Mie Series Research Articles

On Microwave Experiments in Dielectric Media to Simulate Nuclear Volumetric Heating

The use of microwave experiments in normal fluids is proposed for the approximation of the volumetric heating distribution in cryogenic flow of radiation-heated deuterium in the advanced neutron source (ANS) cold neutron source (CNS) planned for construction at Oak Ridge National Laboratory. The potential of such experiments is investigated by solving Maxwell's equations for microwave propgation and absorption in several noncryogenic model fluids. Included are an analytical Mie series solution for an idealized ANS CNS geometry and a solution in a more complex and realistic geometry by the computational finite difference time domain (FDTD) technique. Though data and anecdotal evidence suggest difficulty with specifying a given volumetric heating distribution in a boiling liquid in a microwave cavity, the computational results suggest that CNS-like heating distributions can be obtained by using microwave irradiation. Two aspects of microwave heating are examined. The first is scale dependence of heating across various fluid particle sizes in a potentially complex multiphase flow, and the second is detailed heating distribution across a realistic three-dimensional ANS CNS geometry. By using a Mie series solution in spherical geometries to indicate dependence of microwave heating on fluid particle size for flow scales relevant to ANS CNS flows, several fluids, including n-propanol and n-butanol, are found to show < 20% variation in heating on scales from 0.01 down to 10 -7 m. By using FDTD computations, the expected ANS heating distribution in liquid deuterium is compared with heating distribution under microwave irradiation for several different model fluids. Good qualitative agreement is found between expected ANS heating distribution and microwave heating in the n-propanol and n-butanol fluids including the heating asymmetry expected in ANS CNS flows. By using this simulated heating distribution, volume-heated flow can be investigated. Expected results from such an investigation include flow regime determination, effects of nucleation phenomena, and other physical characteristics such as heating distribution, container shape, and fluid properties

The solution of 3D multiple scattering problems using the boundary element method

The electromagnetic multiple scattering from lossless and lossy three-dimensional objects of arbitrary shape is computed using the boundary element method. Three different formulations starting from a combination of the magnetic field and the electric field integral equation are investigated. The influence of the material properties and the discretization on the solution accuracy is discussed by comparing the numerical results with analytic ones for spheres using the Mie series solution. As an example, the numerical solution of the electromagnetic scattered field of a rain drop model is presented. >

Computation of electromagnetic fields inside strongly inhomogeneous objects by the weak-conjugate-gradient fast-Fourier-transform method

The computation of the electromagnetic field inside a strongly inhomogeneous dielectric object is formulated in terms of a domain-integral equation over the object. We discuss a weak form of the integral equation in which the spatial derivatives are integrated analytically. Doing so, we obtain an equation that is solved efficiently with the advantageous combination of a conjugate-gradient iterative method and a fast-Fourier-transform technique. Numerical computations are performed for a strongly inhomogeneous lossy sphere. For this case we compare the accuracy and the efficiency of the present method with the analytic solution based on the Mie series and the finite-difference time-domain approach. To show that the method is also capable of computing more-complex scattering problems, we assume the incident field to be generated by a (1/2)λ thin-wire dipole. In this case the absorbed power density is presented. All these test cases demonstrate that the weak form of the conjugate-gradient fast-Fourier-transform method can be considered as a comparatively simple and efficient tool for solving realistic electromagnetic wave-field problems.

Boundary element computations of 3D transient scattering from lossy dielectric objects

A method for computing the transient scattering from arbitrarily shaped lossy dielectric objects is presented. Starting from Green's function for lossy media and the surface integral equation formulation of electromagnetic scattering, a combination of integral equations which is capable of dealing with the scattering process in the time domain without instability is given. The discretization of the scatterers using eight-noded boundary elements is discussed. Semidiscontinuous boundary elements are used for objects with edges and vertices. Some examples show the applicability of the method. The results are compared with the analytical solution obtained via the Mie series.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Calcul des champs électromagnétiques diffractés par un obstacle diélectrique tridimensionnel par éléments de frontière en régime transitoire

This paper presents an overview of some time-domain boundary integral methods for the computation of transient scattering of electromagnetic waves from homogeneous dielectric targets of arbitrary shape. The problem is described via the combination of the magnetic field (MFIE) and the electric field integral equation (EFIE). The discretization of the scatterers' surface by means of eight-noded boundary elements is discussed and a numerical solution is given. As an extension of a program package for perfectly conducting bodies with edges or vertices [4], a recently proposed technique [3] is applied which uses eight-noded semidiscontinuous elements. The results are compared with the Mie series solution and results from the literature [2]

Combined field integral equation formulation for inhomogeneous two- and three-dimensional bodies: the junction problem

A combined field integral equation (CFIE) formulation is presented for two- and three-dimensional bodies having discrete dielectric and conducting regions. A three-dimensional case is restricted to bodies of revolution (BORs). The two-dimensional case is analogous to the BOR case when the Fourier mode number is zero. The method of moments (MM) is used to solve the CFIE in terms of two integral operators. It is shown that the CFIE formulation yields accurate answers for scattering problems where the scatterer may be internally resonant. The CFIE results were validated using Mie series results and measured data. The junction problem associated with the CFIE-based formulation is explicated, and several geometries with multiple junctions are used to validate the general CFIE formulation. A number of configurations are tested where the penetrable region consisted of a free-space coating. Extensive numerical studies have shown that such limiting cases are sensitive indicators of the stability of MM solutions and allow direct comparison of different configurations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Importance of normal field continuity in inhomogeneous scattering calculations

The finite-element method with conventional scalar bases is coupled with the moment method to handle the three-dimensional scattering and/or absorption from inhomogeneous, arbitrarily shaped objects. The C/sup 0/ finite-element basis enforces continuity of both normal and tangential E at element boundaries within homogeneous regions. At dielectric interfaces, the continuity of normal D and tangential E are enforced in a strong sense. Excellent agreement between the numerical solution and the Mie series is obtained for both internal and scattered fields for homogeneous and layered spheres under plane wave illumination. Compared to an alternative finite-element method using edge elements which lack strong enforcement of normal field continuity, the present method produces higher-order approximations, especially at dielectric interfaces, with no penalties in computational effort.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Extinction and scattering by soft spheres

On the basis of an approximate relation, the Mie series are replaced by new ones, which can be summed exactly with the aid of various modified and generalized forms of the addition theorem for cylindrical functions. The sums obtained simultaneously simplify the initial expressions for the scattering characteristics and preserve their analytical nature. The conventional approximations for the amplitude functions and the efficiency factors rigorously follow from the new approaches if the optical parameters are properly restricted. The acceptable domains of these approaches contain the long wavelength region and are extensive enough to study the various (including inverse) scattering problems for the real disperse systems, in which the particles are suspended in a medium with similar optical properties.

Scattering of a transient electromagnetic wave by a dielectric sphere

The transient scattered field from a lossless dielectric sphere is evaluated by the inverse Laplace transform of the frequency-domain solution. Two methods of obtaining the inverse transform are employed. In the first approach, direct numerical evaluation of the Bromwich integral is carried out. To overcome the slow convergence of the Mie series at high frequencies (HF), asymptotic expressions are used in the large | s | portion of a properly chosen Bromwich contour, allowing a significant reduction of the computation time. In the second approach, the asymptotically compensated singularity expansion method is employed. In this method, a multiplicative function is chosen to compensate for the asymptotic growth of the field in the right half of the s–plane, leading to a singularity expansion without entire functions. The transient scattered field is then recovered. The numerical results obtained from both methods are in complete agreement. Reference to the physical origins of the various structures appearing in the time-domain response is made by considering the dependence of their magnitude and time of arrival on the permittivity. Verification of the various predictions previously given by frequency-domain analysis can be carried out in a different perspective with new physical insight.

Scattering by a sphere with an anisotropic impedance boundary condition: Mie series solution and uniqueness conditions

The Mie series solution for the scattered field of a sphere with an anisotropic impedance boundary condition (IBC) is derived along with the corresponding scattering cross section. Conditions on the impedance that ensure a unique solution for the scattered field are derived. The anisotropic IBC is derived for a material with an anisotropic electric permittivity and a large anisotropic conductivity. For the special case of an isotropic IBC, a method is presented for testing whether any given impedance leads to a unique solution for the scattered field, and it is shown that, for every incident wave frequency and mode number l, there are exactly two values of the impedance for which the scattered field is not unique.

Three-dimensional electromagnetic scattering from inhomogeneous objects by the hybrid moment and finite element method

The hybrid moment and finite element methods are used to obtain 3-D scattering and/or absorption from inhomogeneous, arbitrarily shaped objects. The surface of the object is approximated by triangles and the volume of the object is approximated by tetrahedrons. The electrical parameters are assumed constant in each tetrahedron. The Galerkin testing procedure is used. To avoid contaminations of spurious mode, a divergenceless vector basis function is used in finite elements. The calculated internal field and scattered field for a homogeneous sphere, a layered sphere, and a lossy prolate spheroid are compared with Mie series solutions and other numerical techniques. The accuracy and rate of convergence of the solution are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Complex resonances of conducting spheres with lossy coatings

The electromagnetic scattering amplitude for metal objects coated with a lossless dielectric exhibits a large number of resonances versus real frequency. These resonances are a function of the object shape and size, coating thickness, and coating electrical properties. Previously, it was shown that for coated spheres and nonspherical bare objects, these resonances can be understood in terms of phase matched circumnavigating surface waves and the objects' complex eigenfrequencies. The effect of dielectric loss in the coating on the complex eigenfrequencies and phase velocities of these surface waves is presented for a metal sphere coated with a uniform homogeneous dielectric coating. It is seen that the positions of the complex resonance frequencies move away from the real frequency axis under the influence of dielectric loss in the coating. The effect of this is shown to correspond to the changes in the backscatter spectrum versus real frequency as computed using a Mie series expansion. The significant difference between the lossless and lossy dielectric-coated sphere cases is seen to be due to a modification of the phase velocities of surface waves, particularly the whispering gallery types, and the increased attenuation of the surface wave modes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Scattering from arbitrarily-shaped objects with impedance boundary conditions

A numerical method is developed to determine the electromagnetic field scattered by homogeneous imperfectly conducting and impedance objects of irregular shape. An integral equation for the induced surface currents on the body surface is derived and solved numerically by the method of moments. Numerical results are given for a sphere with different surface impedances and for lossy finite circular and elliptic cylinders. The results are compared with the Mie series solution for spheres and with the integral equation formulation of bodies of revolution for the finite circular cylinder.

Simplified Mie series calculation of dipole moments for artificial dielectric constituents

Artificial dielectrics are composed of particles embedded in a binding material that fills all of the space between the particles. They have homogeneous macroscopic constitutive parameters but significantly inhomogeneous microscopic properties. The macroscopic electric and magnetic polarizations of such a composite material can often be expressed in terms of the effective dipole moments of its constituents. The dipole moments of spherical multilayered particles can be calculated exactly in terms of Mie series. Except for very simple cases, the calculation of dipole moments by Mie series requires the use of significant computer resources to evaluate complex argument special functions. To alleviate this situation, closed-form solutions for spherical particles have been derived in terms of elementary functions for the case of a semiconductive or reactive coating on a dielectric core. If k is the propagation constant and r is the radius of curvature, the product kr must be greater than 10 for the coating and less than (1)/(2) for the core and surrounding media. These solutions treat the important case of a coating which does not fully shield the core, such that the core properties must be included. The dipole moments calculated by these equations show very close agreement with exhaustive Mie series results. These equations have also proven useful in evaluating Mie series of large complex arguments. The equations are very easily implemented on a desktop computer and facilitate investigation of the mutual coupling between the imposed electric and magnetic fields and the resulting electric and magnetic dipole moments.

Diffraction of an electromagnetic wave at a sphere. Summation of Mie series

The authors suggest a new method for the summation of functional series which diverge absolutely at the edge of the convergence circle. The essence of the method is based on the multiplicative weakening of the logarithmic and power law singularities of functions which are represented by the series. By way of example we sum the Mie series which determine the diffraction of an electromagnetic wave at a sphere when the source and observation points are positioned on the surface of the sphere. It is shown that a double transformation of the original series is, with a good reserve, sufficient for practical purposes.

Identification of resonances of an acoustically rigid sphere

Identification of resonances of an acoustically rigid sphere from simulated input-output data is presented. Scattering from the sphere is formulated using the classic Mie series and the singularity expansion method (SEM). A parametric inverse based on the SEM is discussed. The scattered velocity potential is divided into reflection, first creeping-wave, and second creeping-wave components. The effects on the identification of removing various components of the scattered potential are shown, along with the effects of adding small amounts of noise.

Asymptotic model-based identification of an acoustically rigid sphere

Scattering from an acoustically rigid sphere is formulated using the classic Mie series and the asymptotic series approximation to the reflection. A parametric inverse based on the asymptotic series is discussed, with attention given to the effects of additive noise.

A simple and accurate resonance expansion for the electromagnetic field scattered by a Lossy Dielectric Sphere

A simple and accurate resonance expansion for the space-frequency representation of the electromagnetic field scattered by a homogeneous dielectric sphere with loss is discussed. Results are presented which establish the accuracy of the resonance expansion by comparing with the corresponding Mie series.

Numerical solution of scattering from coated bodies of revolution using different integral equation formulations

The electromagnetic scattering from coated conducting bodies of revolution is computed using the moment method. The equivalent principle is used to generate different formulations for the problem. These formulations are then used to compare the numerically determined solutions with the analytic ones, for coated spheres using the Mie series solution. It is shown that the solution accuracy generally depends on the formulation type. The method is then used to determine the solution for other rotationally symmetric geometries. The results indicate that the resonance effects are less severe for nonspherical geometries.

Relation between the ringing of resonances and surface waves in radar scattering

The exact normal-mode or Mie series for electromagnetic scattering from a conducting sphere has in the previous literature been transformed via the Watson transformation into components corresponding to specularly reflected and to creeping waves, or via the singularity expansion method (SEM) into a series of pole contributions in the complex frequency plane which in the time domain give rise to a series of damped sinusoidal signals. In this work, the connection between the two methods is established by using the Watson transformation for obtaining the specular wave, and by transforming the remainder into the SEM series. From the latter, we obtain the shapes of creeping-wave pulses for the case of an incident \delta -function pulse by evaluating the series via the stationary-phase method.