analytical and numerical examples are given showing the ability of the homogenization method to account for viscosity and finite-amplitude effects.

A Lagrangian formulation of metamaterial homogenization

Many metamaterials may be well described as multi-scale systems where the large scale is represented by the wavelength and the small scales are represented by inhomogeneity lengths. The effective material properties of a metamaterial of this sort may be derived by considering the averaged behavior of material elements that are considered “infinitesimal” relative to the wavelength but is on the order of the inhomogeneity lengths. The averaging of the small-scale features enables a straightforward process to describe the metamaterial mechanics in terms of a Lagrangian formalism, and the macroscopic constitutive equations may be obtained by deriving the Euler-Lagrange equations for the macroscopic field quantities. This energetic approach to homogenization may be used to estimate both linear and nonlinear material properties. This talk describes this homogenization method for one-dimensional systems and compares the resulting effective material properties with other homogenization methods.

Higher Order Multipoles in Metamaterial Homogenization

The higher order multipoles above the electric quadrupole are commonly neglected in metamaterial homogenization. We show that they nevertheless can be significant when second order spatial dispersive effects, such as the magnetic response, are considered. In this respect, they can be equally important as the magnetization and quadrupole terms, and should not automatically be neglected.

Acoustic metamaterial homogenization based on equivalent fluid media with coupled field response

Homogenization schemes for wave propagation in heterogeneous electromagnetic (EM) and elastic materials indicate that EM bianisotropy and elastic momentum-strain and stress-velocity field coupling is required to correctly describe the effective behavior of the medium [Alu, Phys. Rev. B, 84, 075153 (2011); Milton and Willis, Proc. R. Soc. A, 463, 855–880, (2007)]. Further, the determination of material coupling terms in EM resolves apparent violations of causality and passivity which is present in earlier models [A. Alù, Phys. Rev. B, 83, 081102(R) (2011)]. These details have not received much attention in fluid acoustics, but they are important for a proper description of acoustic metamaterial behavior. We derive expressions for effective properties of a heterogeneous fluid medium from expressions for the conservation of mass, the conservation of momentum, and the equation of state and find a physically meaningful effective material response from first-principles. The results show inherent coupling between the ensemble averaged volume strain-momentum and pressure-velocity field. The approach is valid for an infinite periodic lattice of heterogeneities and employs zero-, first-, and second-order tensorial Green’s functions to relate point-discontinuities in compressibility and density to far field pressure and particle velocity fields. [This work was supported by the Office of Naval Research.]

Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization

Metamaterial homogenization may be based on the dominance of a single Floquet–Bloch spatial harmonic in an infinite periodic structure – with the dominance quantified in terms of the relative magnitude of the associated spatial harmonic Poynting vector. For the corresponding finite structure the field is not quasi-periodic and cannot be expanded in Floquet–Bloch spatial harmonics; however, a set of pseudo spatial harmonics can be defined and the dominance of a single such harmonic likewise be used to determine whether the structure can be homogenized. For three different lossless Bragg stack configurations (one of which is magneto-dielectric), we show, using spectral representation, that the field in the finite structure can be accurately expanded in terms of these pseudo spatial harmonics and that the distribution of these agrees very well with the distribution of Floquet–Bloch spatial harmonics of the corresponding infinite Bragg stack. This is even the case for finite Bragg stacks having only two unit cells; thus, the number of unit cells does not influence the homogenizability of this type of configuration.

A Review of the Scattering-Parameter Extraction Method with Clarification of Ambiguity Issues in Relation to Metamaterial Homogenization

The scattering-parameter extraction method of metamaterial homogenization is reviewed to show that the only ambiguity is that related to the choice of the branch of the complex logarithmic function (or the complex inverse cosine function). It is shown that the method has no ambiguity for the sign of the wavenumber and intrinsic impedance. While the method indeed yields two signs for the intrinsic impedance and thus the wavenumber, the signs are dependent. Moreover, both sign combinations lead to the same permittivity and permeability, and are thus permissible. This observation is in distinct contrast to a number of statements in the literature where the correct sign of the intrinsic impedance and wavenumber resulting from the scattering-parameter method is chosen by imposing additional physical requirements, such as passivity. The scattering-parameter method is reviewed through an investigation of a uniform plane wave normally incident on a planar slab in free space. The severity of the branch ambiguity is illustrated through simulations of a known metamaterial realization. Several approaches for proper branch selection are reviewed, and the suitability to metamaterial samples is discussed.

Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures

We discuss the well-known limitations in the retrieval of homogenized constitutive parameters based on reflection and transmission from metamaterial slabs arising near the Fabry-Perot resonances of the samples under analysis. The associated artifacts can significantly affect the retrieval results, causing Lorentzian-like nonphysical features. We discuss the nature of these artifacts and propose a general method to correct them based on a numerical regression procedure. These concepts are applied to a homogeneous slab and a periodic metamaterial array, showing that the proposed correction can provide an improved retrieval technique for metamaterial homogenization purposes.

Restoring the physical meaning of metamaterial constitutive parameters

Metamaterial homogenization is often based on implicit assumptions inspired to natural material models. Retrieved effective permittivity and permeability, however, are often non-physical, especially near the array resonances, of most interest for metamaterial applications. We explain here the nature of typical homogenization artifacts, relating them to an inherent form of magneto-electric coupling associated with the finite phase velocity along metamaterial arrays. Our findings allow restoring the proper definition and physical meaning of local constitutive parameters for metamaterials.

Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets

We describe a simple metamaterial homogenization procedure that calculates a limited set of constitutive parameters that are functions of both the frequency $\ensuremath{\omega}$ and the wave number $\mathbf{k}$. This procedure is based on describing the inclusions of the metamaterial with current sheets excited by the local electromagnetic fields. It is shown that these constitutive parameters correctly predict several measurable properties of a metamaterial crystal including the dispersion relation, reflection from an interface, and emission from a simple directional metamaterial antenna.

Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters

In this work, we develop a systematic and self-consistent approach to homogenize arbitrary nonmagnetic periodic metamaterials. The proposed method does not rely on the solution of an eigenvalue problem and can fully characterize the effects of frequency dispersion, magnetoelectric coupling, and spatial dispersion, even in frequency band gaps or when the materials are lossy. We formulate a homogenization problem to characterize a generic microstructured artificial material, and demonstrate that it is equivalent to an integral-differential system. We prove that this complex system can be reduced to a standard integral equation and solved using standard methods. To illustrate the application of the proposed method, we homogenize several important metamaterial configurations involving split-ring resonators and metallic wires.

CRANKLED S-RING RESONATOR WITH SMALL ELECTRICAL SIZE

Metamaterials with small electrical size are more feasible to be described by the macroscopic parameters and treated as an effective homogenous media. By using geometric optimization, we experimentally realize a metamaterial composed of crankled Sring resonator, whose electrical size is 2.5 times smaller than the conventional S-ring resonator. The overall effective capacitance of the unit structure is greatly increased when viewed from an equivalent circuit model point of view, so the resonant frequency is decreased, and the metamaterial works at a much longer wavelength regime. In addition, we summarize two kinds of other methods that could be used to reduce the electrical size of the structures. Experimental and simulation results are presented, showing the effectiveness of these methods in the metamaterial homogenization.