Effective medium approximation

The effective dielectric function ⑀ e for a medium of anisotropic inclusions embedded in an isotropic host is calculated using the Maxwell Garnett approximation. For uniaxial inclusions, ⑀ e depends on how well the inclusions are aligned. We apply this approximation to study ⑀ e for a model of quasi-one-dimensional organic polymers. The polymer is assumed to be made up of small single crystals embedded in an isotropic host of randomly oriented polymer chains. The host dielectric function is calculated using the effective-medium approximation ͑EMA͒. The resulting frequency-dependent ⑀ e () closely resembles experiment. Specifically, Re⑀ e () is negative over a wide frequency range, while Im⑀ e () exhibits a broad ''surface plasmon'' band at low frequencies, which results from localized electronic excitations within the crystallites. If the host is above the conductivity percolation threshold, Im⑀ e () has a low-frequency Drude peak in addition to the surface plasmon band, and Re⑀ e () is negative over an even wider frequency range. We also calculate the cubic nonlinear susceptibility e () of the polymer, using a nonlinear EMA. At certain frequencies, e () is found to be strongly enhanced above that of the corresponding single crystals. Our results suggest that the electromagnetic properties of conducting polymers can be understood by viewing the material as randomly inhomogeneous on a small scale such that the quasistatic limit is applicable.

We report resistivity measurements on epitaxial Fe 3 O 4 films between 3 and 100 nm thickness grown on polished MgO substrates. The resistivity of the films is larger than the bulk resistivity, and is increasing with decreasing film thickness. This can be explained by the significant decrease of antiphase domain size with decreasing film thickness, as observed by transmission electron microscopy. The domain size decreases from 40 nm for 100-nm-thick films, to 5 nm for 3-nm-thick films. The effective conductivity has been modeled as a function of the bulk and boundary conductivities using the effective medium approximation. It is suggested that the absence of the Verwey transition in the thinnest films is also related to the very small domain size, which inhibits long-range order.

Application of cavity ring down (CRD) spectrometry for measuring the optical properties of pure and mixed laboratory-generated aerosols is presented. The extinction coefficient (α ext ), extinction cross section (σ ext ) and extinction efficiency (Q ext ) were measured for polystyrene spheres (PSS), ammonium sulphate ((NH 4 ) 2 (SO 4 )), sodium 5 chloride (NaCl), glutaric acid (GA), and Rhodamine-590 aerosols. The refractive indices of the different aerosols were retrieved by comparing the measured extinction efficiency of each aerosol type to the extinction predicted by Mie theory. Aerosols composed of sodium chloride and glutaric acid in different mixing ratios were used as model for mixed aerosols of two non-absorbing materials, and their extinction and 10 complex refractive index were derived. Aerosols composed of Rhodamine-590 and ammonium sulphate in different mixing ratios were used as model for mixing of absorbing and non-absorbing species, and their optical properties were derived. The refractive indices of the mixed aerosols were also calculated by various optical mixing rules and a core plus shell Mie model. We found that for non-absorbing mixtures, the 15 linear rule, Maxwell-Garnett rule, extended effective medium approximation (EEMA), and core plus shell model give comparable results, with the linear mixing rule giving a slightly better fit than the others. Overall, calculations for the mixed aerosols are not as good as for single component aerosols. For absorbing mixtures, the differences between the refractive indices calculated using the mixing rules and those retrieved by 20 CRD are generally higher. of aerosols on climate is by absorbing and/or scattering the incoming solar radiation 12348 ACPD Abstract Introduction Conclusions References Tables Figures Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion

For isotropic and homogeneous nonlinear left-handed materials, for which the effective medium approximation is valid, Maxwell's equations for electric and magnetic fields lead naturally, within the slowly varying envelope approximation, to a system of coupled nonlinear Schrödinger equations. This system is equivalent to the well-known Manakov model that under certain conditions, is completely integrable, and admits bright and dark soliton solutions. It is demonstrated that left-and right-handed ͑normal͒ nonlinear media may have compound dark and bright soliton solutions, respectively. These results are also supported by numerical calculations.

The kinetics of capillary imbibition into porous rocks is studied experimentally and theoretically. The Washburn law is modified by introducing various corrections relating to the microstructure of the rocks, such as tortuosity, pore shape (obtained experimentally), and applying the effective medium approximation (EMA) in order to calculate the effective radius that defines the hydraulic conductance and the topology of the capillary imbibition. The application of the EMA shows that capillary imbibition is mainly produced in 1-D, and the pore structure is constituted by different pore throats in series, linked by chamber pores. The capillary process has been discussed as a function of their petrography and pore structure. Our study of the Washburn equation and the addition of correction factors for the pore structure allows a very accurate prediction of the weight rate.

The exact solution of the scattered electromagnetic field from a water droplet containing an arbitrarily located spherical black carbon particle is used to investigate the effect of black carbon on the absorption of solar radiation by clouds. When droplet absorption is averaged over all possible locations of black carbon within a droplet, the averaged absorption is close to the value calculated using the effective medium approximation. The preferential black carbon location on the top or close to the bottom of the droplet leads to an increased absorption. The estimated upper bound on the increased absorption of solar radiation (global and annual average) is 1-3 W/m 2 over the absorption of pure water clouds.

Mineral dust aerosols have complex nonspherical shapes and varying composition. This study utilizes data on morphology (size and shape) and composition of dust particles to determine the extent to which the optical properties of real particles di er from those of spheres. A method for modeling the optical properties of complex particle mixtures is proposed. The method combines dust particle composition-shape-size (CSS) distributions reconstructed from the electron microscopy data, e ective medium approximations and discrete dipole approximation. The method is used to compute optical characteristics of realistic dust mixtures representative of Saharan and Asian dust. We demonstrate that considered CSS distributions result in various di erences in the extinction coe cient, single scattering albedo, asymmetry parameter and the scattering phase function relative to the volume-equivalent spheres and the mixtures of the randomly oriented oblate and prolate spheroids. Implications of these di erences for radiation/climate modeling and remote sensing are discussed.

This paper aims at developing numerically validated models for predicting the through-plane effective index of refraction and absorption index of nanocomposite thin films. First, models for the effective optical properties of such materials are derived from previously reported analysis applying the volume averaging theory ͑VAT͒ to Maxwell's equations. The transmittance and reflectance of nanoporous thin films are computed by solving Maxwell's equations and the associated boundary conditions at all interfaces using finite element methods. The effective optical properties of the films are retrieved by minimizing the root mean square of the relative errors between the computed and theoretical transmittance and reflectance. Nanoporous thin films made of SiO 2 and TiO 2 consisting of cylindrical nanopores and nanowires are investigated for different diameters and various porosities. Similarly, electromagnetic wave transport through dielectric medium with embedded metallic nanowires are simulated. The numerical results are compared with predictions from widely used effective property models including ͑1͒ the Maxwell-Garnett theory, ͑2͒ the Bruggeman effective medium approximation, ͑3͒ the parallel, ͑4͒ series, ͑5͒ Lorentz-Lorenz, and ͑6͒ the VAT models. Very good agreement is found with the VAT model for both the effective index of refraction and absorption index. Finally, the effect of volume fraction on the effective index of refraction and absorption index predicted by the VAT model is discussed. For certain values of wavelengths and volume fractions, the effective index of refraction or absorption index of the composite material can be smaller than that of both the continuous and dispersed phases. These results indicate guidelines for designing nanocomposite materials with desired optical properties.

Samples of silicon nanocrystals on various substrates were prepared by cluster beam deposition of silicon nanoparticles, obtained by laser-induced pyrolysis of silane in a flow reactor. Using optical ellipsometry, the optical properties ͑refractive index and extinction coefficient͒ of the as-prepared silicon nanocrystal layers were determined in the wavelength range from 240 to 700 nm. Two dispersion models were used to describe the silicon nanocrystal optical properties: the Bruggeman effective medium approximation model and the Tauc-Lorentz model. The study showed that while a simple Bruggeman effective medium approximation model could not completely account for the silicon nanocrystal dispersion behavior, the optical response of the silicon nanocrystal layers could be satisfactorily described by a Tauc-Lorentz model. The present study also showed that, as for porous silicon, the silicon nanocrystal optical indexes significantly deviate from those of bulk crystalline and amorphous silicon. It confirms the special behavior of silicon under its nanoscale form.

Submillimeter-wave radiometry is a new technique for determining ice water path (IWP) and particle size in upper tropospheric ice clouds. The first brightness temperatures images of ice clouds above 340 GHz were measured by the Compact Scanning Submillimeter Imaging Radiometer (CoSSIR) during the CRYSTAL-FACE campaign in July 2002. CoSSIR operated with 12 channels from receivers at 183, 220, 380, 487, and 640 GHz. CoSSIR and the nadir viewing 94 GHz Cloud Radar System (CRS) flew on the NASA ER-2 airplane based out of Key West, Florida. A qualitative comparison of the CoSSIR brightness temperatures demonstrates that the submillimeter-wave frequencies are more sensitive to anvil ice cloud particles than the lower frequencies. A Bayesian algorithm, with a priori microphysical information from in situ cloud probes, is used to retrieve IWP and median mass equivalent sphere particle diameter (Dme). Microwave scattering properties of random aggregates of plates and aggregates of frozen droplets are computed with the discrete dipole approximation (DDA) and an effective medium approximation tuned to DDA results. As a test of the retrievals, the vertically integrated 94 GHz radar backscattering is also retrieved from the CoSSIR data and compared to that measured by the CRS. The integrated backscattering typically agrees within 1 to 2 dB for IWP from 1000 to 10,000 g/m , and while the disagreement increases for smaller IWP, it is typically within the Bayesian error bars. Retrievals made with only the three 183 GHz and one 220 GHz channels are generally as good or better than those including 380 ¡ 6.2 and 640 GHz because the CoSSIR submillimeter-wave channels were much noisier than expected. An algorithm to retrieve profiles of ice water content and Dme from CRS and CoSSIR data was developed. This Bayesian algorithm also retrieves the coefficients of an IWC -radar reflectivity power law relation and could be used to evaluate radar only ice cloud retrieval algorithms.

We study the mean time for a random walk to traverse between two arbitrary sites of the Erdős-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-monotonically with p near the percolation transition. Much of this behavior can be understood by simple heuristic arguments.

In our previous study, the refractive indices of freestanding porous silicon (PS) layers were derived using the envelope method, where the computation is based on the values of local minima and maxima in the oscillations of transmission spectra. In the present work, an improved procedure for calculating the optical parameters from the measurements data is described. It is verified by reflection measurements on freestanding samples that optical scattering at the air-PS interface is the main reason for the loss of the transmitted light intensity and thus for the inaccurate results we obtained earlier by the envelope method. This however can be avoided by taking into consideration the relationship between the optical path in the plane-parallel film and the position of extrema in the transmission spectra. The as-determined effective refractive indices show very good matching with the theoretical calculations by the BruggemanÕs effective medium approximation.

Porous silicon rugate filters are fabricated and investigated for their ability to sense chemical species. The durability of the filter is tested by allowing the structure to undergo many cycles of adsorption and desorption of vapor-phase ethanol molecules. The characteristic reflectivity peak of the structure exhibits a relative blueshift of 2.7% after 86 adsorption/desorption cycles. The observed shift is ascribed to the formation of silicon dioxide, which has a lower refractive index than that of silicon. In order to stabilize the structure against oxidation expected from cycling and environmental exposure, the filter is subjected to electrochemical oxidation in an aqueous sulfuric acid electrolyte. The treatment dramatically improves stability of the sensor; a relative blueshift of Ͻ0.4% is observed after 100 adsorption/desorption cycles for this sensor. The sensitivity of the sensor is also affected by electrochemical oxidation: the response to saturated ethanol in air changes from ⌬ = 100 nm to ⌬ = 70 nm, respectively. Theoretical calculations using the Bruggeman effective medium approximation and the characteristic matrix method indicate that up to 15% ͑by volume͒ of silicon is transformed to silicon dioxide by the electrochemical oxidation procedure. This volume ratio is close to that estimated from Auger electron spectroscopy measurements.