Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion

Abstract

The theory of interaction between an arbitrary electromagnetic shaped beam and a sphere with an eccentrically located spherical inclusion is presented. This theory is built as a synthesis between two available theories (i) the generalized Lorenz-Mie theory for a homogeneous sphere (illuminated by an arbitrary shaped beam) and (ii) the theory of interaction between a plane wave and an eccentrically stratified dielectric sphere.     

Scattering of a pulsed wave by a sphere with an eccentric spherical inclusion

A dielectric sphere with an eccentric spherical dielectric inclusion and an incident amplitude-modulated plane electromagnetic wave constitute an exterior radiation problem, which is solved in this paper. A solution is obtained by combined use of the Fourier transform and the indirect-mode-matching method. The analysis yields a set of linear equations for the wave amplitudes of the frequency-domain expansion of the electric-field intensity within and outside the externally spherical inhomogeneous body; that set is solved by truncation and matrix inversion. The shape of the backscattered pulse in the time domain is determined by application of the inverse fast Fourier transform. Numerical results are shown for a pulse backscattered by an acrylic sphere that contains an eccentric spherical cavity. The effects of cavity position and size on pulse spreading and delay are discussed.     

Dyadic Green's function of a sphere with an eccentric spherical inclusion

An exact, analytical solution to the problem of point-source radiation in the presence of a sphere with an eccentric spherical inclusion has been obtained by combined use of the dyadic Green's function formalism and the indirect mode-matching technique. The end result of the analysis is a set of linear equations for the vector wave amplitudes of the electric Green's dyad. The point source can be anywhere, even within the aforesaid nonspherical body, and there is no restriction with regard to the electrical properties in any part of space. Several checks confirm that this solution obeys the energy conservation and reciprocity principles. Numerical results are presented for an electric Hertz dipole radiating from within an acrylic sphere, which contains an eccentric spherical cavity.     

Electromagnetic scattering from a multilayered sphere located in an arbitrary beam

A solution is given for the problem of scattering of an arbitrary shaped beam by a multilayered sphere. Starting from Bromwich potentials and using the appropriate boundary conditions, we give expressions for the external and the internal fields. It is shown that the scattering coefficients can be generated from those established for a plane-wave illumination. Some numerical results that describe the scattering patterns and the radiation-pressure behavior when an incident Gaussian beam or a plane wave impinges on a multilayered sphere are presented.     

Electromagnetic-wave scattering by a sphere with multiple spherical inclusions

An exact solution to the problem of electromagnetic-wave scattering from a sphere with an arbitrary number of nonoverlapping spherical inclusions is obtained by use of the indirect mode-matching technique. A set of linear equations for the wave amplitudes of the electric field intensity throughout the inhomogeneous sphere and in the surrounding empty space is determined. Numerical results are calculated by truncation and matrix inversion of that set of equations. Specific information about the truncation number pertaining to the multipole expansions of the electric field intensity is given. The theory and the accompanying computer code successfully reproduce the results of other pertinent papers. Some numerical results [Borghese et al., Appl. Opt. 33, 484 (1994)] were not reproduced well, and that discrepancy is discussed. Our numerical investigation is focused on an acrylic sphere with up to four spherical inclusions. This is the first time that numerical results are presented for a sphere with more than two spherical inclusions. Interesting remarks are made about the effect that the look direction and the structure of the inhomogeneity have on backscattering by the acrylic host sphere.     

Particle positioning from charge-coupled device images by the generalized Lorenz-Mie theory and comparison with experiment

Three-dimensional position and velocity information can be extracted by direct analysis of the diffraction patterns of seeding particles in imaging velocimetry with real-time CCD cameras. The generalized Lorenz-Mie theory is shown to yield quantitatively accurate models of particle position, such that it can be deduced from typical experimental particle images with an accuracy of the order of 20 mum and an error of 11 gray levels rms, data obtained by comparison of theoretical and experimental images. Both the theory and an experimental verification of the problem presented here are discussed.     

Light scattering from a sliced target through use of the internal field of infinite cylinders: comparison between Mie theory and a sliced sphere

Light scattering from a particle that can be sectioned into circular slices is calculated by performing a coherent integration of the internal field over the volume of the target. The internal field in each slice is taken to be the internal-field solution of an infinite cylinder of radius equal to the radius of the slice. It is shown that for a spherical scatterer with size parameters up to 1.4, the integration leads to results that are in good agreement with those predicted by the Mie theory. Thus, we show the remarkable result that the internal field from an infinite cylinder can reproduce scattering intensities for such a radically different shape as a sphere. This being the case, a wide variety of target shapes between a sphere and a cylinder should be open to evaluation by this technique. The approach also has the benefit of being computationally efficient, requiring a double integration of the internal field over a disk and then coherently adding these calculations. The computations demonstrated in this paper are performed relatively quickly on a computer such as the Macintosh Centris 650, and this efficiency allows us to obtain the scattered fields for many target shapes.     

Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid

The theory of an arbitrarily oriented, shaped, and located beam scattered by a homogeneous spheroid is developed within the framework of the generalized Lorenz-Mie theory (GLMT). The incident beam is expanded in terms of the spheroidal vector wave functions and described by a set of beam shape coefficients (G(m)(n),(TM),G(m)(n),(TE)). Analytical expressions of the far-field scattering and extinction cross sections are derived. As two special cases, plane wave scattering by a spheroid and shaped beam scattered by a sphere can be recovered from the present theory, which is verified both theoretically and numerically. Calculations of the far-field scattering and cross sections are performed to study the shaped beam scattered by a spheroid, which can be prolate or oblate, transparent or absorbing.     

Light scattering fluctuations of a soft spherical particle containing an inclusion

We numerically calculate the light scattering intensity fluctuations and the cross-polarization intensity fluctuations of optically soft spherical particles containing an eccentrically located spherical particle. In all cases the magnitude of the signals tends to increase with particle asymmetry. Such a system approximates a biological cell in solution.     

Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions

The generalized Lorenz-Mie theory describes the electromagnetic scattering of a Gaussian laser beam by a spherical particle. The most intensive computational aspect of the theory concerns the evaluation of the beam-shape coefficients in the general case of an off-axis location of the scatterer. These beam-shape coefficients can be computed starting from the set of beam-shape coefficients for an on-axis location by using the addition theorem for the spherical vector wave functions of the first kind under a translation of the coordinate origin.     

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.     

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.     

Detachment of an elastic matrix from a rigid spherical inclusion

An approximate theoretical treatment is given for detachment of an elastomer from a rigid spherical inclusion by a tensile stress applied to the elastomeric matrix. The inclusion is assumed to have an initially-debonded patch on its surface and the conditions for growth of the patch are derived from fracture energy considerations. Catastrophic debonding is predicted to occur at a critical applied stress when the initial debond is small. The strain energy dissipated as a result of this detachment, and hence the mechanical hysteresis, are also evaluated. When a reasonable value is adopted for Young's modulus E of the elastomeric matrix, it is found that detachment from small inclusions, of less than about 0.1 mm in diameter, will not occur, even when the level of adhesion is relatively low. Instead, rupture of the matrix near the inclusion becomes the preferred mode of failure at an applied stress given approximately by E/2. For still smaller inclusions, of less than about 1 m in diameter, rupture of the matrix becomes increasingly difficult, due to the increasing importance of a surface energy term. These considerations account for the general features of reinforcement of elastomers. Small-particle fillers become effectively bonded to the matrix, whereas larger inclusions induce fracture near them, or become detached from the matrix, at applied stress that can be calculated from the particle diameter, the strength of adhesion, and the elasticity of the matrix material.     

Off-axis scattering of an ultrasound bessel beam by a sphere

In this paper, the scattering of an ultrasound zero-order Bessel beam by a rigid sphere in the off-axis configuration is studied. The beam is described through the partial wave expansion. The beam-shape coefficients which represent the amplitude of each multipole mode of the partial wave expansion are computed by numerical quadrature. Calculations are presented for both near- and far-field off-axis scattering. The far-field scattering is examined in both Rayleigh and geometrical acoustic limits. Results demonstrate that the scattered pressure in the off-axis case may significantly deviate from that in the on-axis configuration. In addition, the directive pattern of the scattered pressure is highly dependent on the relative position of the beam to the sphere.     

Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration

Calculation of the radiation trapping force in laser tweezers by use of generalized Lorenz-Mie theory requires knowledge of the shape coefficients of the incident laser beam. The localized version of these coefficients has been developed and justified only for a moderately focused Gaussian beam polarized in the x direction and traveling in the positive z direction. Here the localized model is extended to a beam tightly focused and truncated by a high-numerical-aperture lens, aberrated by its transmission through the wall of the sample cell, and incident upon a spherical particle whose center is on the beam axis. We also consider polarization of the beam in the y direction and propagation in the negative z direction to be able to describe circularly polarized beams and reflected beams.     

Electromagnetic scattering by an infinite cylinder of material or metamaterial coating eccentrically a dielectric cylinder

The electromagnetic scattering by an infinite cylinder of dielectric material or metamaterial, coating eccentrically another infinite dielectric cylinder, is treated in this work. The problem is solved using classical separation of variables techniques. No use is made of the translational addition theorem. For small eccentricities h = d/a(? 1), where d is the distance between the axes of the cylinders and a the radius of the outer cylinder, we use instead the cosine and the sine laws to satisfy the boundary conditions at the surface of the outer cylinder. Keeping terms up to the order h2 we finally obtain exact, closed-form expressions for the expansion coefficients g(1) and g(2) in the relation S(h) = S(0)[1 + g(1)h + g(2)h2 + O(h3)], giving the scattered field and the scattering cross sections of the problem, where S(0) corresponds to the coaxial geometry, with h = 0 (d = 0). Both polarizations are considered for normal incidence. Numerical results are given for various values of the parameters, corresponding to materials or metamaterials. Our method is an alternative of the one using the translational addition theorem in the case of small eccentricities h.