Bruggeman Formalism Research Articles

Negative phase velocity in isotropic dielectric‐magnetic media via homogenization: Part II

AbstractThe scope for negative‐phase‐velocity (NPV) propagation in a particulate composite material, based upon isotropic dielectric‐magnetic constituent material phases which do not themselves support NPV propagation, is examined using the extended Bruggeman homogenization formalism. Unlike the conventional Bruggeman formalism, the extended formalism takes into account the dimensions of the constituent particles. Representative calculations show that increasing the particulate dimensions diminishes the scope for NPV propagation in the composite material. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 709–712, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21451.

Negative phase velocity in isotropic dielectric-magnetic media via homogenization

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 media; the real parts of their permittivities/permeabilities are negative-valued, whereas the real parts of their permeabilities/permittivities are positive-valued. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 47: 313–315, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21155

Extended effective medium model for refractive indices of thin films with oblique columnar structure

The refractive indices of thin films, containing dielectric and voids in an oblique columnar structure, are modeled by extended effective medium in the quasi-static limit. The dielectric function is shown to be strongly dependent on the angle of incidence and on the columnar orientation for p-polarized light. This model is applied to model ZrO2 thin films with oblique columnar structures and the computed results, with the Maxwell Garnett, the Bragg–Pippard, and the Bruggeman formalisms, have been given.

A limitation of the Bruggeman formalism for homogenization

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 physically reasonable when the component mediums are weakly dissipative; furthermore, both the Wiener bounds and the Hashin–Shtrikman bounds exhibit strong resonances.

Enhanced group velocity in metamaterials

The Bruggeman formalism is implemented to estimate the refractive index of an isotropic, dielectric, homogenized composite medium (HCM). Invoking the well--known Hashin--Shtrikman bounds, we demonstrate that the group velocity in certain HCMs can exceed the group velocities in their component materials. Such HCMs should therefore be considered as metamaterials.

Bragg–Pippard formalism for bianisotropic particulate composites

AbstractHere the Bragg‐Pippard (BP) formalism is extended in order to homogenize linear bianisotropic‐in‐bianisotropic particulate composites, assuming the inclusion particles to be ellipsoidal. The BP formalism yields satisfactory results as the inclusion volume fraction f → 1, unlike the Maxwell Garnett (MG) formalism but like the Bruggeman formalism. However, the BP formalism is asymmetric with respect to the two component mediums in the same way as the MG formalism is, but the Bruggeman formalism can be symmetric. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 33: 40–44, 2002; DOI 10.1002/mop.10225

Nominal model for structure-property relations of chiral dielectric sculptured thin films

The Bruggeman formalism is implemented to estimate the permittivity dyadics of chiral dielectric sculptured thin films (STFs), called thin-film helicoidal bianisotropic mediums (TFHBMs), modelled as helicoidally arranged stacks of ellipsoidal inclusions in vacuum. Spectral maximums of several observable optical properties of axially excited TFHBM slabs are examined as functions of inclusion shape, volume fraction and orientation. Changes in the transverse aspect ratio of the ellipsoidal inclusions significantly affect the predicted optical properties of the TFHBM. A critical value for the transverse aspect ratio of the ellipsoids can be found such that optical activity disappears when all other parameters are fixed. The orientation of the inclusions significantly affects the value of optical property maximums, and can be made to suppress optical activity completely. A distribution of orientations decreases various measures of optical activity.

Homogenisation of similarly oriented, metallic, ellipsoidal inclusions using the bilocally approximated strong-property-fluctuation theory

The homogenisation of a two-phase composite medium comprising metallic, ellipsoidal inclusions embedded in a non-conducting host medium is carried out via the bilocally approximated strong-property-fluctuation theory. Comparison is made against the results obtained via the Bruggeman formalism. The correlated actions of neighbouring inclusions result in a considerable mitigation of the percolation threshold anisotropy due to inclusion shape alone. Furthermore, the choice of covariance function may only have a secondary influence in relation to the effects of the correlation length.

Application of strong permittivity fluctuation theory for isotropic, cubically nonlinear, composite mediums

A simple formula derived from the strong permittivity fluctuation theory (SPFT) is shown to provide a correction to the Bruggeman formalism for isotropic, cubically nonlinear, composite mediums by making use of the correlation length, a second-order statistic of the distribution of the two component material phases. The larger the correlation length, the larger are the differences between the estimates of the linear and the nonlinear dielectric properties of the homogenized composite medium provided by the two formalisms. The SPFT predicts attenuation and downgrades the possibility of nonlinearity enhancement, relative to the Bruggeman formalism.

On the inverse homogenization problem of linear composite materials

The standard problem of homogenization consists of a derivation or estimation of the electromagnetic constitutive properties of a composite material. In that approach, the constitutive properties of the constituent materials (as specified by their respective permittivity, permeability, and magnetoelectric dyadics), their mixing ratios, and certain geometrical properties pertaining to the constituents are known. Here, the inverse problem is pursued: What information about the electromagnetic constitutive parameters of one of the constituent materials in a two-component mixture can be extracted from a knowledge of the constitutive properties of the homogenized composite medium and those of the other component material? This approach is called the inverse homogenization problem, and it is studied within the framework of linear homogenization through the Maxwell Garnett and the Bruggeman formalisms. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 28: 421–423, 2001.

Incremental and differential Maxwell Garnett formalisms for bi-anisotropic composites

We present, compare and contextualize two approaches to the homogenization of bi-anisotropic-in-bi-anisotropic particulate composite medias: (i) the incremental Maxwell Garnett (IMG) formalism, in which the composite medium is built incrementally by adding the inclusions in N discrete steps to the host medium; and (ii) the differential Maxwell Garnett (DMG) formalism, which is obtained from the IMG in the limit N→∞ . Both formalisms are applicable to arbitrary inclusion concentration and are well-suited for computational purposes. Either of the two formalisms may be used as an alternative to the well-known Bruggeman formalism. Numerical results for the homogenization of a uniaxial dielectric composite medium and of a chiroferrite are presented.

Homogenization of similarly oriented, metallic, ellipsoidal inclusions using the Bruggeman formalism

The effective relative permittivity dyadic of a composite material made by randomly embedding parallel, ellipsoidal, isotropic, metallic inclusions in an homogeneous, isotropic, dielectric host material is computed using the Bruggeman formalism with exact depolarization dyadics. Numerical calculations carried out for iron inclusions at 670 nm free-space wavelength indicate that the inclusion volume fraction at the percolation threshold is direction-dependent, being lower in those directions that the ellipsoids have longer extents. When the longest principal semi-axis of an ellipsoidal inclusion exceeds a certain relative size – about three to five times as large as the other two principal semi-axes – the Bruggeman estimate of the effective permittivity along this direction does not indicate a percolation threshold, becoming merely a simple weighting of the permittivities of the two materials in relation to their volume fractions.

Low-permittivity nanocomposite materials using sculptured thin film technology

The feasibility of using sculptured thin film technology to engineer multiphase low-permittivity (also called low-k) materials for microelectronic and electronic packaging applications is established via a numerical study. The chosen films are theoretically modeled as multiphase, uniaxial, dielectric nanocomposite materials and their anisotropic relative permittivity tensors are estimated using the Bruggeman formalism.

Correction to ?Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium?

In this paper, a new design technique has been proposed for microwave nonhomogeneous media filters. The main attraction of this kind of filter is in its use for matching purposes. These components, in fact, can be used to realize a wide frequency hand matching. Via the new expression of the reflection coefficient, very fast design formulas have been carried out, which can be successfully used either for analysis or synthesis purposes. In the first case, as shown in Table 1, the CPU time used for computing the reflection coefficient Ž . strongly drops down using the new closed-form 12 instead Ž . of the other one 4 . In the second case, as shown in Table 2, physical parameters of the filter can be straightforwardly computed.

Homogenization of linear bianisotropic particulate composite media – Numerical studies

We present here the application of the Maxwell Garnett (MG) and the Bruggeman (Br) formalisms to homogenize very general linear bianisotropic-in-bianisotropic particulate composite media with ellipsoidal inclusions. Both formalisms involve the calculation of certain depolarization dyadics, which generally amounts to the numerical evaluation of several two-dimensional integrals. The MG estimate of the constitutive dyadic of the homogenized composite medium can then be obtained by straightforward matrix manipulations. The Br estimate, however, involves nonlinear equations which have to be solved iteratively. We present an iteration scheme that converged rapidly in most cases tested. Numerical results are given for those composite media for which the Bruggeman formalism so far has not been implemented: spherical chiral inclusions in a uniaxial dielectric host medium, spheroidal chiral inclusions in free space, spherical chiral inclusions in a biaxial dielectric host medium, and spherical voids in a gyrotropic dielectric host medium.

Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium

We present here the application of Maxwell Garnett and Bruggeman formalisms to homogenize very general bianisotropic-in-bianisotropic particulate composites, assuming the inclusion particles to be ellipsoidal. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 263–266, 1997.

The role of anisotropy in the Maxwell Garnett and Bruggeman formalisms for uniaxial particulate composite media

A particulate composite medium in which the host and the spherical inclusions are made of uniaxial dielectric - magnetic media with parallel crystallographic axes is considered. The polarizability dyadics of a single inclusion are obtained, showing in detail the significant role played therein by the degree of anisotropy of the host medium. The derived expressions are used to set up the Maxwell Garnett and the Bruggeman formalisms for homogenization of the particulate composite medium. Numerical results are presented to show that the degree of anisotropy of the homogenized composite medium (HCM) is a non-trivial function of the degrees of anisotropy of the constituent media. Shifts in axial and transverse Fröhlich resonances are also predicted.

Bruggeman formalism for two models of uniaxial composite media: Dielectric properties

The Bruggeman formalism is rigorously implemented on two models for homogenizing a composite medium—wherein identical, parallel, isotropic dielectric, cylindrical inclusions of electrically small cross-section are randomly dispersed in a homogeneous, isotropic dielectric host medium. The inclusion medium is supposed to be dispersed in the homogenized composite medium (HCM) as cylindrical particles in both models. However, the host medium is dispersed as cylindrical particles in the first model and as spherical particles in the second. Numerical results show that the Wiener bounds are complied with by both models, although they yield different estimates of the anisotropic dielectric properties of the HCM. Both models also exhibit percolation. Benchmarking against careful experiments appears to be necessary to evaluate the adequacy of either model. The described models may eventually find use for on-line inspection and control of manufacturing processes for aligned fibrous composite media.

Glass-metal nanocomposite synthesis by metal organic route

Glass-metal nanocomposites incorporating ultrafine particles of iron, nickel, cobalt and manganese, respectively, in a silica glass matrix have been prepared by heat treatment of a gel derived from a sol containing silicon tetraethoxide and a suitable metal organic compound. Metal particles in all the nanocomposites are isolated and spherical-shaped with diameters ranging from 3 to 10 nm. Films of these nanocomposites with thickness of the order of a few micrometres have been prepared on glass slides by a simple dip-and-pull technique. Optical absorption spectra of the nanocomposite films have been measured over the wavelength range 200 to 2000 nm. Effective medium theories of Maxwell-Garnett and Bruggeman, respectively, have been used to calculate theoretically the optical absorption of these materials. The Maxwell-Garnett theory gives results which are in better agreement with experimental data than those obtained from Bruggeman formalism. The filling factor f as estimated from the least-squares fit of the experimental results with the Maxwell-Garnett theory has a value in the range 1 to 4%.