Depolarization Dyadics Research Articles

Bruggeman homogenization of a particulate composite material comprising truncated spheres and spheroids

Closed-form expressions were established for depolarization dyadics for a truncated sphere and a truncated spheroid, both electrically small, immersed in a uniaxial dielectric ambient medium. These depolarization dyadics were used to develop the Bruggeman homogenization formalism to predict the relative permittivity dyadic of a homogenized composite material (HCM) arising from a randomly distributed mixture of oriented particles shaped as truncated spheres and spheroids. Unlike other homogenization formalisms, most notably the Maxwell Garnett formalism, the Bruggeman formalism is not restricted to composites containing dilute volume fractions of constituent particles. Numerical investigations highlighted the anisotropy of the HCM and its relation to the shapes of the constituent particles and their volume fractions. Specifically, greater degrees of HCM anisotropy arise from constituent particles whose shapes deviate more from spherical, especially for mid-range volume fractions.

Homogenization of a random mixture of identically oriented superspheroidal particles

Depolarization dyadics were computed for superspheroidal particles made of an isotropic dielectric material and immersed in a uniaxial dielectric material. Superspheroidal shapes of two kinds were considered: one intermediate between spheroidal and cylindrical, and the other intermediate between spheroidal and cuboidal. These depolarization dyadics were employed in the Maxwell Garnett and Bruggeman homogenization formalisms to estimate the relative permittivity dyadics of homogenized composite materials (HCMs) that are random mixtures of identically oriented superspheroidal particles. The anisotropy of the HCMs was exposed through numerical investigations which revealed the dependencies of the HCM's anisotropy upon the shapes of the constituent particles and their volume fractions. The largest degrees of HCM anisotropy arose when the shape of the constituent particles deviated most from spherical, especially at mid-range volume fractions. Estimates of HCM anisotropy from the Maxwell Garnett and Bruggeman formalisms were broadly similar over the volume-fraction range appropriate for the Maxwell Garnett formalism, with modest differences being most apparent for particle shapes that deviated most from spherical.

Anisotropic homogenized composite mediums arising from truncated spheres, spheroids, and ellipsoids

Closed-form expressions were recently derived for depolarization dyadics for truncated spheres and truncated spheroids, and the formalism was extended to truncated ellipsoids. These results were exploited to develop an implementation of the Maxwell Garnett homogenization formalism for the relative permittivity parameters of homogenized composite mediums (HCMs) arising from an isotropic host medium impregnated with isotropic inclusions that are truncated spheres, spheroids, and ellipsoids. In so doing, the anisotropy of the HCM was related to the geometry of the inclusions: in general, the more the shape of the inclusions deviated from spherical, the greater was the degree of anisotropy exhibited by the HCM.

Erratum to “Depolarization Dyadics for Truncated Spheres, Spheroids, and Ellipsoids”

Erratum to “Depolarization Dyadics for Truncated Spheres, Spheroids, and Ellipsoids”

Depolarization Dyadics for Truncated Spheres, Spheroids, and Ellipsoids

Depolarization Dyadics for Truncated Spheres, Spheroids, and Ellipsoids

Sensitivity of surface plasmon resonance sensors based on metallic columnar thin films in the spectral and angular interrogations

Surface Plasmon Resonance (SPR) from metallic Columnar Thin Films (CTFs) of porosity as high as 0.5 was experimentally and theoretically investigated. The CTF layers were prepared by the Glancing Angle Deposition (GLAD) method. The SPR features were investigated in both the angular and the spectral modes. In the angular interrogation, increasing the porosity causes broadening to the dip width, shift to larger resonance angles, and increase of the sensitivity to analyte refractive index (RI) changes by about threefold compared with closed metal films. In the spectral interrogation, on the other hand, the resonance wavelengths are red-shifted for porous films; hence their spectral sensitivities are higher than those of closed films under the same experimental conditions. Nevertheless, the sensitivity behavior versus the resonance wavelength is similar to that of SPR sensors based on dense film layers. The shapes of the nanostructures constituting the CTF are described as ellipsoidal inclusions in which the effective permittivity dyadic of the composite material is calculated using the Bruggeman formalism with exact depolarization dyadics. The correlation between the sensitivity enhancement and the electromagnetic field intensity at the interface between the metallic film and the analyte was examined. Electromagnetic fields analyses were performed using the general 4 × 4 propagation matrices of general homogenous biaxial layers.

Effective constitutive parameters of linear nanocomposites in the long-wavelength regime

By the process of homogenization, judiciously designed nanocomposite materials can offer unprecedented degrees of material enhancement and/or novel material properties that may be usefully exploited in nanophotonic applications. Recently, there have been significant developments in the theory of such homogenized nanocomposite materials (HNMs), within the context of linear bianisotropic scenarios for particulate nanocomposites. These developments involve: the incorporation of depolarization dyadics, which represent component particles of nonzero volume, and the implementation of the strong-property-fluctuation theory wherein scattering interactions between neighboring component particles are treated on a statistical basis. Four recent areas of application are notable: (i) HNMs that support the propagation of plane waves with negative phase velocity (while their component materials do not); (ii) HNMs that support Voigt wave propagation (while their component materials do not); (iii) modeling the infiltration of certain sculptured thin films with a view to optical sensing applications; and (iv) simulation of the electromagnetic properties of vacuum in curved spacetime via HNMs as Tamm mediums. Forward homogenization is implemented in applications (i) and (ii); inverse homogenization is implemented in application (iv); and both forward and inverse homogenization are implemented in application (iii).

Depolarization regions of nonzero volume in bianisotropic homogenized composites

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 which depolarization regions of nonzero volume and ellipsoidal geometry are accommodated. Therein, the size, shape and spatial distribution of the component phase particles are taken into account. The analysis is developed within the most general linear setting of bianisotropic homogenized composite mediums (HCMs). Numerical results are presented for a representative example of a bianisotropic HCM, namely a Faraday chiral medium. These studies reveal that estimates of the HCM constitutive parameters in relation to volume fraction, particle eccentricity, particle orientation and correlation length are all significantly influenced by the size of the component phase particles.

Reply on: Comment on: On Improper Integrals in the Depolarization Dyadics of Lossless Gyrotropic Media, and the Maxwell Garnett Formalism by Akhlesh Lakhtakia

Summary Some applications of approximation as a research method in electrodynamics are discussed.

Comment on: On Improper Integrals in the Depolarization Dyadics of Lossless Gyrotropic Media, and the Maxwell Garnett Formalism by Dmitri A. Marakassov (in AEÜ, Vol. 55 (2001), 252–259)

A “strange and unexpected result” presented recently by Marakassov [AEÜ 55 (2001), 252–259] is intrinsically incorrect.

Needles and pillboxes in anisotropic mediums

We establish a connection between the singularities of dyadic Green functions in biaxial-dielectric media arising from eigenfunction expansions in cylindrical coordinates and general expressions of depolarization dyadics. The results are then extended to general anisotropic-dielectric media.

On Improper Integrals in the Depolarization Dyadics of Lossless Gyrotropic Media, and the Maxwell Garnett Formalism

Summary Improper integrals in the depolarization dyadic for the ellipsoidal-shaped inclusion embedded in an environment medium are considered and a method for obtaining their values in the lossless limit is suggested. The last is used to investigate the depolarization dyadics for a gyrotropic environment. Some key properties of mixtures with gyrotropic background are outlined.

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.

Electromagnetic depolarization dyadics and elliptic integrals

Depolarization dyadics are essential for the characterization of electromagnetic fields in source regions. As such they are key ingredients in the formulation of homogenization theories of composite media. Explicit expressions for the depolarization dyadics for a biaxial dielectric anisotropic medium (encompassing orthorhombic, monoclinic and triclinic crystallographic classes) are presented here. The results are expressed in terms of elliptic functions of the first and second kind.

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.

New expressions for depolarization dyadics in an axially uniaxial bianisotropic medium

Novel expressions for the depolarization dyadics in an axially uniaxial bianisotropic medium (AUBM) are presented. Explicit formulae for the depolarization dyadics are derived for cubical, cylindrical and spherical geometries and their dependence on the magnetoelectric coupling parameter is studied graphically.

New expressions for depolarization dyadics in uniaxial dielectric-magnetic media

We present some novel expressions for the depolarization dyadics in uniaxial dielectric-magnetic media. These expressions were obtained by generalizing the Fikioris approach for extracting the singular behaviour of integrals involving the dyadic Green functions and the source current density distributions. Cubical, cylindrical and spherical geometries serve as examples for a discussion of the depolarization dyadics' dependence on geometry and anisotropy.