Volume Equivalence Principle Research Articles

A Review on the Application of Integral Equation‐Based Computational Methods to Scattering Problems in Plasmonics

AbstractMany computational methods have been developed and used for modeling, understanding, and tailoring extreme optical effects at the nanoscale. Among them, this review focuses on the integral equation‐based methods: within the local response limit, a potential‐based boundary integral equation (BIE) formalism and a field‐based volume integral equation (VIE) formalism; within the nonlocal hydrodynamic model, a potential‐based BIE formalism. These formalisms are derived from macroscopic electrodynamics (together with appropriate constitutive relations). The derivations are based on three pillars: the Green function, the field relation(s) (for the VIE formalism, the incident—scattered—total field relation; for the BIE formalism, the interface conditions connecting the fields at two sides of the interface), and the field equivalence principle (for the VIE formalism, the volume equivalence principle; for the BIE formalism, the Huygens principle). By applying the method of moments (MoM) algorithm, the derived integral equations are converted into matrix equations, with possible problems in the implementation being discussed. Levels of solutions, including the eigenmode and natural mode solutions, and group representation theory are introduced as powerful post‐processing steps. Many examples are shown to demonstrate the effectiveness of the reviewed algorithms.

Accelerated excitation‐independent characteristic basis function method with multiscale adaptive cross approximation for monostatic radar cross section of dielectric objects

AbstractAn accelerated excitation‐independent characteristic basis function method (CBFM) combined with the multiscale adaptive cross approximation (MSACA) is proposed to analyze electromagnetic scattering from dielectric objects with volume equivalence principle. In CBFM, the adaptive cross approximation (ACA) is hybridized to improve the filling of mutual‐impedance matrices, except for the mutual‐impedance matrices of adjacent blocks, which are calculated by the time‐consuming method of moments (MoM). In this study, a new hybrid CBFM with MSACA is proposed to improve the filling of the mutual‐impedance matrix, which achieves excellent effect. Moreover, an accelerated approach is adopted to remove the redundant excitations in advance by the ACA‐singular value decomposition, leading to great reduction on the CPU time of the generation of the characteristic basis functions. The validity of the proposed method is illustrated with the systematic analysis of canonical geometries, and numerical results are given to demonstrate its efficiency and accuracy.

Electromagnetic scattering analysis of normal chiral, metamaterials chiral and chiral nihility materials

ABSTRACTScattering of electromagnetic plane waves from normal chiral materials, chiral metamaterials, chiral nihility materials are presented. Based on the volume equivalence principle, the integral equations are represented in terms of a pair of coupled bi-isotropic polarized volume electric and magnetic flux densities. The volume integral equations (VIE) are solved using the method of moments (MoM) combined with tetrahedral mesh. Then the adaptive cross approximation (ACA) algorithm is combined with the fast dipole method (FDM) based on the equivalent dipole method (EDM) is first extended to analyze the scattering of chiral materials by solving the VIE. The co-polarized and cross-polarized bistatic radar cross-scattering (RCS) of normal chiral media, chiral metamaterials and chiral nihility materials are given, and the scattering properties of them are discussed.

A Method of Moments for Analysis of Electromagnetic Scattering From Inhomogeneous Anisotropic Bodies of Revolution

This paper proposes an efficient method to analyze scattering from an inhomogeneous body of revolution (BOR) with anisotropic electromagnetic properties. Both the permittivity and the permeability are assumed to be generalized tensors while there are no restrictions on the geometry of the BOR. The proposed method involves three stages. First, the volume equivalence principle is used to replace the BOR with volume electric and magnetic current densities. Requisite volume integral equations (VIEs) are, then, derived with the electric and magnetic flux densities inside the BOR being the unknown quantities. Finally, the VIEs are solved by Galerkin’s method of moments. This is done by expanding the unknown flux densities in terms of appropriate basis functions compatible with the problem symmetry, and hence, reducing the integral equations to a matrix equation. The formation of the resultant matrix equation is elucidated in detail, whereby it is shown that the computation burden is remarkably reduced as no double volumetric-type integrals are to be computed. The efficiency of the proposed method is demonstrated by comparing the simulation results of several case studies with those available in the literature, or computed by commercial numerical codes.

Simulation of the Scattered Field From a Vibrating Tumor Inside the Tissue Using 3D-FDTD Method

A simulation method for calculating the scattered field from a small, vibrating breast tumor is given. The Volume Equivalence Principle together with subcell type finite difference time domain formulation is used for obtaining the scattered field from the tumor. The Doppler component of the scattered field is calculated using the simulation results for the undisplaced and the displaced positions of the tumor. The method is validated by comparing the results obtained on a sample problem with the results of a semi-analytical analysis.

COMPREHENSIVE SOLUTION TO SCATTERING BY BIANOSOTROPIC OBJECTS OF ARBITRARY SHAPE

This paper presents a method of moments (MoM) solution for the problems of electromagnetic scattering by inhomogeneous three- dimensional bianisotropic scatterers of any shape. The electromagnetic response of bianisotropy has been described by the constitutive relations of the most general form composed of four 3 £ 3 matrices or tensors. The volume equivalence principle is used to obtain a set of mixed potential formulations for a proper description of the original scattering problem. Here, the total flelds are separated into the incident flelds and the scattered flelds. The scattered flelds are related to the electric and magnetic potentials which are excited by electric and magnetic bound charges and polarization currents. The body of the scatterer is meshed through the use of tetrahedral cells with face-based functions used to expand unknown quantities. At last, the Galerkin test method is applied to create a method of moments (MoM) matrix from which the numerical solution is obtained. Implemented in a MATLAB program, the numerical formulation is evaluated and verifled for various types of scatterers. The results are compared with those of previous work, and a good agreement is observed. Finally, a scattering from a two-layered dispersive chiroferrite sphere is presented as the most general example.

A Marching-on-in-Degree Solution of Volume Integral Equations for Transient Electromagnetic Scattering by Bi-Isotropic Objects

A time-domain volume integral equation solved by the marching-on-in-degree scheme is presented for the analysis of transient electromagnetic scattering from three-dimensional inhomogeneous bi-isotropic objects. Based on the volume equivalence principle, the volume-based coupled integral equations for electric and magnetic flux densities are formulated. The transient variations of the electric and magnetic flux densities are expanded in terms of weighted Laguerre polynomials. With the use of the Galerkin spatial and temporal testing procedure, the proposed algorithm overcomes the late-time instability problem that often occurs in the marching-on-in-time approach. Numerical results are presented to illustrate the accuracy and versatility of the proposed algorithm.

An Efficient Volume Integral Equation Solution to EM Scattering by Complex Bodies With Inhomogeneous Bi-Isotropy

A generalized volume integral equation method is formulated for electromagnetic scattering by arbitrarily shaped complex bodies with inhomogeneous bi-isotropy. Based on the volume equivalence principle, the integral equations are represented in terms of a pair of coupled bi-isotropic polarized volume electric and magnetic flux densities. Reduction of the integral equations into the corresponding matrix equations is obtained using the method of moments (MoM) combined with the tetrahedral mesh. In the MoM solution, the three-dimensional solenoidal function is incorporated as the basis function defined over each tetrahedral element and the details of implementation, particularly the treatment of integral singularities, will be elucidated. The efficiency and accuracy of the proposed method are validated by illustratively supported examples.

Scattering from 3-D Inhomogeneous Chiral Bodies of Arbitrary Shape by the Method of Moments

A method of moments (MoM) solution is presented for the problem of electromagnetic scattering by a three-dimensional (3D) inhomogeneous chiral scatterer illuminated by an arbitrary incident field. The volume equivalence principle was used to obtain coupled integral equations for equivalent volume currents. These integral equations were then solved numerically using MoM. The volume of the scatterer was modeled by tetrahedral cells, and face-based expansion functions were used to approximate the equivalent currents. Computed results are in very good agreement with exact data or other published data.

CG‐FFT algorithm for EM scattering by small dielectric particles with high permittivity and permeability

AbstractWe have proposed a fast algorithm to study the electromagnetic (EM) scattering problems involving three‐dimensional dielectric targets with inhomogeneous permittivity and permeability. Based on the volume equivalence principle, coupled volume integral equations are derived for the equivalent electric and magnetic current densities. Such integral equations are then solved using the conjugate gradient method combined with the fast Fourier transform (FFT), where the memory requirement is of the order N ‐ the number of unknowns and the computational complexity is of the order N log N. The accuracy and efficiency of the proposed algorithm are verified by comparing numerical results with analytical results from the Mie series for dielectric spheres. We have applied the algorithm to investigate the EM scattering by small dielectric particles with high permittivity and permeability, which may be important to realize the artificial left‐handed materials. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 305–310, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22116

Fast Multipole Acceleration of a MoM Code for the Solution of Composed Metallic/Dielectric Scattering Problems

Abstract. An existing method of moments (MoM) code for the solution of complex scattering bodies has been accelerated by means of a multilevel fast multipole method (MLFMM). We demonstrate the usage of this technique both for metallic structures (wires and surfaces) and for dielectric bodies (volume and surface equivalence principle). Aspects like the effect of the type of integral equation, preconditioning schemes, or iterative solution techniques are discussed. But also limitations are addressed, which are encountered when for instance attempting to model highly lossy dielectric bodies with a high permittivity. Several validation and application examples demonstrate the usefulness of this method, both with regard to the obtained accuracy, but also with respect to the potential saving in memory and run-time as compared to a standard MoM formulation.

A fast time domain integral equation based scheme for analyzing scattering from dispersive objects

A fast integral equation-based scheme for analyzing transient scattering from inhomogeneous dispersive bodies is presented. A time domain integral equation in terms of the electric flux inside the body is constructed by invoking the volume equivalence principle, i.e., by equating the sum of the incident electric field and that radiated by equivalent volume polarization currents to the total electric field. The proposed algorithm for solving this time domain integral equation incorporates a recursive convolution scheme that updates the polarization current from knowledge of the electric flux. To reduce the computational cost and memory requirement of the resulting marching-on-in-time scheme, it is augmented with the plane wave time domain algorithm. Numerical results that validate the proposed approach and demonstrate its accuracy are presented.

Two‐dimensional time‐domain volume integral equations for scattering of inhomogeneous objects

This paper proposes a time‐domain volume integral equation based method for analyzing the transient scattering from a two‐dimensional inhomogeneous cylinder by invoking the volume equivalence principle for both the transverse magnetic and electric cases. The cylinder is discretized into triangular cells, and the electric flux is chosen as the unknown. For the transverse magnetic case, the electric flux is defined on the surfaces of the triangles. For the transverse electric case, because of the electric charges induced inside and on the surface of the cylinder, the electric flux is defined on the edges of the triangles, and expanded in space in terms of two‐dimensional surface roof‐top basis functions. The time‐domain volume integral equation is solved by using a marching‐on‐in‐time scheme. Numerical results obtained using this method are in excellent agreement with the data obtained using the finite‐difference time‐domain method.

Time-domain volume integral equation for transient scattering from inhomogeneous objects—2D TE case

This letter proposes a time-domain volume integral equation based method for analyzing the transient scattering from a 2D inhomogeneous cylinder by involking the volume equivalence principle for the transverse electric case. This integral equation is solved by using an MOT scheme. Numerical results obtained using this method agree very well with those obtained using the FDTD method.

Volume‐integral‐equation‐based analysis of transient electromagnetic scattering from three‐dimensional inhomogeneous dielectric objects

A novel technique for analyzing transient electromagnetic scattering from three‐dimensional inhomogeneous dielectric targets is proposed. An integral equation for the electric flux density throughout the scatterer is constructed by invoking the electromagnetic volume equivalence principle. This equation is solved using a marching‐on‐in‐time scheme in which the electric flux density is expanded in space by volumetric rooftop basis functions defined on a tetrahedral mesh and in time by piecewise polynomials. The proposed method is validated for representative dielectric structures by comparison via Fourier transformation of scattering data obtained with this method and various frequency domain techniques.

A potential integral equations method for electromagnetic scattering by penetrable bodies

A method for computing the electromagnetic scattering by general inhomogeneous penetrable bodies is presented. The method is based on the volume equivalence principle and it uses the electromagnetic potentials as unknowns. The resulting coupled integral equations system is solved by the method of moments in combination with cubical and curvilinear meshes in the special case of purely dielectric scatterers. To show the accuracy of the method, numerical results of the transmitted and of the scattered fields are compared with existing analytical and experimental results.

Efficient prism modeling for arbitrary-shape antennas printed on finite-size dielectric substrate in EFIE analysis

The volume equivalence principle is used for 3-D arbitrary geometry analysis, printed on a finite-size dielectric slab. Simple basis functions associated with triangular prism elements are studied for substrate modeling. The results of convergence on input impedance are presented. Experiments on printed monopoles confirm the theoretical results, on a wide band of frequency, for different geometries and permittivity values. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 17: 370–375, 1998.

Microwave Imaging of Dielectric Cylinder in Layered Media

Microwave imaging of a lossless two-dimensional dielectric object buried in layered media using time domain scattering data is considered in this paper. The method is based on volume equivalence principle and moment method. Born type approximation is adopted. It is an iterative method. In each iteration, both the direct and inverse problems are solved once. The direct problem is solved by finite-difference time domain method. To overcome the ill-posedness of the inverse problem, the pseudoinverse technique is adopted. The Green function in layered media excited by a unit line current source is derived by plane wave expansion method. Several examples are given to show the ability of this method to invert arbitrarily shaped permittivity profiles. Good imaging solutions are obtained.

A Novel Method for Microwave Imaging of Dielectric Cylinder in Layered Media

A novel method for microwave imaging of lossless two-dimensional dielectric targets buried in layered media using time-domain scattering data with or without the effect of random noise is presented in this paper. It is an iterative method derived using volume equivalence principle, variation principle, Fourier transform. We name it as time-domain variational iterative method (TVIM). In each iteration, both the direct and inverse problems are solved once. The direct problem is solved by finite-difference time domain (FDTD) method and the inverse problem is solved iteratively. Numerical results show that TVIM performs much better than our former method. Those factors which make TVIM perform better are also investigated.

A numerical formulation of dyadic Green's functions for planar bianisotropic media with application to printed transmission lines

An integral equation (IE) method with numerical solution is presented to determine the complete Green's dyadic for planar bianisotropic media. This method follows directly from the linearity of Maxwell's equations upon applying the volume equivalence principle for general linear media. The Green's function components are determined by the solution of two coupled one-dimensional IE's, with the regular part determined numerically and the depolarizing dyad contribution determined analytically. This method is appropriate for generating Green's functions for the computation of guided-wave propagation characteristics of conducting transmission lines and dielectric waveguides. The formulation is relatively simple, with the kernels of the IE's to be solved involving only linear combinations of Green's functions for an isotropic half-space. This method is verified by examining various results for microstrip transmission lines with electrically and magnetically anisotropic substrates, nonreciprocal ferrite superstrates, and chiral substrates. New results are presented for microstrip embedded in chiroferrite media.