Here, we present an overview of a first principles homogenization theory of periodic metamaterials. It is shown that in a rather general context it is possible to formally introduce effective parameters that describe the time evolution of macroscopic (slowly-varying in space) initial states of the electromagnetic field using an effective medium formalism. The theory is applied to different types of "wire metamaterials" characterized by a strong spatial dispersion in the long wavelength limit. It is highlighted that the spatial dispersion may tailor in unique ways the wave phenomena in wire metamaterials leading to exotic tunneling effects and broadband lossless anomalous dispersion.
Homogenization of metamaterials is a crucial issue as it allows to describe their optical response in terms of effective wave parameters as e.g. propagation constants. In this paper we consider the possible homogenization of chiral metamaterials. We show that for meta-atoms of a certain size a critical density exists above which increasing coupling between neighboring meta-atoms prevails a reasonable homogenization. On the contrary, a dilution in excess will induce features reminiscent to photonic crystals likewise prevailing a homogenization. Based on Bloch mode dispersion we introduce an ana…
We investigate the ab-initio homogenization of separable metamaterials with factorized dielectric permittivity profile which can be achieved through suitable grey-scale permittivity design techniques. Separability allows such metamaterials to be physically regarded as the superposition of three fictitious 1D generating media. We prove that, in the long-wavelength limit, separable metamaterials admit simple and analytical description of their electromagnetic bi-anisotropic response which can be reconstructed from the properties of the 1D generating media. Our approach provides a strategy which …
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.
A unified homogenization procedure for split ring metamaterials taking into account time and spatial dispersion is introduced. The procedure is based on two coupled systems of equations. The first one comes from an approximation of the metamaterial as a cubic arrangement of coupled LC circuits, giving the relation between currents and local magnetic field. The second equation comes from macroscopic Maxwell equations, and gives the relation between the macroscopic magnetic field and the average magnetization of the metamaterial. It is shown that electromagnetic and magnetoinductive waves propag…
We apply the method of asymptotic homogenization to metamaterials with microscopically bianisotropic inclusions to calculate a full set of constitutive parameters in the long wavelength limit. Two different implementations of electromagnetic asymptotic homogenization are presented. We test the homogenization procedure on two different metamaterial examples. Finally, the analytical solution for long wavelength homogenization of a one dimensional metamaterial with microscopically bi-isotropic inclusions is derived.
We theoretically investigate the homogenization of the dielectric response to transverse electric waves of a transverse grating characterized by the Kapitza condition, i.e. the permittivity is rapidly modulated with a modulation depth scaling as the large wavelength to modulation period ratio. We show that the resulting effective dielectric permittivity, in addition to the standard average of the underlying dielectric profile, has a further contribution arising from the fast and deep dielectric modulation. Such contribution turns out to be comparable with the other one and hence can provide an…
We develop, from first principles, a general and compact formalism for predicting the electromagnetic response of a metamaterial with non-magnetic inclusions in the long wavelength limit, including spatial dispersion up to the second order. Specifically, by resorting to a suitable multiscale technique, we show that medium effective permittivity tensor and the first and second order tensors describing spatial dispersion can be evaluated by averaging suitable spatially rapidly-varying fields each satysifing electrostatic-like equations within the metamaterial unit cell. For metamaterials with ne…
Based on the recently introduced homogenization theory developed in [Phys. Rev. B 84, 075153 (2011)], we propose a generalized retrieval method that allows extracting physically meaningful bulk effective parameters from conventional scattering measurements of periodic metamaterial samples composed of subwavelength inclusions. We show that, compared to conventional approaches, our method is able to capture the anomalous physics in the wave interaction with resonant metamaterials and return physically meaningful homogenized parameters that retain local properties in the long-wavelength limit. As…
A rigorous homogenization theory of metamaterials -- artificial periodic structures judiciously designed to control the propagation of electromagnetic waves -- is developed. All coarse-grained fields are unambiguously defined and effective parameters are then derived without any heuristic assumptions. The theory is an amalgamation of two concepts: Smith & Pendry's physical insight into field averaging and the mathematical framework of Whitney-Nedelec-Bossavit-Kotiuga interpolation. All coarse-grained fields are defined via Whitney forms and satisfy Maxwell's equations exactly. The new approach…
We considered the experimental realization of a Tamm medium that is optically equivalent to the collision of two linearly polarized gravitational plane waves as a piecewise homogeneous metamaterial. Our formulation was based on the homogenization of remarkably simple arrangements of oriented ellipsoidal nanoparticles of isotropic dielectric-magnetic mediums. The inverse Bruggeman homogenization was used to estimate the constitutive parameters, volume fractions, and shape parameters for the component mediums. The presented formulation is appropriate for the regions of spacetime where the two gr…
It is well known that defining a local refractive index for a metamaterial requires that the wavelength be large with respect to the scale of its microscopic structure (generally the period). However, the converse does not hold. There are simple structures, such as the infinite, perfectly conducting wire medium, which remain non-local for arbitrarily large wavelength-to-period ratios. In this work we extend these results to the more realistic and relevant case of finite wire media with finite conductivity. In the quasi-static regime the metamaterial is described by a non-local permittivity whi…
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-di…
We present a consistent theoretical approach for calculating effective nonlinear susceptibilities of metamaterials taking into account both frequency and spatial dispersion. Employing the discrete dipole model, we demonstrate that effects of spatial dispersion become especially pronounced in the vicinity of effective permittivity resonance where nonlinear susceptibilities reach their maxima. In that case spatial dispersion may enable simultaneous generation of two harmonic signals with the same frequency and polarization but different wave vectors. We also prove that the derived expressions fo…
Reflection and refraction of electromagnetic waves by artificial periodic composites (metamaterials) can be accurately modeled by an effective medium theory only if the boundary of the medium is explicitly taken into account and the two effective parameters of the medium -- the index of refraction and the impedance -- are correctly determined. Theories that consider infinite periodic composites do not satisfy the above condition. As a result, they cannot model reflection and transmission by finite samples with the desired accuracy and are not useful for design of metamaterial-based devices. As…
We study the eigenmodes of a slab of a wire metamaterial and demonstrate that such a waveguiding structure supports deep-subwavelength propagating modes exhibiting properties of a single-mode waveguide at any fixed frequency below the plasma frequency of metal wires. We compare our analytical results with the dispersion relations extracted from the experimental measurements.
We study numerically and experimentally magnetic metamaterials based on cut-wire pairs instead of split-ring resonators. The cut-wire pair planar structure is extended in order to create a truly two-dimensional metamaterial suitable for scaling to optical frequencies. We fabricate the cut-wire metamaterial operating at microwave frequencies with lattice spacing around 10% of the free-space wavelength, and find good agreement with direct numerical simulations. Unlike the structures based on split-ring resonators, the nearest-neighbor coupling in cut-wire pairs can result in a magnetic stop-band…
Strong diamagnetic response of wire medium with finite wire radius is reported. Contrary to the previous works where it was assumed that the wire medium exhibits only the electric response, we show that the non-zero magnetic susceptibility has to be taken into account for proper effective medium description of the wire medium. Analytical and numerical results are supported by the experimental measurements.
Anomalous optical coupling properties between two silicon wires in a silicon slot waveguide embedded in epsilon-near-zero (ENZ) metamaterials are proposed and demonstrated. The dependences of optical field enhancement in the slot region and transverse optical force on the slot size and the permittivity of surrounding material are studied in details. It is demonstrated that the optical field in the slot region is significantly enhanced due to the giant index contrast at the slot interface between silicon wires and ENZ metamaterials, but the optical mode coupling between silicon wires is greatly…
The scattering parameter extraction method of metamaterial homogenization is reviewed to show that the only ambiguity is the one related to the choice of the branch of the complex logarithmic function (or the complex inverse cosine function), whereas it has no ambiguity for the sign of the wave number and intrinsic impedance. While the method indeed yields two signs of the intrinsic impedance, and thus the wave number, the signs are dependent, and moreover, both sign combinations lead to the same permittivity and permeability, and are thus permissible. This observation is in distinct contrast …
We study the optical properties of metamaterials made from cut-wire pairs or plate pairs. We obtain a more pronounced optical response for arrays of plate pairs -- a geometry which also eliminates the undesired polarization anisotropy of the cut-wire pairs. The measured optical spectra agree with simulations, revealing negative magnetic permeability in the range of telecommunications wavelengths. Thus, nanoscopic plate pairs might serve as an alternative to the established split-ring resonator design.
We theoretically and numerically investigate metamaterials composed of coupled resonators with indirect coupling. First, we theoretically analyze a mechanical model of coupled resonators with indirect coupling. The theoretical analysis shows that an electromagnetically induced transparency (EIT)-like phenomenon with a transparency bandwidth narrower than the resonance linewidths of the constitutive resonators can occur in the metamaterial with strong indirect coupling. We then numerically examine the characteristics of the metamaterial composed of coupled cut-wire pairs using a finite-differen…
We propose and investigate a new type of metamaterial structure composing of several interpenetrating wire meshes. Calculated band structures show that they exhibit index ellipsoids locating at nonzero k-point in long wavelength limit. We can comprehend these quasistatic modes by Poison's equation and find these modes do not rely on the detailed geometry of the wires but the connectivities of the wires. We can engineer the locations of index ellipsoid by designing the connectivities of the wire meshes.
Classical theory of crystals states that a medium to be considered homogeneous must satisfy the following requirements: a) the dimension of the elementary cell must be much smaller than the incident wavelength; b) the sample must contain a large number of elementary cells, i.e. it must be macroscopic with respect to wavelength. Under these conditions, macroscopic quantities can be introduced in order to describe the optical response of the medium. We analytically demonstrate that for a symmetric elementary cell those requirements can be relaxed, and it is possible to assign a permittivity and …
Manipulation of radiation is required for enabling a span of electromagnetic applications. Since properties of antennas and scatterers are very sensitive to a surrounding environment, macroscopic artificially created materials are good candidates for shaping their characteristics. In particular, metamaterials enable controlling both dispersion and density of electromagnetic states, available for scattering from an object. As the result, properly designed electromagnetic environment could govern waves' phenomena. Here electromagnetic properties of scattering dipoles, situated inside a wire medi…
We employ a homogenization technique based on the Lorentz electronic theory to show that planar chiral structures (PCSs) can be described by an effective dielectric tensor similar to that of biaxial elliptically dichroic crystals. Such a crystal is shown to behave like a PCS insofar as it exhibits its characteristic optical properties, namely, co-rotating elliptical polarization eigenstates and asymmetric, direction-dependent transmission for left/right-handed incident wave polarization.
Infrared (IR) light is considered important for short-range wireless communication, thermal sensing, spectroscopy, material processing, medical surgery, astronomy etc. However, IR light is in general much harder to transport than optical light or microwave radiation. Existing hollow-core IR waveguides usually use a layer of metallic coating on the inner wall of the waveguide. Such a metallic layer, though reflective, still absorbs guided light significantly due to its finite Ohmic loss, especially for transverse-magnetic (TM) light. In this paper, we show that metal-wire based metamaterials ma…
We report single and multiband linear polarizers for terahertz (THz) frequencies using cut-wire metamaterials (MM). The MMs are designed by finite element method, fabricated by electron beam lithography, and characterized by THz time-domain spectroscopy. The MM unit cells consist of single or multiple length cut-wire pads of gold on semi-insulating Gallium Arsenide for single or multiple band polarizers. The dependence of the resonance frequency of the single band polarizer on the length of the cut-wires is explained based a transmission line model.
Asymmetric Negative Refraction in Nonlocal Double-Wire Metamaterials
Instituto de Telecomunicações and Department of Electrical and Computer Engineering, University of Coimbra（科英布拉大学电信研究所与电气与计算机工程系）；Photonics Initiative, Advanced Science Research Center, City University of New York（纽约市立大学高级科学研究中心光子学倡议项目）；Physics Program, The Graduate Center, City University of New York（纽约市立大学研究生院物理学项目）；University of Lisbon–Instituto Superior Técnico and Instituto de Telecomunicações（里斯本大学高等技术学院与电信研究所）
The optical properties of periodic electromagnetic metamaterials are considered as functions of their relative unit cell size $d / λ$. The reflection $R$ and transmission $T$ coefficients are numerically calculated for some realistic metamaterials in a wide range of their relative unit cell size values that comprises different operating regimes of the metamaterials. Peculiarities in $R$ and $T$ behavior are discussed and the causes of those peculiarities are outlined. The obtained results support the opinion on inapplicability of the very homogenization concept to metamaterials whose unit cell…
We design three-dimensional (3D) metallic nanowire media with different structures and numerically demonstrate that they can be homogeneous effective indefinite anisotropic media by showing that their dispersion relations are hyperbolic. For a finite slab, a nice fitting procedure is exploited to obtain the dispersion relations from which we retrieve the effective permittivities. The pseudo focusing for the real 3D wire medium agrees very well with the homogeneous medium having the effective permittivity tensor of the wire medium. Studies also show that in the long-wavelength limit, the hyperb…
In this chapter we describe a selection of mathematical techniques and results that suggest interesting links between the theory of gratings and the theory of homogenization, including a brief introduction to the latter. By no means do we purport to imply that homogenization theory is an exclusive method for studying gratings, neither do we hope to be exhaustive in our choice of topics within the subject of homogenization. Our preferences here are motivated most of all by our own latest research, and by our outlook to the future interactions between these two subjects. We have also attempted, …
The homogenization of a metamaterial made of a collection of scatterers periodically disposed is studied from three different points of view. Specifically tools for multiple scattering theory, functional analysis, differential geometry and optimization are used. Detailed numerical results are given and the connections between the different approaches are enlightened.
Recently, a new energy density formula for electromagnetic waves in the metamaterial consisting of arrays of conducting wires and split-ring-resonators (SRR) has been derived [Phys. Rev. E 80, 046601 (2009), arXiv:0909.1043]. According to that formula, the positiveness is obvious for the electric part of the energy density but not clear for the magnetic part. In this paper, I show that the magnetic energy density is also positively definite.
We introduce a general implementation of the recently proposed homogenization theory [Tsukerman, J. Opt. Soc. Am. B 28, 577 (2011)] allowing one to retrieve all 36 linear constitutive parameters of any 3D metamaterial with parallelepipedal unit cells. The effective parameters are defined directly as linear relations between pairs of coarse-grained fields, in contrast with methods where these parameters are obtained from reflection/transmission data or other indirect considerations. The method is applied to plasmonic metamaterials with spherical gold particles and split-ring resonators (SRR), r…
In this paper we develop homogenization theory for spatiotemporally modulated wire medium. We first solve for the modal waves that are supported by this composite medium, we show peculiar properties such as extraordinary waves that propagate at frequencies below the cut-off frequency of the corresponding stationary medium. We explain how these unique solutions give rise to an extreme Fresnel drag that exists already with weak and slow spatiotemporal modulation. Next, we turn to derive the effective material permittivity that corresponds to each of the first few supported modes, and write the a…
We study theoretically the enhancement of spontaneous emission in wire metamaterials. We analyze the dependence of the Purcell factor dependence on wire dielectric constant for both electric and magnetic dipole sources, and find an optimal value of the dielectric constant for maximizing the Purcell factor for the electric dipole. We obtain analytical expressions for the Purcell factor and also provide estimates for the Purcell factor in realistic structures operating in both microwave and optical spectral range.
We present an analytical model which permits the calculation of effective material parameters for planar metamaterials consisting of arbitrary unit cells (metaatoms) formed by a set of straight wire sections of potentially different shape. The model takes advantage of resonant electric dipole oscillations in the wires and their mutual coupling. The pertinent form of the metaatom determines the actual coupling features. This procedure represents a kind of building block model for quite different metaatoms. Based on the parameters describing the individual dipole oscillations and their mutual co…
Engineering plasmonic metamaterials with anisotropic optical dispersion enables us to tailor the properties of metamaterial-based waveguides. We investigate plasmonic waveguides with dielectric cores and multilayer metal-dielectric claddings with hyperbolic dispersion. Without using any homogenization, we calculate the resonant eigenmodes of the finite-width cladding layers, and find agreement with the resonant features in the dispersion of the cladded waveguides. We show that at the resonant widths, the propagating modes of the waveguides are coupled to the cladding eigenmodes and hence, are …
We develop a theory of nonlocal homogenization for metamaterial consisting of parallel metallic wires with dielectric coating. It is demonstrated that manipulation of dielectric contrast between wire dielectric shell and host material results in switching of metamaterial dispersion regime from elliptic to the hyperbolic one, i.e. the topological transition takes place. We confirm our theoretical predictions by full-wave numerical simulations.
Photonic crystals and metamaterials represent two seemingly different classes of artificial electromagnetic media but often they are composed of similar structural elements arranged in periodic lattices. The important question is how to distinguish these two types of periodic photonic structures when their parameters, such as dielectric permittivity and lattice spacing, vary continuously. Here, we discuss transitions between photonic crystals and all-dielectric metamaterials and introduce the concept of a phase diagram and an order parameter for such structured materials, based on the physics …
We present a planar design of a metamaterial exhibiting electromagnetically induced transparency that is amenable to experimental verification in the microwave frequency band. The design is based on the coupling of a split-ring resonator with a cut-wire in the same plane. We investigate the sensitivity of the parameters of the transmission window on the coupling strength and on the circuit elements of the individual resonators, and we interpret the results in terms of two linearly coupled Lorentzian resonators. Our metamaterial designs combine low losses with the extremely small group velocity…
