Bruggeman Homogenization Formalism 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.

Toward morphologically induced anisotropy in thermally hysteretic dielectric properties of vanadium dioxide

The Bruggeman homogenization formalism was used to numerically investigate the dielectric properties of a columnar thin film (CTF) made from vanadium dioxide. For visible and near-infrared wavelengths, the CTF is electromagnetically equivalent to a homogeneous orthorhombic material. Over the 58–72 °C temperature range, the eigenvalues of the CTF’s relative permittivity dyadic are highly sensitive to temperature and vary according to whether the CTF is being heated or cooled. The anisotropy revealed through the eigenvalues and the anisotropy of the associated hysteresis were investigated in relation to temperature for CTFs of different porosities and columnar cross sections. When the free-space wavelength is 800 nm, the CTF is a dissipative dielectric material that exhibits temperature-dependent anisotropy and anisotropic hysteresis. In contrast, when the free-space wavelength is 1550 nm, the CTF can be a dissipative dielectric material, a hyperbolic material, or a metal-like material, depending on the temperature and the porosity of the CTF. As the porosity of the CTF decreases from 0.55 to 0.3, the anisotropy of the CTF becomes more pronounced, as does the anisotropy of the hysteresis. Only relatively modest variations in anisotropy and hysteresis arise in response to varying the columnar cross-sectional shape, as compared to the variations induced by varying the porosity.

Structural characterization of ZnO:N thin films using the optical optimization method

Recently, we introduced a method based on Bruggeman homogenization formalism in assistance with particle swarm optimization (PSO) to predict the optimized structure of the thin film that has a desired optical property. In this work, the same method was utilized for the characterization of ZnO:N thin films fabricated by annealing of Zn films with flow of a mixture of oxygen and nitrogen gases with different stoichiometry. The optimized simulated multilayer structure was proposed for each thin film based on obtaining the best agreement between experimental and theoretical optical spectra. The minimum value of the cost functions of PSO algorithm used as the criteria for determining the thickness of each thin film. The method introduced in this work produced results in good agreement with experimental data, hence it can be suggested that it is a powerful technique for structural characterization of thin films with complex structures.

Optical analyses of annealed copper films by a method based on Bruggeman homogenization formalism: Influence of deposition angle and annealing temperature

Normal and oblique angle depositions were used to produce copper thin films which were subsequently annealed with flow of nitrogen gas at different temperatures (100, 180 and 350 °C). Optical responses of these samples obtained at near normal incidence angle using both s- and p-polarized lights. Field emission scanning electron microscope (FESEM) analyses of the samples provided the physical structure and morphology of the samples. Bruggeman homogenization formalism calculations were used to propose a method for prediction of the reflection spectra which were compared and fitted to the experimental results. The most effective factors in this type of work were found to be the film thickness, the material inclusion (void fraction) and the thickness ratio of copper nitride layer formed on top of copper layer to the thickness of copper layer which was estimated from comparison of theoretical optical spectra with those of experimental results. Results showed that the latter ratio increases with annealing temperature and the highest temperature of 350 °C caused a drastic change in the optical spectra of the samples.

A computational approach for engineering optical properties of multilayer thin films: Particle swarm optimization applied to Bruggeman homogenization formalism

An algorithm based on Particle Swarm Optimization (PSO) and Bruggeman Homogenization Formalism was presented to design a model structure for a thin film with various sublayers to obtain a desired optical reflectance/transmittance spectrum. In principle, the results showed that the designed structure is strongly affected by the input parameters of the algorithm such as film thickness, number of sublayers, incident light characteristic wavelength and utilized materials, and the method has its ability to find the best optimized structure based on desired optical properties. The results of calculations were in good agreement with the physical principles of classical optics and the method could be useful in both design and characterization purposes.

Gain and loss enhancement in active and passive particulate composite materials

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 relative permittivities of both component materials are dissimilar in magnitude.

Simultaneous amplification and attenuation in isotropic chiral materials

The electromagnetic field phasors in an isotropic chiral material (ICM) are superpositions of two Beltrami fields of different handedness. Application of the Bruggeman homogenization formalism to two-component composite materials delivers ICMs wherein Beltrami fields of one handedness attenuate whereas Beltrami fields of the other handedness amplify. One component material is a dissipative ICM, the other an active dielectric material. The range of the volume fraction of the active component material for which simultaneous amplification and attenuation is exhibited decreases—but does not vanish—as the ICM component becomes more dissipative and as its chirality parameter reduces in magnitude.

Dyakonov–Tamm waves-based optical sensing using sculptured nematic thin film

Dyakonov–Tamm (DT) waves are highly sensitive to the constitutive properties of the partnering materials near the interface. DT waves are excited at the interface of two dielectric materials of which at least one is anisotropic and periodically nonhomogeneous normal to their interface. Sculptured nematic thin film (SNTF) is a good candidate for the periodically nonhomogeneous dielectric partner for optical sensing of a fluid due to its porosity. The nanoscale parameters of an uninfiltrated SNTF obtained from the inverse Bruggeman homogenization formalism were used in the forward Bruggeman homogenization formalism to determine the constitutive parameters for the infiltrated SNTF. The sensitivity of DT waves to the refractive index was analyzed for two possible sensing modalities and it was found that the sensitivity was comparable to that of the chiral sculptured thin films (STFs) made of the same material as of the SNTF. This implies that the sensing with DT waves is robust, is independent of the morphology of the partnering nonhomogeneous dielectric material and could make the sensing easier since SNTFs are easier to fabricate than the chiral STFs.

Towards metamaterials with giant dielectric anisotropy via homogenization: An analytical study

A random mixture of two isotropic dielectric materials, one composed of oriented spheroidal particles of relative permittivity ϵa and the other composed of oriented spheroidal particles of relative permittivity ϵb, 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 degree of anisotropy of the HCM, as gauged by the ratio of the eigenvalues of the HCM's permittivity dyadic, increased as the shape of the constituent particles became more eccentric. The greatest degrees of HCM anisotropy were achieved for the limiting cases wherein the constituent particles were shaped as needles or discs. In these instances explicit formulas for the HCM anisotropy were derived from the dyadic Bruggeman equation. Using these formulas it was found that the degrees of HCM anisotropy are proportional to ϵb or ϵb, at fixed values of volume fraction and ϵa, for ϵb>ϵa. Thus, in principle, metamaterials can be conceptualized via homogenization with extremely large degrees of anisotropy, by controlling the geometries and orientations of remarkably simple constituent particles. In practice, the degree of anisotropy would be limited by the available value of ϵb (and/or ϵa).

Controlling Voigt waves by the Pockels effect

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 na. 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 shapes, sizes, and orientations of the pores, as well as the refractive index na. These findings may be particularly significant for potential technological applications of Voigt waves such as in optical sensing.

On the sensitivity of directions that support Voigt wave propagation in infiltrated biaxial dielectric materials

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 degrees of anisotropy and dissipation of the porous biaxial material. The orientations, shapes and sizes of the particles which constitute the infiltrating material and the porous biaxial material exerted only a secondary influence on the maximum sensitivities achieved. Also, for the parameter ranges considered, the degree of porosity of the biaxial material had little effect on the maximum sensitivities achieved. These numerical findings bode well for the possible harnessing of VWP for optical sensing applications.

Surface plasmon resonance of Cu nanowires in polycarbonate template

The Cu nanowires were electrodeposited in polycarbonate track-etched (PCT) membrane. SEM, TEM and XPS techniques were used to characterize the morphology, structure, and size of nanowires as well as chemical composition. The absorption spectrum of copper nanowires embedded in PCT was measured and calculated for different incident angles and wavelengths. Our results showed that there is a broad peak due to excitation surface plasmons at θ=70° for wavelength λ=730nm. We applied the transfer matrix method and the Bruggeman homogenization formalism for optical modeling. The results of absorption spectra showed that there exists good agreement between the experimental and our used model. The results of this work may be useful in the study of surface plasmon resonance of copper nanowires.

Modeling Chiral Sculptured Thin Films as Platforms for Surface-Plasmonic-Polaritonic Optical Sensing

Biomimetic nanoengineered metamaterials called chiral sculptured thin films (CSTFs) are attractive platforms for optical sensing because their porosity, morphology, and optical properties can be tailored to order. Furthermore, their ability to support more than one surface-plasmon-polariton (SPP) wave at a planar interface with a metal offers functionality beyond that associated with conventional SPP-based sensors. An empirical model was constructed to describe SPP-wave propagation guided by the planar interface of a CSTF-infiltrated with a fluid which supposedly contains analytes to be detected-and a metal. The inverse Bruggeman homogenization formalism was first used to determine the nanoscale model parameters of the CSTF. These parameters then served as inputs to the forward Bruggeman homogenization formalism to determine the reference relative permittivity dyadic of the infiltrated CSTF. By solving the corresponding boundary-value problem for a modified Kretschmann configuration, the characteristics of the multiple SPP modes at the planar interface were investigated as functions of the refractive index of the fluid infiltrating the CSTF and the rise angle of the CSTF. The SPP sensitivities thereby revealed bode well for the implementation of fluid-infiltrated CSTFs as SPP-based optical sensors.

Voigt waves in homogenized particulate composites based on isotropic dielectriccomponents

Homogenized composite materials (HCMs) can support a singular form of opticalpropagation, known as Voigt wave propagation, while their component materials do not. Inessence, Voigt waves represent coalescent degenerate eigenmodes of the correspondingpropagation matrix. Crucially, the existence of Voigt waves stems from the non-Hermitiannature of this propagation matrix, which in turn is a manifestation of the dissipative natureof the HCM. This phenomenon was investigated for biaxial HCMs arising fromnondissipative isotropic dielectric component materials. The biaxiality of these HCMsstems from the oriented spheroidal shapes of the particles which make up the componentmaterials. An extended version of the Bruggeman homogenization formalism wasused to investigate the influence of component particle orientations, shapes andsizes, as well as volume fractions of the component materials, upon Voigt wavepropagation. Our numerical studies revealed that the directions in which Voigtwaves propagate are highly sensitive to the orientations of the component particlesand to the volume fractions of the component materials, but less sensitive tothe shapes of the component particles and less sensitive still to the sizes of thecomponent particles. Furthermore, whether or not an HCM supports Voigt wavepropagation at all is critically dependent upon the sizes of the component particlesand, in certain cases, upon the volume fractions of the component materials.

Effects of dispersion and dissipation on the optical rotation and ellipticity of chiral sculptured thin films

The optical rotation and ellipticity spectra from a chiral sculptured thin film were calculated using the piecewise homogeneity approximation method and Bruggeman homogenization formalism. The effects of dispersion and dissipation on the spectra were investigated using experimental dielectric functions for bulk zirconia (ZrO2) and titania (TiO2). The results showed that, in the event of weak frequency dependence of the optical properties of the raw material, effects on the transmission optical activity spectra can be discounted.

Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial

While metamaterials offer the potential to realize Tamm mediums which represent vacuous spacetime subjected to gravitational fields, practical formulations for suitable metamaterials have not hitherto been developed. This matter is addressed by establishing a metamaterial formulation for the Tamm medium representing Schwarzschild-(anti-)de Sitter spacetime. Our formulation is remarkably simple and does not involve a complex nanostructure of the type that is often associated with metamaterials. Instead it is based on the homogenization of only isotropic dielectric and isotropic magnetic component mediums, which are distributed randomly as oriented spheroidal particles. Using the inverse Bruggeman homogenization formalism, we demonstrated that a wide range of constitutive parameter values for the Tamm medium may be accessed through varying the particle shape, volume fraction or relative permittivity and relative permeability of the component mediums. The presented formulation is appropriate for the regions of spacetime which lie outside the event horizon for Schwarzschild spacetime and inside the event horizon for de Sitter spacetime; there are no such restrictions for anti-de Sitter spacetime.

On chemiluminescent emission from an infiltrated chiral sculptured thin film

The theory describing the far-field emission from a dipole source embedded inside a chiral sculptured thin film (CSTF), based on a spectral Green function formalism, was further developed to allow for infiltration of the void regions of the CSTF by a fluid. In doing so, the extended Bruggeman homogenization formalism — which accommodates constituent particles that are small compared to wavelength but not vanishingly small — was used to estimate the relative permittivity parameters of the infiltrated CSTF. For a numerical example, we found that left circularly polarized (LCP) light was preferentially emitted through one face of the CSTF while right circularly polarized (RCP) light was preferentially emitted through the opposite face, at wavelengths within the Bragg regime. The centre wavelength for the preferential emission of LCP/RCP light was red shifted as the refractive index of the infiltrating fluid increased from unity, and this red shift was accentuated when the size of the constituent particles in our homogenization model was increased. Also, the bandwidth of the preferential LCP/RCP emission regime decreased as the refractive index of the infiltrating fluid increased from unity.

Optical spectra of graded nanostructured TiO 2 chiral sculptured thin films

The optical properties of graded chiral sculptured TiO 2 thin films in axial and non-axial excited states are calculated using the rigorous coupled wave analysis method (RCWA) in conjunction with the Bruggeman homogenization formalism. The filtering frequency and polarization selectivity of these graded nanostructured TiO 2 sculptured thin films showed dependence on both structural and deposition parameters. The results achieved are consistent with the experimental data [K. M. Krause and M. J. Bret, Adv. Funct. Mater. 18 (2008) 3111].

Modeling columnar thin films as platforms for surface–plasmonic–polaritonic optical sensing

Via exploitation of surface plasmon polaritons (SPPs), columnar thin films (CTFs) are attractive potential platforms for optical sensing as their relative permittivity dyadic and porosity can be tailored to order. Nanoscale model parameters of a CTF were determined from its measured relative permittivity dyadic, after inverting the Bruggeman homogenization formalism. These model parameters were then used to determine the relative permittivity dyadic of a fluid-infiltrated CTF. Two boundary-value problems were next solved: the first relating to SPP-wave propagation guided by the planar interface of a semi-infinitely thick metal and a semi-infinitely thick CTF, and the second to the plane-wave response of the planar interface of a finitely thick metallic layer and a CTF in a modified Kretschmann configuration. Numerical studies revealed that SPP waves propagate at a lower phase speed and with a shorter propagation length, if the fluid has a larger refractive index. Furthermore, the angle of incidence required to excite an SPP wave in a modified Kretschmann configuration increases as the refractive index of the fluid increases.

Determination of constitutive and morphological parameters of columnar thin films by inverse homogenization

A dielectric columnar thin film (CTF), characterized macroscopically by a relative permittivity dyadic, was investigated theoretically. The CTF was assumed, on the nanoscale, to be 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 volume fraction occupied by the deposited material, from a knowledge of relative permittivity dyadic of the CTF. A modified Newton-Raphson technique was implemented to solve the inverse Bruggeman equations. Numerical studies revealed how the three nanoscale parameters of CTFs vary as functions of the vapor incidence angle.