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 HCM is uniaxial. Representative numerical results for composite materials containing dielectric--dielectric dimers demonstrate close agreement between the estimates delivered by the Bruggeman and Maxwell Garnett formalisms. For composite materials containing metal--dielectric dimers with moderate degrees of dissipation, the estimates of the two formalisms are in broad agreement, provided that the dimer volume fractions are relatively low. In general, the effects of intradimer coupling on the estimates of the HCM permittivity are relatively modest but not insignificant, these effects being exacerbated by anisotropy when all dimers are identically oriented.
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…
The Bruggeman formalism provides an estimate $\eps^{Br}_{hcm}$ of the relative permittivity of a homogenized composite material (HCM), arising from two component materials with relative permittivities $\eps_a$ and $\eps_b$. It can be inverted to provide an estimate of $\eps_a$, from a knowledge of $\eps^{Br}_{hcm}$ and $\eps_b$. Numerical studies show that the inverse Bruggeman estimate $\eps_a$ can be physically implausible when (i) $Re\{\eps^{Br}_{hcm}\}/Re\{\eps_b\} > 0 $ and the degree of HCM dissipation is moderate or greater; or (ii) $Re\{\eps^{Br}_{hcm}\}/Re\{\eps_b\} < 0 $ regardless o…
The Bruggeman formalism provides an estimate of the effective permittivity of a particulate composite medium comprising two component mediums. The Bruggeman estimate is required to lie within the Wiener bounds and the Hashin-Shtrikman bounds. Considering the homogenization of weakly dissipative component mediums characterized by relative permittivities with real parts of opposite signs, we show that the Bruggeman estimate may not be not physically reasonable when the component mediums are weakly dissipative; furthermore, both the Wiener bounds and the Hashin-Shtrikman bounds exhibit strong res…
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…
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.
Two interpretations of the Bruggeman approach for the homogenization of isotropic chiral mixtures are shown to lead to different results. Whereas the standard interpretation is shown to yield the average-polarizability-density approach, a recent interpretation turns out to deliver a null-excess-polarization approach. The difference between the two interpretations arises from differing treatments of the local field
The homogenization of vaterite impregnated with gold nanoparticles was accomplished using the extended Maxwell Garnett formalism. The extended formalism takes into account the intrinsic anisotropy of vaterite as well as the size, shape, and orientation of the nanoparticles. Size-dependent permittivity was used for the gold nanoparticles. Numerical studies revealed that the homogenized composite material's permittivity parameters are acutely sensitive to the size, shape, orientation, and volume fraction of the gold nanoparticles.
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…
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…
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.
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…
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. …
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…
Within the framework of quantum electrodynamics (QED), vacuum is a nonlinear medium which can be linearized for a rapidly time-varying electromagnetic field with a small amplitude subjected to a magnetostatic field. The linearized QED vacuum is a uniaxial dielectric-magnetic medium for which the degree of anisotropy is exceedingly small. By implementing an affine transformation of the spatial coordinates, the degree of anisotropy may become sufficiently large as to be readily perceivable. The inverse Bruggeman formalism can be implemented to specify a homogenized composite material (HCM) which…
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…
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…
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…
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…
The Bruggeman formalism for the homogenization of particulate composite materials is used to predict the effective permittivity dyadic of a two-constituent composite material with one constituent having the ability to display the Pockels effect. Scenarios wherein the constituent particles are randomly oriented, oriented spheres, and oriented spheroids are numerically explored. Thereby, homogenized composite materials (HCMs) are envisaged whose constitutive parameters may be continuously varied through the application of a low-frequency (dc) electric field. The greatest degree of control over t…
An effective-medium theory (EMT) is developed to predict the effective permittivity ε_eff of dense random dispersions of high optical-conductivity metals such as Ag, Au and Cu. Dependence of ε_eff on the volume fraction ϕ, a microstructure parameter κrelated to the static structure factor and particle radius a is studied. In the electrostatic limit, the upper and lower bounds of κcorrespond to Maxwell-Garnett and Bruggeman EMTs respectively. Finite size effects are significant when |β^2(ka/n)^3| becomes O(1) where β, k, and n denote the nanoparticle polarizability, wavenumber and matrix refrac…
Voigt wave propagation was investigated in a homogenized composite material (HCM) arising from a porous electro--optic host material infiltrated by a fluid of refractive index $n_a$. The constitutive parameters of the HCM were estimated using the extended Bruggeman homogenization formalism. Numerical studies revealed that the directions which support Voigt wave propagation in the HCM could be substantially controlled by means of an applied dc electric field. Furthermore, the extent to which this control could be achieved was found to be sensitive to the porosity of the host material, the shape…
Voigt wave propagation (VWP) was considered in a porous biaxial dielectric material which was infiltrated with a material of refractive index $n_a$. The infiltrated material was regarded as a homogenized composite material in the long-wavelength regime and its constitutive parameters were estimated using the extended Bruggeman homogenization formalism. In our numerical studies, the directions which support VWP were found to vary by as much as $300^\circ$ per RIU as the refractive index $n_a$ was varied. The sensitivities achieved were acutely dependent upon the refractive index $n_a$ and the d…
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…
