Mie Series Research Articles

A meshless regularized method of fundamental solution for electromagnetic scattering problems of three-dimensional perfect electric conductor targets

In this study, a meshless regularized method of fundamental solution (RMFS) is proposed to solve the electromagnetic scattering problem for three-dimensional perfect electric conductor (PEC) targets. The method is based on integral equation and constructed solutions using de-singularized subtraction and add-back techniques to regularize the singularity and hyper-singularity of the fundamental solution. The performance of the RMFS in terms of convergence and error is evaluated and compared with the Mie series solution, MFS, and the EM simulation software FEKO on the PEC sphere. Finally, several test examples are given. The results show the potential of RMFS in solving three-dimensional electromagnetic scattering problems.

Analysis of electromagnetic scattering from hypersonic vehicle coated with non-uniform plasma sheath based on FVTD method

The finite volume time domain (FVTD) method is proposed to accurately calculate the electromagnetic (EM) scattering of hypersonic vehicles coated with non-uniform plasma sheath. The plasma sheath in the actual flight environment is accurately established by solving the Navier–Stokes equations and combining appropriate thermochemical models. The radar cross section (RCS) of metal and uniform medium coated metal targets calculated by FVTD are in good agreement with those simulated by the software FEKO (MoM) and the Mie series. The calculated RCS of the vehicle under the reentry condition is consistent with the flight test data, too. The calculated electron number density distribution is consistent with the flight experiment results. The body fitted structure EM grid is adopted. The EM grid independence of the target coated with plasma sheath is studied. Then, the scattering and plasma EM parameters of the vehicle during the actual reentry are studied. The reentry altitude is from 78 to 55 km, and the reentry velocity is from 6406 to 6350 m/s. With the reentry altitude decreasing, the plasma angular frequency and collision frequency increased gradually. The scattering of different incident conditions and different wall catalytic conditions is analyzed in depth. This paper provides a valuable reference for vehicles' detection, recognition, and stealth during reentry flights.

Method of Moments and T-Matrix Hybrid

Hybrid computational schemes combining the advantages of a method of moments formulation of a field integral equation and T-matrix method are developed in this article. The hybrid methods are particularly efficient when describing the interaction of electrically small complex objects and electrically large objects of canonical shapes such as spherical multilayered bodies where the T-matrix method is reduced to the Mie series making the method an interesting alternative in the design of implantable antennas or exposure evaluations. Method performance is tested on a spherical multilayer model of the human head. Along with the hybrid method, an evaluation of the transition matrix of an arbitrarily shaped object is presented and the characteristic mode decomposition is performed, exhibiting fourfold numerical precision compared to conventional approaches.

Exact Solution of New Magnetic Current Based Surface-Volume-Surface EFIE and Analysis of Its Spectral Properties

A novel magnetic current based Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE-M) is presented for the problem of scattering on homogeneous non-magnetic dielectric objects. The exact Galerkin Method of Moments (MoM) utilizing both the rotational and irrotational vector spherical harmonics as orthogonal basis and test functions according to the Helmholtz decomposition is implemented to solve SVS-EFIE-M analytically for the case of dielectric sphere excited by an electric dipole. The field throughout the sphere is evaluated and compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. This exact solution validates the rigorous nature of the new SVS-EFIE-M formulation. It also reveals the spectral properties of its individual operators, their products and their linear combination. The spectrum of the MoM impedance matrix is also obtained. It is shown that upon choosing basis and test functions in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}(S)$</tex-math></inline-formula> space and evaluating testing inner products in the same space, the MoM impedance matrix features bounded condition number with increasing order of discretization and/or at low frequencies. This makes the proposed SVS-EFIE-M formulation free of oversampling and low-frequency breakdowns giving it advantage both over its SVS-EFIE-J predecessor and classical double-source integral equations such as PMCHWT, Muller, and others suffering from this type of numerical instabilities inherent to their inferior spectral properties.

Analytic Solution of Surface–Volume–Surface Electric Field Integral Equation on Dielectric Sphere and Analysis of Its Spectral Properties

The exact solution of the surface–volume–surface electric field integral equation (SVS-EFIE) is presented for the problem of radiation in the vicinity of the homogeneous dielectric sphere. The solution is obtained via the Galerkin method of moments (MoM) utilizing rotational and irrotational complete sets of orthogonal vector spherical harmonics as basis and test functions according to the Helmholtz decomposition. In the case of radial or tangential electric dipole radiation, the electric field throughout the sphere evaluated via the analytic MoM solution of the SVS-EFIE is compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. The exact solution confirms and validates the rigorous nature of the SVS-EFIE formulation. It also reveals the spectral properties of its individual operators, their products, and their linear combination, as well as the spectrum of the MoM impedance matrix. It is shown that, upon choosing basis and test functions in the Sobolev space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{1/2}(S)$ </tex-math></inline-formula> and performing testing inner products evaluation in the space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{-1/2}(S)$ </tex-math></inline-formula> , with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> being the surface of the sphere, the MoM impedance matrix features bounded condition number with increasing order of discretization similar with analogous exact MoM solution of the surface EFIE (S-EFIE) on the perfectly electrically conducting (PEC) sphere.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part II: recursive computations by the boundary integral equation method

PurposeThe purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).Design/methodology/approachThe elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.FindingsAs the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

Permittivity Estimates of Dielectric Spheres Using Radar Scattering Measurements

Broadband coherent radar scattering measurements were used to estimate the complex permittivity of homogeneous isotropic spheres. A Nelder-Mead optimization minimized differences between measured coherent scattering data and a corresponding forward solution provided by the dielectric Mie series. The method was applied across a range of dielectric materials to include Teflon, polystyrene foam, alumina oxide, neoprene rubber, and shungite. Dielectric constant estimates were achieved to within 0.1-6.85% of published values using traditional waveguide or cavity approaches. The method could not satisfactorily estimate dielectric loss tangent for low loss materials (e.g., foam and Teflon) but increased in accuracy with increasing dielectric loss.

New Magnetic Current Based Surface-Volume-Surface Electric Field Integral Equation For Solution of Scattering Problems on Dielectric Objects

New Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE) based on magnetic current single source field representation (SVS-EFIE-M) is proposed for the problems of scattering on homogeneous non-magnetic dielectric objects. It's analytic MoM solution for a dielectric sphere is compared against the Mie series expansion to validate the exact nature of this new magnetic current based SVS-EFIE formulation. This analytic solution also allows to rigorously study spectral properties of the SVS-EFIE- M operators and to determine proper functional spaces in which MoM solutions of it should be sought in order to maintain bounded condition number of the pertinent matrix equation.

Nonparaxial Mie Theory of Image Formation in Optical Microscopes and Characterization of Colloidal Particles

We derive an explicit partial-wave (Mie) series for the image of a dielectric microsphere collected by a typical infinity-corrected microscope. We model the propagation of the illumination and scattered vector fields through the optical components of the microscope by using the angular-spectrum theorem with the help of Wigner rotation matrix elements, allowing us to identify the contribution from spin-orbit helicity reversal. We consider a high numerical aperture objective well beyond the validity range of the paraxial approximation. The spherical aberration introduced by refraction at the planar interface between the sample and the glass slide is fully taken into account. By comparing our theoretical model with images of colloidal particles placed at different positions with respect to the objective focal plane, we characterize their radii and refractive index. We employ polystyrene microspheres with a known refractive index in order to fit the transverse attenuation length describing the transmission loss of the scattered field. As an application, we measure the radius and refractive index of individual silica beads. We compare the result for the radius with an independent measurement using high-resolution scanning electron microscopy. To validate the result for the refractive index, we develop a second method, independent of the theoretical model, based on the image contrast in glycerin-water solutions. In all cases we find very good agreement between our method and the validation procedures. In addition, the nonparaxial theory provides a reliable description of the images found for all focal-plane positions and for both polystyrene and silica microspheres. Our approach allows a common optical microscope to be used to measure the refractive index and radius of spherical particles covering the entire size range from the Rayleigh regime to the ray optics one.

On the edge element boundary element method/finite element method coupling for time harmonic electromagnetic scattering problems

AbstractOver recent decades coupled MoM/FEM formulation emerged as dominant method for solving EM scattering problems over inhomogeneous domains. Initial efforts on coupling of edge element BEM with edge element FEM have lost momentum and somehow disappeared from recent literature, possibly because of mathematical and other issues. This article, however, presents correct coupling of edge element BEM with edge element FEM. Such coupling is convenient because edge basis functions are used for modeling both the surface and the interior of the problem. We show that singularity extraction is simpler when compared with singularity extraction in MoM/FEM. Because BEM part of edge element BEM/FEM formulation uses simpler Green's function, the method is easier to implement than MoM/FEM which uses Dyadic Green's functions. The accuracy of the method is demonstrated by comparison with Mie series and with MoM/FEM and MoM/VSIE methods. Convergence of the method with respect to number of surface and interior edges was investigated by comparing 20 computational models with Mie series. Computational results suggest that edge‐element BEM/FEM performs well for scattering problems involving objects with sharp corners. Finally, the ability of the method to cope with relatively large number of elements was demonstrated on the example of human head illuminated by 3.5 GHz EM plane wave.

Hybrid Surface Electric Field Volume Magnetic Field Integral Equations for Electromagnetic Analysis of Heterogeneous Dielectric Bodies With Embedded Electrically Conducting Structures

In this article, hybrid surface electric field volume magnetic field integral equations (SEVMIE) for electromagnetic analysis of heterogeneous dielectric bodies with embedded perfect electrical conductor (PEC) structures are presented. These are then solved using a quasi-exact vector moment method employing mixed order edge-based divergence conforming basis functions to expand the current on the PEC surface and node-based linear Lagrangian basis functions to expand the magnetic field within the dielectric region. For an incident plane wave scattered by the combined structure, this procedure results in a linear matrix equation for the surface currents and the volume magnetic fields. The method is demonstrated for three spherical test cases and the results compared to those obtained by a semi-analytic Mie series expansion. The performance is compared to that of the well-known volume surface integral equation (VSIE) method and found to provide more accurate results with approximately one-third the number of degrees of freedom. The method is then demonstrated on the scattering cross section calculation of a heterogeneous composite structure. This new method enables the efficient and accurate analysis of composite structures with large dielectric regions.

Algorithm 1009

MieSolver provides an efficient solver for the problem of wave propagation through a heterogeneous configuration of nonidentical circular cylinders. MieSolver allows great flexibility in the physical properties of each cylinder, and the cylinders may have opaque or penetrable cores, as well as an arbitrary number of penetrable layers. The wave propagation is governed by the two-dimensional Helmholtz equation and models electromagnetic, acoustic, and elastic waves. The solver is based on the Mie series solution for scattering by a single circular cylinder and hence is numerically stable and highly accurate. We demonstrate the accuracy of our software with extensive numerical experiments over a wide range of frequencies (about five orders of magnitude) and up to 60 cylinders.

Efficient computation of arbitrary beam scattering on a sphere

Electromagnetic scattering on a sphere is one of the most fundamental problems, which has a closed form analytical solution in the form of Mie series. Being initially formulated for a plane incident wave, the formalism can be extended to more complex forms of incident illumination. Here we present a fast calculation approach to address the scattering problem in the case of arbitrary illumination, incident on a spherical scatterer. This method is based on the plane wave decomposition of the incident illumination and weighted integration of Mie solutions, rotated to a global coordinate system. Tabulated solutions, sampled with an accurately level of sparsity, and efficient rotational transformations allow performing fast calculations on electrically large structures, outperforming capabilities relatively to other numerical techniques. Our approach is appropriate for real-time analysis of electromagnetic scattering from electrically large objects, which is essential for monitoring and control applications, such as optomechanical manipulation, scanning microscopy, and fast optimization algorithms.

Numerically stable formulation of Mie theory for an emitter close to a sphere.

Numerical implementations of Mie theory make extensive use of spherical Bessel functions. These functions are, however, known to overflow/underflow (grow too large/small for floating point precision) for orders much larger than the argument. This is not a problem in applications such as plane wave excitation, as the Mie series converge before these numerical problems arise. However, for an emitter close to the surface of a sphere, the scattered field in the vicinity of the sphere is expressed as slowly converging series, with multipoles up to order 1000 required in some cases. These series may be used to calculate experimentally relevant quantities such as the decay rate of an emitter near a sphere. In these cases, overflow/underflow prevents any calculation in double precision using Mie theory, and alternatives are either computationally intensive (e.g., arbitrary precision calculations) or not accurate enough (e.g., the electrostatics approximation). We present here a formulation of Mie theory that overcomes these limitations. Using normalized Bessel functions where the large growth/decay is extracted as a prefactor, we re-express the Mie coefficients for scattering by spheres in a normalized form. These normalized expressions are used to accurately compute the series for the electric field and decay rate of a dipole emitter near a spherical surface, in cases where the Mie coefficients would normally overflow before any degree of accuracy can be obtained.

A Ternary Parallelization Approach of MLFMA for Solving Electromagnetic Scattering Problems With Over 10 Billion Unknowns

In this paper, we present a flexible and efficient ternary parallelization approach for the multilevel fast multipole algorithm (MLFMA) for the solution of extremely large 3-D scattering problems with over 10 billion unknowns. In the ternary parallelization approach, a binary clustering tree of all the message passing interface (MPI) processes is built first. Then, the MLFMA tree is categorized into plane wave partitioning, hierarchical-structure partitioning, and box partitioning (BP) levels via a top–down approach to match the process clustering tree. Compared with the bottom–up approach, interprocess communications in the far interaction are reduced. An inner-group transition level is designed for switching partitions on the intermediate level between the hierarchical-structure partitioning and BP levels. The ternary strategy can realize as high parallel efficiency as the hierarchical partitioning strategy while maintaining flexibility in selecting the total number of processes. An auxiliary-tree-based parallel mesh refinement technique and a hybrid octree storage strategy are designed to facilitate the realization of fast full-wave simulation on a scale of over 10 billion unknowns with the ternary MLFMA. The accuracy of the solutions is validated by comparing the radar cross sections (RCSs) of a sphere of diameter 3042 wavelengths with 10 773 415 728 unknowns via the MLFMA and the Mie series, which is the largest number of unknowns to be computed via full wave numerical methods to date. Furthermore, the solutions for complicated objects, namely a ship and an aircraft model, both of which have maximum lengths that exceed 10 000 wavelengths and over 10 billion unknowns, are presented.

A new approach to vector scattering: the 3s boundary source method.

This paper describes a novel Boundary Source Method (BSM) applied to the vector calculation of electromagnetic fields from a surface defined by the interface between homogenous, isotropic media. In this way, the reflected and transmitted fields are represented as an expansion of the electric fields generated by a basis of orthogonal electric and magnetic dipole sources that are tangential to, and evenly distributed over the surface of interest. The dipole moments required to generate these fields are then calculated according to the extinction theorem of Ewald and Oseen applied at control points situated at either side of the boundary. It is shown that the sources are essentially vector-equivalent Huygens' wavelets applied at discrete points at the boundary and special attention is given to their placement and the corresponding placement of control points according to the Nyquist sampling criteria. The central result of this paper is that the extinction theorem should be applied at control points situated at a distance d = 3s (where s is the separation of the sources) and consequently we refer to the method as 3sBSM. The method is applied to reflection at a plane dielectric surface and a spherical dielectric sphere and good agreement is demonstrated in comparison with the Fresnel equations and Mie series expansion respectively (even at resonance). We conclude that 3sBSM provides an accurate solution to electromagnetic scattering from a bandlimited surface and efficiently avoids the singular surface integrals and special basis functions proposed by others.

Validation of a volume integral equation method for indoor propagation modelling

A full-wave indoor propagation model based on the two-dimensional (2D) volume electric field integral equation (VEFIE) is presented. The model is discretised with the method of moments (MoM). The linear system arising from the MoM discretisation is solved with an iterative solver accelerated by using the fast Fourier transform and a reduced operator. The 2D VEFIE is validated against the analytical Mie series solution for scattering from a dielectric cylinder before being applied to a typical indoor propagation environment and compared against measurements. It is also compared against current popular empirical models and ray tracing. The ability of the frequency domain VEFIE method to produce time domain information is demonstrated with an investigation of numerical considerations that must be made.

Detection of multiple impedance obstacles by non-iterative topological gradient based methods

We investigate a fast, one-step imaging method of multiple 2D and 3D acoustic obstacles fully-coated by a complex surface impedance with either monochromatic or multi-frequency noisy data. Introducing the topological gradient of the misfit functional as a limit of shape derivatives, closed-form expressions of the obstacle indicator are derived using Fourier and Mie series expansions of the radiating solution. We provide a wide variety of numerical experiments that assesses the performance and limitations of the one step single and multi-frequency imaging strategies when dealing both with full and limited aperture measurements.

$\mathcal {H}$-Matrix Accelerated Solution of Surface–Volume–Surface EFIE for Fast Electromagnetic Analysis on 3-D Composite Dielectric Objects

An efficient fast direct algorithm based on the hierarchical ( $\mathcal {H}$ -) matrices is presented for solution of the radiation problems on piecewise homogeneous dielectric objects using Method of Moment (MoM) discretization of the surface–volume–surface electric field integral equation (SVS-EFIE). The SVS-EFIE for the composite objects introduces independent surface electric current density on the boundary of each region. Therefore, different from the traditional Poggio–Miller–Chang–Harrington–Wu–Tsai formulation, in the SVS-EFIE, the object regions can be meshing independently according to their local properties which improves the flexibility and efficiency of the proposed method. It also makes the proposed algorithms appropriate for the analysis of both multiscale and large-scale composite structures. The numerical results from the proposed fast method are provided for the high-loss biological tissues from bioelectromagnetics applications and agree well with the analytical Mie series solution and commercial software. The CPU time and memory cost of the required $\mathcal {H}$ -matrix operations are analyzed in details and verified through several numerical experiments. The new computational framework allows for fast direct solution of 3-D radiation and scattering problems of moderate electrical size with $O(P^{\alpha } \log ^2 P)$ CPU time and $O(P^{\alpha } \log P)$ memory complexity, $P$ being the number of surface unknowns produced by the MoM discretization, and $1\leq \alpha \leq 1.5$ being a geometry-dependent parameter.

Formulation of Surface-Volume-Surface-EFIE for Solution of Three-Dimensional Scattering Problems on Composite Dielectric Objects

A novel formulation of the surface-volume-surface electric field integral equation (SVS-EFIE) is introduced for the solution of the scattering and radiation problems on the composite dielectric objects made of arbitrary number of piecewise homogeneous nonmagnetic dielectric regions. The regions composing the scatterer may or may not share common boundaries. Due to the fact that SVS-EFIE introduces the independent fictitious surface electric current densities on the boundary of each region, the proposed formulation allows for independent meshing of the scatterer regions according to its permittivity. As a result, the new SVS-EFIE formulation presents no complications in method-of-moments discretization at the material junctions. The proposed formulation exhibits the same CPU time and the memory complexities as traditional surface integral equation formulations when fields throughout the scatterer are to be computed. The new equation is validated through comparison of its solution against the Mie series and fields computed using the commercial software.