In conventional approaches to the homogenization of random particulate composites, both the distribution and size of the component phase particles are often inadequately taken into account. Commonly, the spatial distributions are characterized by volume fraction alone, while the electromagnetic response of each component particle is represented as a vanishingly small depolarization volume. The strong-permittivity-fluctuation theory (SPFT) provides an alternative approach to homogenization wherein a comprehensive description of distributional statistics of the component phases is accommodated. The bilocally-approximated SPFT is presented here for the anisotropic homogenized composite which arises from component phases comprising ellipsoidal particles. The distribution of the component phases is characterized by a two-point correlation function and its associated correlation length. Each component phase particle is represented as an ellipsoidal depolarization region of nonzero volume. The effects of depolarization volume and correlation length are investigated through considering representative numerical examples. It is demonstrated that both the spatial extent of the component phase particles and their spatial distributions are important factors in estimating coherent scattering losses of the macroscopic field.
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…
Using the multiple-scale homogenization method, we derive generalized sheet transition conditions (GSTCs) for electromagnetic fields at the surface of a metafilm. The scatterers that compose the metafilm are of arbitrary shape and are embedded between two different magneto-dielectric media. The parameters in these boundary conditions are interpreted as effective electric and magnetic surface susceptibilities, which themselves are related to the geometry of the scatterers that constitute the metafilm.
We report on a strategy for achieving negative phase velocity (NPV) in a homogenized composite medium (HCM) conceptualized using the Bruggeman formalism. The constituent material phases of the HCM do not support NPV propagation. The HCM and its constituent phases are isotropic dielectric-magnetic mediums; the real parts of their permittivities/permeabilities are negative-valued whereas the real parts of their permeabilities/permittivities are positive-valued.
The Bergman-Milton bounds provide limits on the effective permittivity of a composite material comprising two isotropic dielectric materials. These provide tight bounds for composites arising from many conventional materials. We reconsider the Bergman-Milton bounds in light of the recent emergence of metamaterials, in which unconventional parameter ranges for relative permittivities are encountered. Specifically, it is demonstrated that: (a) for nondissipative materials the bounds may be unlimited if the constituent materials have relative permittivities of opposite signs; (b) for weakly dissi…
The Maxwell Garnett and Bruggeman formalisms were applied to estimate the effective permittivity dyadic of active dielectric composite materials. The active nature of the homogenized composite materials (HCMs) arises from one of the component materials which takes the form of InAs/GaAs quantum dots. Provided that the real parts of the permittivities of the component materials have the same sign, the Maxwell Garnett and Bruggeman formalisms give physically plausible estimates of the HCM permittivity dyadic that are in close agreement. However, if the real parts of the permittivities of the comp…
An implementation of the strong-permittivity-fluctuation theory (SPFT) is presented in order to estimate the constitutive parameters of a homogenized composite material (HCM) which is both cubically nonlinear and anisotropic. Unlike conventional approaches to homogenization, the particles which comprise the component material phases are herein assumed to be small but not vanishingly small. The influence of particle size on the estimates of the HCM constitutive parameters is illustrated by means of a representative numerical example. It is observed that, by taking the nonzero particle size into…
In conventional approaches to the homogenization of random particulate composites, the component phase particles are often treated mathematically as vanishingly small, point-like entities. The electromagnetic responses of these component phase particles are provided by depolarization dyadics which derive from the singularity of the corresponding dyadic Green functions. Through neglecting the spatial extent of the depolarization region, important information may be lost, particularly relating to coherent scattering losses. We present an extension to the strong-property-fluctuation theory in whi…
A random mixture of two isotropic dielectric materials, one composed of oriented spheroidal particles of relative permittivity $ε_a$ and the other composed of oriented spheroidal particles of relative permittivity $ε_b$, was considered in the long wavelength regime. The permittivity dyadic of the resulting homogenized composite material (HCM) was estimated using the Bruggeman homogenization formalism. The HCM was an orthorhombic biaxial material if the symmetry axes of the two populations of spheroids were mutually perpendicular and a uniaxial material if these two axes were mutually aligned. …
The transmission and reflection spectra from a right-handed chiral sculptured zirconia thin film are calculated using the piecewise homogeneity approximation method and the Bruggeman homogenization formalism by considering that the propagation of both dispersive and non-dispersive dielectric function occurs for axial and non-axial states. The comparison of spectral results shows that the dispersion of the dielectric function has a considerable effect on the results. In axial excitation of cross-polarized reflectances and co-polarized transmittances the dispersion effect becomes more pronounced…
The homogenization of a composite material comprising three isotropic dielectric materials was investigated. The component materials were randomly distributed as spherical particles, with the particles of two of the component materials being coupled to form dimers. The Bruggeman and Maxwell Garnett formalisms were developed to estimate the permittivity dyadic of the homogenized composite material (HCM), under the quasi-electrostatic approximation. Both randomly oriented and identically oriented dimers were accommodated; in the former case the HCM is isotropic, whereas in the latter case the HC…
An effective cylindrical cloak may be conceptualized as an assembly of adjacent local neighbourhoods, each of which is made from a homogenized composite material (HCM). The HCM is required to be a certain uniaxial dielectric-magnetic material, characterized by positive-definite constitutive dyadics. It can arise from the homogenization of remarkably simple component materials, such as two isotropic dielectric-magnetic materials, randomly distributed as oriented spheroidal particles. By carefully controlling the spheroidal shape of the component particles, a high degree of HCM anisotropy may be…
We show that artificial magnetism of periodic dielectric or metal/dielectric structures has limitations and is subject to at least two "uncertainty principles". First, the stronger the magnetic response (the deviation of the effective permeability tensor from identity), the less accurate ("certain") the predictions of any homogeneous model. Second, if the magnetic response is strong, then homogenization cannot accurately reproduce the transmission and reflection parameters and, simultaneously, power dissipation in the material. These principles are general and not confined to any particular me…
We consider an anisotropic homogenized composite medium (HCM) arising from isotropic particulate component phases based on ellipsoidal geometries. For cubically nonlinear component phases, the corresponding zeroth-order strong-permittivity-fluctuation theory (SPFT) (which is equivalent to the Bruggeman homogenization formalism) and second-order SPFT are established and used to estimate the constitutive properties of the HCM. The relationship between the component phase particulate geometry and the HCM constitutive properties is explored. Significant differences are highlighted between the esti…
A dielectric columnar thin film (CTF), characterized macroscopically by a relative permittivity dyadic, was investigated theoretically with the assumption that, on the nanoscale, it is an assembly of parallel, identical, elongated ellipsoidal inclusions made of an isotropic dielectric material that has a different refractive index from the bulk material that was evaporated to fabricate the CTF. The inverse Bruggeman homogenization formalism was developed in order to estimate the refractive index of the deposited material, one of the two shape factors of the ellipsoidal inclusions, and the volu…
The propagation of electromagnetic waves through disordered layered system is considered in the paradigm of Maxwell's equations homogenization. In spite of the impossibility to describe the system in terms of effective dielectric permittivity and/or magnetic permeability the unified way to describe the propagation and Anderson localization of electromagnetic waves is proposed in terms of the introduced effective wave vector (effective refractive index). It is demonstrated that both real and imaginary parts of the effective wave vector (contrary to effective dielectric permittivity and/or magne…
We calculate the static dielectric tensor of a periodic system of aligned anisotropic dielectric cylinders. Exact analytical formulas for the effective dielectric constants for the E- and H- eigenmodes are obtained for arbitrary 2D Bravais lattice and arbitrary cross-section of anisotropic cylinders. It is shown that depending on the symmetry of the unit cell photonic crystal of anisotropic cylinders behaves in the low-frequency limit like uniaxial or biaxial natural crystal. The developed theory of homogenization of anisotropic cylinders is applied for calculations of the dielectric propertie…
In a long wavelength regime, the effective properties of particulate composites, including nanocomposites, may be estimated using one of various homogenization formalisms, such as the Bruggeman and Maxwell Garnett formalisms, and the approach of the strong--property--fluctuation theory (SPFT). In the conventional implementations of these formalisms, the constituent particles are treated as point--like scattering centres. However, extended formalisms have been established--which involve integral formulations--that take account of the spatial extent of the constituent particles. In particular, t…
Under certain circumstances, the group velocity in a homogenized composite medium (HCM) can exceed the group velocity in its component material phases. We explore this phenomenon for a uniaxial dielectric HCM comprising isotropic component material phases distributed as oriented spheroidal particles. The theoretical approach is based upon the Bruggeman homogenization formalism. Enhancement in group velocity in the HCM with respect to the component material phases is shown to be sensitively dependent upon the shape of the component spheroids and their alignment relative to the direction of prop…
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…
The strong-property-fluctuation theory (SPFT) provides a sophisticated means of estimating the effective constitutive parameters of a homogenized composite material (HCM), which takes account of the statistical distribution of the component particles. We present an extended version of the third-order SPFT in which the component particles are represented as depolarization regions of nonzero volume. Numerical results are provided for a bianisotropic homogenization scenario wherein the HCM is a Faraday chiral medium. Thereby, convergence of the extended SPFT at the second-order level of approxima…
We explain the Lorentz resonances in plasmonic crystals that consist of 2D nano dielectric inclusions as the interaction between resonant material properties and geometric resonances of electrostatic nature. One example of such plasmonic crystals are graphene nanosheets that are periodically arranged within a non-magnetic bulk dielectric. We identify local geometric resonances on the length scale of the small scale period. From a materials perspective, the graphene surface exhibits a dispersive surface conductance captured by the Drude model. Together these phenomena conspire to generate Loren…
We investigate a high-order homogenization (HOH) algorithm for periodic multilayered stacks. The mathematical tool of choice is a transfer matrix method. Expressions for effective permeability, permittivity and magnetoelectric coupling are explored by frequency power expansions. On the physical side, this high-order homogenization uncovers a magnetoelectric coupling effect (odd order approximation) and artificial magnetism (even order approximation) in moderate contrast photonic crystals. Comparing the effective parameters' expressions of a stack with three layers against that of a stack with …
Two active dielectric materials may be blended together to realize a homogenized composite material (HCM) which exhibits more gain than either component material. Likewise, two dissipative dielectric materials may be blended together to realize an HCM which exhibits more loss than either component material. Sufficient conditions for such gain/loss enhancement were established using the Bruggeman homogenization formalism. Gain/loss enhancement arises when (i) the imaginary parts of the relative permittivities of both component materials are similar in magnitude and (ii) the real parts of the re…
Homogenized composite materials (HCMs) can support a singular form of optical propagation, known as Voigt wave propagation, while their component materials do not. This phenomenon was investigated for biaxial HCMs arising from nondissipative isotropic dielectric component materials. The biaxiality of these HCMs stems from the oriented spheroidal shapes of the particles which make up the component materials. An extended version of the Bruggeman homogenization formalism was used to investigate the influence of component particle orientation, shape and size, as well as volume fraction of the comp…
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.
Realization of the anomalous refraction effects predicted by Huygens' metasurfaces (HMS) have required tedious and time-consuming trial-and-error numerical full-wave computations. It is shown herein that these requirements can be alleviated for transverse magnetic (TM) propagation by a periodic dielectric-based HMS consisting of an electrically thick array of cascaded Fabry-Pérot etalons. This "Fabry-Pérot HMS" (FP-HMS) is easily designed to mimic the local scattering coefficients of a standard zero-thickness HMS (ZT-HMS) which, according to homogenization theory, should result in the desired …
Anisotropic homogeneous metamaterials that are neither wholly dissipative nor wholly active at a specific frequency are permitted by classical electromagnetic theory. Well-established homogenization formalisms indicate that such a metamaterial may be realized quite simply as a random mixture of electrically small (possibly nanoscale) spheroidal particles of at least two different isotropic dielectric materials, one of which must be dissipative but the other active. The dielectric properties of this metamaterial are influenced by the volume fraction, spatial distribution, particle shape and siz…
Two different formalisms for the homogenization of composite materials containing oriented ellipsoidal particles of isotropic dielectric materials are being named after Bruggeman. Numerical studies reveal clear differences between the two formalisms which may be exacerbated: (i) if the component particles become more aspherical, (ii) at mid-range values of the volume fractions, and (iii) if the homogenized component material is dissipative. The correct Bruggeman formalism uses the correct polarizability density dyadics of the component particles, but the other formalism does not.
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 …
Two different formalisms for the homogenization of composite materials containing ellipsoidal inclusions based on Bruggeman's original formula for spherical inclusions can be found in the literature. Both approximations determine the effective macroscopic permittivity of such an idealized composite assuming randomly distributed dielectric particles of equal shape and differ only in the definition of the depolarization factors. The two approaches are applied to analyze ellipsometric Mueller matrix spectra acquired in the visible and near-infrared spectral region from metal and semiconductor sla…
