Electromagnetic scattering by multiple three-dimensional scatterersburied under multilayered media. I. Theory

A general procedure is developed for the analysis of electromagnetic (EM) scattering by multiple three-dimensional (3D) dielectric and/or conducting objects buried under one-dimensional (1D) multilayered media. In this first part of a two-part paper, general closed-form formulations for the electric fields excited by an arbitrarily oriented electric dipole under the layered media are first presented, from which electric-field integral equations for the buried dielectric objects, pure conducting objects, and their combinations are then obtained, and the scattered electric fields in the upper space are formulated. Finally, the physical significance of the above formulations is discussed. In the second part, numerical implementations for these integral equations and the scattered fields are investigated     

Two-dimensional surface integral representations for a rotationallyinvariant anisotropic medium: applications to scattering

The Huygens' principle is presented for an electromagnetic field in a rotationally invariant anisotropic region. The representation is investigated by deriving surface integral equations for scattering, resulting, for instance, in scattering formulations for an impedance body and for a perfectly conducting electric sheet (both embedded in the anisotropic material). Validation is accomplished via application to a canonical geometry     

The Solution of Waveguide Scattering Problems by Application of an Extended Huygens Formulation

The implementation of a recent new hybrid integral-equation/vector finite-element method formulation applicable to inhomogeneous obstacle scattering in hollow waveguide, requiring discretization just of the obstacle, is presented. The integral equation links the given incident modes with the discontinuity-surface electric and magnetic fields. The finite-element equation is expressed in terms of the entire magnetic and surface electric field of the obstacle. Compatible vector finite-element basis function expansions are inserted, resulting in a pair of matrix equations soluble for the unknown electric and magnetic basis coefficients. Corresponding two-port scattering parameters are further derived. Test cases of posts in the$ TE_10$waveguide, with details of the matrix constructions, are described. Numerical results verified against an established commercial code are given. The ability to model inhomogeneous, lossy, and multiple scatterers is demonstrated.     

A frequency-domain differential equation formulation forelectromagnetic scattering from inhomogeneous cylinders

A procedure is described for calculating scattering cross-section data based on the numerical solution of differential equations in the frequency domain. The scatterers considered are two-dimensional and consist of lossy, inhomogeneous dielectric, magnetic, and perfectly conducting material. The appropriate wave equation is combined with an approximate local absorbing boundary condition and discretized with the finite-element method. Results for conducting and dielectric scatterers are presented to illustrate the accuracy and generality of the approach     

Finite element solution for a class of unbounded geometrics [EMscattering]

An efficient and systematic procedure is described for the finite-element solution of a class of electromagnetic radiation and scattering problems involving unbounded geometries. The numerical procedure is well suited for analyzing infinite metallic structures with cavity regions filled with inhomogeneous and anisotropic media. The formulation is based upon an approach that combines the finite-element method (FEM) with the surface integral equation to truncate the mesh region. The efficiency of the proposed technique arises from the use of the Green's function of the first kind in the surface integral. Illustrative numerical representations that demonstrate the validity, versatility, and efficiency of the method are included     

Transient scattering by conducting surfaces of arbitrary shape

The time-domain electric field integral equation (EFIE) is used along with the method of moments to develop a simple and efficient numerical procedure for treating problems of transient scattering by arbitrary shaped conducting objects. The conducting surface is modeled by planar triangular patches for numerical purposes. Because the EFIE is used in the solution procedure, the method is applicable to both open and closed bodies. the EFIE approach is applied to the scattering problems of Gaussian plane wave illumination of a flat square plate and sphere. Comparisons of surface current densities and far-scattered fields are made with previous computations and good agreement is obtained in each case     

Time-domain electromagnetic computations using a modified EFIE kernel and field-source equilibrium relations

A form for the electric-field dyadic Green's function for free space is derived that allows explicit time evolution of the modified electric-field integral equation (EFIE) applied to surface scattering. The modified EFIE kernel, here called a "source function," has an integrable singularity in the source region, and is shown to be equivalent, in the frequency domain, to the standard dyadic Green's function. With definitions of "local" and "non-local" fields at a conductor surface, both electric and magnetic versions of the relations between non-local fields and equilibrium surface sources (currents and charges) are derived. These field-source equilibrium (FSE) relations are exact if all the non-local fields are included: the interaction fields, as well as the usual incident fields from distant sources. When the interaction fields are neglected, the magnetic-field version of the FSE relations becomes the usual physical optics approximation. Source functions and the FSE relations were used in two three-dimensional, time-domain numerical simulations to compute radiation patterns from a conical helical antenna driven at a fixed frequency, and scattering of a CW plane wave by a perfectly conducting sphere. This surface-scattering simulation was explicit but remained stable. Excellent agreement between the computed and known results validated the approach.     

Moment method analysis of the magnetic shielding factor of aconducting TM shield at ELF

This paper presents an integral equation and method of moments (MoM) solution to the problem of TM transmission by a metallic conducting shield at extremely low frequencies (ELF). In order to accurately compute the total fields interior to the shield, equivalent problems are formulated which avoid the numerically difficult problem of computing the total fields as the sum of the incident plus scattered fields. In particular, the total electric field on the interior surface of the shield is obtained by a volume current equivalent problem, and then the total magnetic field interior to the shield is formulated in terms of equivalent magnetic surface currents flowing on the interior surface of the shield replaced by a perfect conductor     

Hybrid FDTD-MPIE method for the simulation of locally inhomogeneous multilayer LTCC structure

A new hybrid finite-difference time-domain (FDTD) and mixed potential integral equation (MPIE) method is proposed for the modeling of multilayer planar circuits with locally inhomogeneous objects. By using equivalence principle, the original problem can be decomposed into two kinds of regions. The FDTD method is employed to model the locally inhomogeneous objects and construct an interaction matrix to be used in the subsequent model coupling procedure. The MPIE method with less singular kernels is applied to model the layered structure with possible perfect electric conductors. The FDTD model and the MPIE model are coupled together by enforcing the continuity of the tangential electric and magnetic fields on the equivalent surface using a Galerkin testing procedure. Numerical results are presented to validate the proposed hybrid FDTD-MPIE method.     

An exact line integral representation of the physical opticsscattered field: the case of a perfectly conducting polyhedral structureilluminated by electric Hertzian dipoles

An exact line integral representation of the electric physical optics scattered field is presented. This representation applies to scattering configurations with perfectly electrically conducting polyhedral structures illuminated by a finite number of electric Hertzian dipoles. The positions of the source and observation points can be almost arbitrary. The line integral representation yields the exact same result as the conventional surface radiation integral; however, it is potentially less time consuming and particularly useful when the physical optics field can be augmented by a fringe wave contribution as calculated from physical theory of diffraction equivalent edge currents. The final expression for the line integral representation is lengthy but involves only simple functions and is thus suited for numerical calculation. To illustrate the exactness of the line integral representation, comparisons of numerical results obtained from the surface and the line integral representations are performed     

Electromagnetic scattering from arbitrary shaped conducting bodiescoated with lossy materials of arbitrary thickness

A simple and efficient numerical technique is presented to solve the electromagnetic scattering problem of coated conducting bodies of arbitrary shape. The surface equivalence principle is used to formulate the problem in terms of a set of coupled integral equations involving equivalent electric and magnetic surface currents which represent boundary fields. The conducting structures and the dielectric materials are modeled by planar triangular patches, and the method of moments is used to solve the integral equations. Numerical results for scattering cross sections are given for various structures and compared with other available data. These results are proved accurate by a number of representative examples     

Scattering analysis of a large body with deep cavities

A numerical scheme is presented for simulating electromagnetic scattering from a large and arbitrarily shaped body, coated with inhomogeneous composite materials, with large and deep cavities. This numerical scheme employs the higher order vector finite-element method (FEM) to discretize the fields inside the cavities and coatings and the higher order boundary integral (BI) method to terminate the FEM computational domain. A highly efficient special solver is designed to eliminate the unknowns inside the cavities, which yields a computed relation (CR) matrix over the cavity's aperture between the tangential electric and magnetic fields. This CR matrix is then combined with the finite element-boundary integral (FE-BI) matrix equation to form a complete linear system for the discrete fields everywhere in the computational domain. The resulting system is solved iteratively using a novel preconditioner derived by replacing the BI with a corresponding absorbing boundary condition (ABC).     

Dual-surface integral equations in electromagnetic scattering

A brief review is given of the derivation and application of dual-surface integral equations, which eliminate the spurious resonances from the solution to the original electric-field and magnetic-field integral equations applied to perfectly electrically conducting scatterers. Emphasis is placed on numerical solutions of the dual-surface electric-field integral equation for three-dimensional perfectly electrically conducting scatterers.     

High-frequency fields excited by a line source located on a concave cylindrical impedance surface

A previous study of high-frequency currents induced by a line source on a perfectly conducting concave cylindrical surface is extended to the case of nonvanishing surface impedanceZ_{s}. Alternative field representations are formulated and evaluated asymptotically as combinations of ray-optical, whispering gallery (WG) mode, surface wave, continuous spectrum, and canonical integral contributions. Numerical calculations provide an insight into the accuracy and utility of the various formulations. Sufficiently far from the source point, a combination of ray optical fields and tightly bound WG modes was previously found to be a most appealing form whenZ_{s} = 0. As the surface impedance becomes more dissipative, the WG modes axe weakened by attenuation and eventually render the ray optical fields adequate by themselves. A representation in terms of rays and a canonical integral is found to be useful for all parameter ranges. The canonical integral has been evaluated numerically and tabulated.     

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.     

EFIE time-marching scattering from bodies of revolution and itsapplications

The authors treat the “entire-tangent” representation of a time-domain electric-field integral equation (TDEFIE) for solving transient scattering problems involving perfectly conducting (PEC) bodies of revolution (BOR). This entire-tangent TDEFIE reduces three-dimensional scattering problems with axial symmetry to two-dimensional problems, thus facilitating the numerical treatment and significantly reducing CPU requirements. Additional refinements and algorithm efficiencies are described. Some numerical results and applications are presented     

Transient fields of a horizontal electric dipole on a multilayereddielectric medium

The transient electric fields of a horizontal electric dipole excited by a short pulse current and located on a layered dielectric medium were analyzed using the Cagniard-de-Hoop method. The fields are expressed as the convolution of the exciting current with the layered medium response. The layered medium response is obtained directly from the integral representation for the electric fields in the frequency domain and is expressed as a finite integral. In contrast to the conventional frequency synthesis approach, the Cagniard-de-Hoop (1960) method proves to be computationally more efficient and numerically more stable. Compared with the asymptotic approach, the solution involves no approximation. The nature of the various waves, reflected waves (guided wave and leaky wave), and lateral waves can be easily recognized on the Cagniard integral path. Numerical results are obtained to provide a rigorous forward modeling for the geo-radar operating on layered media     

Singular Value Decomposition of the Radiation Operator—Application to Model-Order and Far-Field Reduction

The time-harmonic electromagnetic scattering or radiation problem is considered. The singular value decomposition (SVD) is applied to the radiation operator that maps the set of electric and magnetic currents defined on the surface of an inhomogeneous object onto the set of the far-fields scattered (or radiated) from this object. The SVD yields orthonormal bases for both sets. Because the radiation operator is compact and regularizing, it is demonstrated that the far-field calculated from the series expansions of the currents on these bases converges exponentially fast to the exact one if a sufficient number of terms is considered in these series. This number is closely related to the degrees of freedom that characterize the far-field. The latter can be computed from a reduced number of unknowns in the discretized integral equation that links electric and magnetic surface currents by writing it in the new currents bases. Also, it allows the reduction of the far-field in a given angular sector. The numerical complexity of this technique is addressed, and 2D numerical examples are presented that illustrate its potentialities.     

TM scattering by an impedance sheet extension of a paraboliccylinder

An integral equation and method of moments (MM) solution are presented for the two-dimensional (2-D) problem of transverse magnetic (TM) scattering by an impedance-sheet extension of a perfectly conducting parabolic cylinder. An integral equation is formulated for a dielectric cylinder of general cross section in the presence of a perfectly conducting parabolic cylinder. It is then shown that the solution for a general dielectric cylinder considerably simplifies for the special case of TM scattering by a thin multilayered dielectric strip that can be represented as an impedance sheet. The solution is termed an MM/Green's function solution, where the unknowns in the integral equation are the electric surface currents flowing in the impedance sheet; the presence of the parabolic cylinder is accounted for by including its Green's function in the kernel of the integral equation. The MM solution is briefly reviewed, and expressions for the elements in the matrix equation and the scattered fields are given. Sample numerical results are provided     

A Spectral Integral Method and Hybrid SIM/FEM for Layered Media

This paper first presents a spectral integral method (SIM) for electromagnetic scattering from homogeneous dielectric and perfectly electric conducting objects straddling several layers of a multilayered medium. It then uses this SIM as an exact radiation boundary condition to truncate the computational domain in the finite-element method (FEM) to form a hybrid SIM/FEM, which is applicable to arbitrary inhomogeneous objects. Due to the high accuracy of the SIM, the sampling density on the radiation boundary requires less than five points per wavelength to achieve 1% accuracy. The efficiency and accuracy of the developed methods have been demonstrated with several numerical experiments for the TM_z case. The TE_z case can be obtained by duality