Mie scattering by a spherical particle in an absorbing medium

We show that the large-size parameter limit of the scattering efficiency of a spherical particle of relative refractive index m(r) embedded in an absorbing medium is equal to [m(r) - 1[2/]m(r) + 1]2 and not to zero as has been claimed in a recent article [J. Appl. Opt. 40, 1354-1361 (2001)].     

Mie theory for light scattering by a spherical particle in an absorbing medium

Analytic equations are developed for the single-scattering properties of a spherical particle embedded in an absorbing medium, which include absorption, scattering, extinction efficiencies, the scattering phase function, and the asymmetry factor. We derive absorption and scattering efficiencies by using the near field at the surface of the particle, which avoids difficulty in obtaining the extinction based on the optical theorem when the far field is used. Computational results demonstrate that an absorbing medium significantly affects the scattering of light by a sphere.     

Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium

The radiative properties of bubbles or particles embedded in an absorbing medium are investigated. We aim first to determine the conditions under which absorption by the surrounding medium must be accounted for in the calculation of the efficiency factors by comparing results from Mie theory and the far-field and near-field approximations. Then, we relate these approximations for a single particle to the effective radiation characteristics required for solving the radiative transfer in an ensemble of scatterers embedded in an absorbing medium. The results indicate that the efficiency factors for a spherical particle can differ significantly from one model to another, in particular for large particle size parameter and matrix absorption index. Moreover, the effective scattering coefficient should be expressed based on the far-field approximation. Also, the choice of the absorption efficiency factor depends on the model used for estimating the effective absorption coefficient. However, for small void fractions, absorption by the matrix dominates, and models for the absorption coefficient and efficiency factor are unimportant. Finally, for bubbles in water, the conventional Mie theory can be used between 0.2 and 200 mum except at some wavelengths at which absorption by water must be accounted for.     

Improved version of the eikonal model for absorbing spherical particles

Abstract

We present a new expression for the scattering amplitude, valid for spherical absorbing objects, which leads to an improved version of the eikonal method outside the diffraction region. Limitations of this method are discussed and numerical results are presented and compared successfully with the Mie theory.     

Finite-difference time-domain solution of light scattering and absorption by particles in an absorbing medium

The three-dimensional (3-D) finite-difference time-domain (FDTD) technique has been extended to simulate light scattering and absorption by nonspherical particles embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. When computing the single-scattering properties of a particle in an absorbing dielectric medium, we derive the single-scattering properties including scattering phase functions, extinction, and absorption efficiencies using a volume integration of the internal field. A Mie solution for light scattering and absorption by spherical particles in an absorbing medium is used to examine the accuracy of the 3-D UPML FDTD code. It is found that the errors in the extinction and absorption efficiencies from the 3-D UPML FDTD are less than approximately 2%. The errors in the scattering phase functions are typically less than approximately 5%. The errors in the asymmetry factors are less than approximately 0.1%. For light scattering by particles in free space, the accuracy of the 3-D UPML FDTD scheme is similar to a previous model [Appl. Opt. 38, 3141 (1999)].     

Interfacial debonding of a spherical inclusion embedded in an infinite medium under remote stress

In the study of strength of particle reinforced composites, it is important to understand the energy release rate due to interfacial debonding between the particle and the matrix which is induced by manufacturing imperfection. This paper is aimed at the investigation of the critical condition for growth of the interfacial debonding and the corresponding volume increase due to void formation. The model used in the study is an isotropic elastic spherical inclusion embedded in an infinite isotropic elastic matrix under remote stress. Initial axisymmetrical interfacial debondings are assumed to exist in the vicinity of poles of the spherical inclusion. Axisymmetrical deformations of the matrix and the inclusion are analyzed based on the theory of three-dimensional elasticity in spherical coordinates. In order to avoid oscillatory stress singularity at the interfacial debonding front between two dissimilar materials, a condition of free slipping without friction at the interface is imposed. A Fredholm integral equation of the first kind is formulated based on the continuity conditions in the normal components of stress and displacement at the contact interface. The kernel function of the integral equation is expressed in terms of an infinite series of Legendre functions. Two types of remote stresses are considered in this study. The first type is the remote tension in the axial direction of the spherical inclusion and the second type is the remote compression in the transverse direction with respect to the axis of the spherical inclusion. Energy release rate is determined according to the rate of change of work done by remote stresses. In this paper, energy release rate and volume of the deformed void due to debonding are computed for any given size of initial interfacial debonding.     

Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium

The conventional Lorenz-Mie formalism is extended to the case for a coated sphere embedded in an absorbing medium. The apparent and inherent scattering cross sections of a particle, derived from the far field and near field, respectively, are different if the host medium is absorptive. The effect of absorption within the host medium on the phase-matrix elements associated with polarization depends on the dielectric properties of the scattering particle. For the specific cases of a soot particle coated with a water layer and an ice sphere containing an air bubble, the phase-matrix elements -P12/P11 and P33/P11 are unique if the shell is thin. The radiative transfer equation for a multidisperse particle system embedded within an absorbing medium is discussed. Conventional multiple-scattering computational algorithms can be applied if scaled apparent single-scattering properties are applied.     

Photon diffusion coefficient in an absorbing medium

A number of investigators have recently claimed, based on both analysis from transport theory and transport-theory-based Monte Carlo calculations, that the diffusion coefficient for photon migration should be taken to be independent of absorption. We show that these analyses are flawed and that the correct way of extracting diffusion theory from transport theory gives an absorption-dependent diffusion coefficient. Experiments by two different sets of investigators give conflicting results concerning whether the diffusion coefficient depends on absorption. The discrepancy between theory and the earlier set of experiments poses an interesting challenge.     

Light scattering by an infinite circular cylinder immersed in an absorbing medium

Analytic solutions are developed for the single-scattering properties of an infinite dielectric cylinder embedded in an absorbing medium with normal incidence, which include extinction, scattering and absorption efficiencies, the scattering phase function, and the asymmetry factor. The extinction and scattering efficiencies are derived by the near-field solutions at the surface of the particle. The normalized scattering phase function is obtained by use of the far-field approximation. Computational results show that, although the absorbing medium significantly reduces the scattering efficiency, it has little effect on absorption efficiency. The absorbing medium can significantly change the conventional phase function. The absorbing medium also strongly affects the polarization of the scattered light. However, for large absorbing particles the degrees of polarization change little with the medium's absorption. This implies that, if the transmitting lights are strongly weakened inside the particle, the scattered polarized lights can be used to identify objects even when the absorption property of the host medium is unknown, which is important for both active and passive remote sensing.     

Light scattering by a coated infinite cylinder in an absorbing medium

The scattering formulation for a coated infinite cylinder in an absorbing medium is presented in this paper. The cylinder is subjected to an arbitrarily polarized plane wave propagating in a general direction at the cylinder. The refractive index and magnetic permeability of the host medium, as well as those for the core and coating of the cylinder, can be real or complex. The scattering and extinction efficiencies and the scattering amplitudes are derived for both the near field and the far field. As the medium is absorbing, the "true" extinction and scattering efficiencies are derived based on the radiative energy outflow at the surface of the cylinder. The radiative efficiencies in the far field are denoted as "apparent" properties because they include absorption by the intervening medium. The influence of the refractive index and permeability of the host medium on the scattering properties of a coated cylinder is illustrated by numerical examples.     

Modified Eshelby tensor for an ellipsoidal inclusion imperfectly embedded in an infinite piezoelectric medium

The modified Eshelby tensor for predicting the effective moduli of particle-reinforced piezoelectric composites is derived for the problem of an ellipsoidal inclusion which is imperfectly bonded to the matrix. A linear interface relation is adopted, which involves discontinuities of the mechanical displacements and electric potential across the interface, and assumes that the corresponding jumps are proportional to the continuous stresses and electric displacements at the interface. The piezoelectric field induced by a uniform eigenstrain given only in the inclusion is deduced analytically. As the induced piezoelectric field is no longer uniform, the average strains and electric displacements are calculated, and the modified piezoelectric Eshelby tensor is evaluated by both an iterative method and a direct method. By comparison, it is shown that the iterative method yields rapidly convergent results.     

The numerical solution of radiative heat transfer for grey bodies in an absorbing medium

A new approximate analytical method for solving the integral radiation equations [1, 2] is used for the numerical calculation and investigation of the local and average characteristics of radiative heat transfer in systems of grey bodies separated by an isothermal absorbing medium.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 3, pp 520–525, September, 1969.     

Penny-shaped crack in a sphere embedded in an infinite medium

We consider the problem of determining the stress intensity factor and the crack energy in an Isotropie, homogeneous elastic sphere embedded in an infinite Isotropie, homogeneous elastic medium when there is a diametrical crack in the sphere. We assume that the crack is opened by an internal pressure and the sphere is bonded to the surrounding material. The problem is reduced to the solution of a Fredholm integral equation of the second kind in the auxiliary function φ($\text{t}$). Expressions for the stress intensity factor and the crack energy are obtained in terms of φ($\text{t}$). The integral equation is solved numerically and the numerical values of the stress intensity factor and the crack energy are graphed.     

Improvement of Mishchenko's T-matrix code for absorbing particles

The use of Gaussian elimination with backsubstitution for matrix inversion in scattering theories is discussed. Within the framework of the T-matrix method (the state-of-the-art code by Mishchenko is freely available at http://www.giss.nasa.gov/-crmim), it is shown that the domain of applicability of Mishchenko's FORTRAN 77 (F77) code can be substantially expanded in the direction of strongly absorbing particles where the current code fails to converge. Such an extension is especially important if the code is to be used in nanoplasmonic or nanophotonic applications involving metallic particles. At the same time, convergence can also be achieved for large nonabsorbing particles, in which case the non-Numerical Algorithms Group option of Mishchenko's code diverges. Computer F77 implementation of Mishchenko's code supplemented with Gaussian elimination with backsubstitution is freely available at http://www.wave-scattering.com.     

Melting behaviour of cadmium particles embedded in an aluminium matrix

An Al-4.5% Cd alloy has been manufactured by melt spinning to produce a microstructure of 14–150 nm diameter faceted cadmium particles embedded in an aluminium matrix. The melting behaviour of the cadmium particles has been investigated by differential scanning calorimetry. The melting point of 20 and 14 nm diameter cadmium particles embedded are depressed by 7 and 9 K respectively, below the bulk equilibrium cadmium melting point, because of Gibbs-Thomson capillarity effects. The average solid cadmium particle/aluminium matrix interfacial energy is 27 mJ m^–2 higher than the average liquid cadmium particle/aluminium matrix interfacial energy. No significant superheating is needed to nucleate cadmium particle melting.     

Weight function for an elliptical planar crack embedded in a homogeneous elastic medium

The method of simultaneous dual integral equations is used for obtaining the exact analytical solution for the weight function for an elliptical crack embedded in an infinite elastic solid. We show that the solution is unique and can be reduced to the known solutions for a number of particular cases.     

Sensitivity studies for imaging a spherical object embedded in a spherically symmetric, two-layer turbid medium with photon-density waves

We present analytic expressions for the amplitude and phase of photon-density waves in strongly scattering, spherically symmetric, two-layer media containing a spherical object. This layered structure is a crude model of multilayered tissues whose absorption and scattering coefficients lie within a range reported in the literature for most tissue types. The embedded object simulates a pathology, such as a tumor. The normal-mode-series method is employed to solve the inhomogeneous Helmholtz equation in spherical coordinates, with suitable boundary conditions. By comparing the total field at points in the outer layer at a fixed distance from the origin when the object is present and when it is absent, we evaluate the potential sensitivity of an optical imaging system to inhomogeneities in absorption and scattering. For four types of background media with different absorption and scattering properties, we determine the modulation frequency that achieves an optimal compromise between signal-detection reliability and sensitivity to the presence of an object, the minimum detectable object radius, and the smallest detectable change in the absorption and scattering coefficients for a fixed object size. Our results indicate that (l) enhanced sensitivity to the object is achieved when the outer layer is more absorbing or scattering than the inner layer; (2) sensitivity to the object increases with the modulation frequency, except when the outer layer is the more absorbing; (3) amplitude measurements are proportionally more sensitive to a change in absorption, phase measurements are proportionally more sensitive to a change in scattering, and phase measurements exhibit a much greater capacity for distinguishing an absorption perturbation from a scattering perturbation.     

Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient

We address the problem of the modeling of the extinction coefficient into an absorbing medium, including a random distribution of identical scatterers of arbitrary size. We show that the extinction coefficient, including losses in the host medium, can be derived from a diagrammatic expansion arising from the rigorous multiple-scattering theory of electromagnetic waves in random media. While in previous approaches the contribution to the extinction coefficient due to the absorption in the host medium and due to the absorption and scattering by the particles were evaluated separately and heuristically, our approach is based on a derivation from first principles.     

Wave propagation from a spherical cavity imbedded in an elasto-plastic medium

Summary The problem of an impulsively applied pressure acting on the surface of a spherical cavity imbedded in an elasto-plastic medium governed by a bilinear stress-strain law is considered. The problem is solved by using a certain iterative finite difference scheme which prevents almost all the numerical oscillations which usually occur in the region behind the discontinuity at the elastic-plastic boundary, when a standard finite difference scheme is applied.