On the application of homogenization formalisms to active dielectric composite materials

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 component materials have different signs then there are substantial differences between the Bruggeman and Maxwell Garnett estimates. These differences are slightly less pronounced if the relative permittivity of the metallic component material is described by the extended version of the Drude formula appropriate to very small particles. However, these differences become enormous - with the Bruggeman estimate being physically implausible - as the imaginary parts of the permittivities of the component materials tend to zero.     

Slow and Stopped Light in Active Gain Composite Materials of Metal Nanoparticles: Ultralarge Group Index‐Bandwidth Product Predicted

Chip‐compatible slow light devices with large group index‐bandwidth products and low losses are of great interest in the community of modern photonics. In this work, active gain materials containing metal nanoparticles are proposed as the slow and stopped light materials. Gain‐assisted high field enhancement in metal nanoparticles and the resultant strong dispersion lead to such phenomena. From the Maxwell‐Garnett model, it is revealed that the metal nanocomposite exhibits the infinitely large group index when the gain of the host medium and the filling factor of metal nanoparticles satisfy a critical condition. For the gain of the host above the critical value, one can observe slowing down effect with amplification of light pulses. Significantly large group index‐bandwidth products, which vary from a few to several thousand or even infinity depending on the gain value of the host medium, have been numerically predicted in active silica glasses containing spheroidal metal nanoparticles, as examples. The proposed scheme inherently provides the widely varying operating spectral range by changing the aspect ratio of metal nanoparticles and chip‐compatibility with low cost.     

The Generalized Maxwell Garnett Approximation for Textured Matrix Composites with Coated Inclusions

Doklady Physics - A generalization of the Maxwell Garnett approximation for a textured matrix composite consisting of ellipsoidal inclusions with a shell is proposed; using it, an expression for an...     

Tailoring the microwave permittivity and permeability of composite materials

The microwave permittivity (ɛ_r) and permeability (μ_r) of composite materials are tailored by adding various loading agents to a host plastic and are subsequently modeled using the Maxwell Garnett theory and second order polynomials. With the addition of manganese zinc ferrite, strontium ferrite, nickel zinc ferrite, barium tetratitanate and graphite powders, materials with values of ɛ′, e″, μ′, μ″ as high as 22, 5, 2.5 and 1.7 have been obtained. Permittivity and permeability data are calculated at 2.0245 GHz from reflection and transmission measurements performed in a 7 mm coaxial test line. The Maxwell Garnett (MG) theory successfully models ɛ_r if the filling factor is less than 0.30 and ratio |ɛ₁| (host)/ |ɛ₂| (powder) is greater than 0.04. As this ratio decreases, the MG theory is shown to be independent of ɛ₂ and second order polynomials are used to effectively model the dielectric constant. Polynomials are also used for the ferrite composites because it was determined that the MG theory was unable to model μ_r. This deficiency is attributed to the difference of domain structures that exist in powdered and sintered ferrites.     

On Tailoring of Nonlinear Spectral Properties of Nanocomposites Having Maxwell Garnett or Bruggeman Structure

Nonlinear spectral properties of two different types of conjugated polymers (polythiophene PT10 and polysilane PDHS) with nanoscale TiO₂ particles forming a Maxwell Garnett or a Bruggeman composite are studied. According to the present simulations, it is possible both to enhance and to tailor the magnitude and the spectral properties, respectively, of the effective degenerate third-order nonlinear susceptibility of the composite. This is done simply by tuning the volume fraction of TiO₂ inclusions and by changing the topology of the composite.     

Study of the relationship between porous ratio and effective index in nanoporous film

In this paper, a theory model called 'composite media in parallel' is proposed to explain the connection between the porous ratio and effective index of the nanoporous film. This model comes from theory of composite media's effective dielectric constant in solid material field. From this model, a function relationship between the porous ratio and effective index of the nanoporous film is obtained. And the Finite Difference Time Domain (FDTD) method is used to construct the model of nanoporous film, simulate the propagation of lightwave in the film and calculate some effective indices according to different ratios of pore. Compared with Maxwell Garnett theory model and Bruggeman theory model, it is found that the function curve based on the composite media in parallel model is most consistent with the results simulated by FDTD method.     

Application of Bruggeman and Maxwell Garnett homogenization formalisms to random composite materials containing dimers

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 and metal–metal 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’s permittivity are relatively modest but not insignificant, these effects being pronounced by anisotropy when all dimers are identically oriented.     

Modeling of ferrite-based materials for shielding enclosures

An analytical model for a magneto-dielectric composite material is presented based on the Maxwell Garnett rule for a dielectric mixture, and on Bruggeman's effective medium theory for permeability of a ferrite powder embedded in a dielectric. In order to simultaneously treat frequency-dispersive permittivity and permeability of a composite in a full-wave FDTD code, a new algorithm based on discretized auxiliary differential equations has been implemented. In this paper, numerical examples of modeling structures containing different magneto-dielectric mixtures are presented.     

Linear and nonlinear optical properties of highly concentrated gold nanoparticles

Linear and nonlinear (NL) optical properties of composite materials containing high concentration of gold nanoparticles (NPs) were studied using the Maxwell–Garnett model and the degenerated electron gas model. High values of the linear refraction index of the composite, NL shift of the plasmon resonance peak and reversal sign of the real and imaginary parts of the NL third-order susceptibility were observed. Figures of merit for photonic devices were calculated and fulfilled depending of the filling factor and NPs size.     

Generalized Clausius–Mossotti Formula for Random Composite with Circular Fibers

An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius–Mossotti (Maxwell–Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius–Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays (“shaking” geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.     

Minimal spaser threshold within electrodynamic framework: Shape,size and modes

It is known (yet often ignored) from quantum mechanical or energetic considerations, that the threshold gain of the quasi‐static spaser depends only on the dielectric functions of the metal and the gain material. Here, we derive this result from the purely classical electromagnetic scattering framework. This is of great importance, because electrodynamic modelling is far simpler than quantum mechanical one. The influence of the material dispersion and spaser geometry are clearly separated; the latter influences the threshold gain only indirectly, defining the resonant wavelength. We show that the threshold gain has a minimum as a function of wavelength. A variation of nanoparticle shape, composition, or spasing mode may shift the plasmonic resonance to this optimal wavelength, but it cannot overcome the material‐imposed minimal gain. Furthermore, retardation is included straightforwardly into our framework; and the global spectral gain minimum persists beyond the quasi‐static limit. We illustrate this with two examples of widely used geometries: Silver spheroids and spherical shells embedded in and filled with gain materials.     

Enhancement of nonlinear optical properties of compounds of silica glass and metallic nanoparticle

The aim of this paper is to introduce a method for enhancing the nonlinear optical properties in silica glass by using metallic nanoparticles. First, the T-matrix method is developed to calculate the effective dielectric constant for the compound of silica glass and metallic nanoparticles, both of which possess nonlinear dielectric constants. In the second step, the Maxwell–Garnett theory is exploited to replace the spherical nanoparticles with cylindrical and ellipsoidal ones, facilitating the calculation of the third-order nonlinear effective susceptibility for a degenerate four-wave mixing case. The results are followed by numerical computations for silver, copper and gold nanoparticles. It is shown, graphically, that the maximum and minimum of the real part of the reflection coefficient for nanoparticles of silver occurs in smaller wavelengths compared to that of copper and gold. Further, it is found that spherical nanoparticles exhibit greater figure-of-merit compared to those with cylindrical or ellipsoidal geometries.     

Dispersion Theory of Liquids Containing Optically Linear and Nonlinear Maxwell Garnett Nanoparticles

Kramers-Kronig relations and sum rules for effective linear permittivity of the Maxwell Garnett liquid-nanosphere system were obtained using complex analysis and general expression of the effective permittivity. When reflectance from optically linear and nonlinear Maxwell Garnett nanoparticles was calculated it was observed that the reflectance, in the case of attenuated total reflection, depends on the fill fraction of the nanospheres and their nonlinear susceptibility. According to simulations a good sensitivity of the nonlinear contribution on the reflectance can be obtained by using a probe wavelength corresponding to the resonance frequency of the nanosphere system. In Kretchmann#x0027;s configuration it was observed that the surface-plasmon-resonance angle depends on the fill fraction and on the intensity of the incident light. By using reflectance, it is possible to detect optically nonlinear nanospheres in liquids.     

Interference of scattered waves and mixing rules for group velocities in nanocomposite materials

Mixing rules for group velocities in nanocomposite materials with different architecture, including lamellar-inhomogeneous nanotextures, Maxwell Garnett structures, and one-dimensional photonic crystals, are derived and analyzed. The group velocity can be controlled for such composite structures by changing nanocrystal sizes and varying the dielectric properties and the volume-filling fractions of the constituent materials. The interference of scattered waves in structures with a spatial scale of optical inhomogeneities comparable to the radiation wavelength gives rise to new physical phenomena that cannot be described in terms of the effective-medium approximation.     

Mixing rules for group velocities in nanocomposite materials and photonic crystals

Mixing rules for group velocities in nanocomposite materials with different architecture, including lamellarinhomogeneous nanotextures, Maxwell Garnett structures, and one-dimensional photonic crystals, are derived and analyzed. The group velocity can be controlled for such composite structures by changing nanocrystal sizes and varying the dielectric properties and the content of the constituent materials. The interference of scattered waves in structures with a spatial scale of optical inhomogeneities comparable to the radiation wavelength gives rise to new physical phenomena that cannot be described in terms of the effective-medium approximation.     

Fluctuating dimension in a discrete model for quantum gravity based on the spectral principle

The spectral principle of Connes and Chamseddine is used as a starting point to define a discrete model for Euclidean quantum gravity. Instead of summing over ordinary geometries, we consider the sum over generalized geometries where topology, metric, and dimension can fluctuate. The model describes the geometry of spaces with a countable number n of points, and is related to the Gaussian unitary ensemble of Hermitian matrices. We show that this simple model has two phases. The expectation value , the average number of points in the Universe, is finite in one phase and diverges in the other. We compute the critical point as well as the critical exponent of . Moreover, the space-time dimension delta is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, < 2.     

Numerical modeling of the propagation of electromagnetic radiation in 3D photonic crystals

We describe the mathematical model and program BAL 3D that enables one to solve numerically three-dimensional Maxwell equations in inhomogeneous media. We present the results of calculations of the electromagnetic-radiation propagation in a 3D quasiperiodic medium (photonic crystal). We found the positions of stop bands in photonic crystals and calculate the reflection coefficients for composite materials consisting of artificial opals with pores filled with water. We compare our results with the experimental data.     

Symplectic and multisymplectic numerical methods for Maxwell’s equations

In this paper, we compare the behaviour of one symplectic and three multisymplectic methods for Maxwell’s equations in a simple medium. This is a system of PDEs with symplectic and multisymplectic structures. We give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments. We also show that these numerical methods preserve the divergence. Furthermore, we extend the discussion on dispersion for (multi)symplectic methods applied to PDEs with one spatial dimension, to include anisotropy when applying (multi)symplectic methods to Maxwell’s equations in two spatial dimensions. Lastly, we demonstrate how varying the Courant–Friedrichs–Lewy (CFL) number can cause the (multi)symplectic methods in our comparison to behave differently, which can be explained by the study of backward error analysis for PDEs.