Shifted-frequency internal equivalence

The shifted frequency internal equivalence (SFIE) theorem involving inhomogeneous regions is introduced and proven. For a lossless inhomogeneous region using a vector Green's theorem and potential formulation, it is shown that the frequency-domain electromagnetic field at frequency ω inside the region can be obtained using a set of equivalent volume and surface currents radiating in free space and at the different frequency ω₀. The equivalent currents thus obtained are functions of the two frequencies, electric- and magnetic-volume-type sources of the original problem, material parameters, and the original field phasors at ω, and they only exist inside the region and on its boundary. A direct application of this equivalence is that it can be used to construct an internal equivalence at a shifted frequency for electromagnetic scattering problems if data are needed in a band of frequency. ω₀ can be kept constant while the incident field frequency changes and, as a result, full computation of fields at each different frequency for volume-type equivalent sources can be avoided     

A boundary-element solution of the Leontovitch problem

A boundary-element method is introduced for solving electromagnetic scattering problems in the frequency domain relative to an impedance boundary condition (IBC) on an obstacle of arbitrary shape. The formulation is based on the field approach; namely, it is obtained by enforcing the total electromagnetic field, expressed by means of the incident field and the equivalent electric and magnetic currents and charges on the scatterer surface, to satisfy the boundary condition. As a result, this formulation is well-posed at any frequency for an absorbing scatterer. Both of the equivalent currents are discretized by a boundary-element method over a triangular mesh of the surface scatterer. The magnetic currents are then eliminated at the element level during the assembly process. The final linear system to be solved keeps all of the desirable properties provided by the application of this method to the usual perfectly conducting scatterer; that is, its unknowns are the fluxes of the electric currents across the edges of the mesh and its coefficient matrix is symmetric     

New Physical Optics Approach for an Efficient Treatment of Multiple Bounces in Curved Bodies Defined by an Impedance Boundary Condition

A new physical optics (PO) formulation is presented to treat radiation and scattering problems of curved bodies, including multiple reflections between the body parts. The approach uses electric and magnetic PO currents that are expanded as exponential terms. These terms are defined by spatially slow-varying amplitude and exponent functions. All the reflections of a multiple bounce contribution are computed by using PO, considering a very efficient recursive scheme to evaluate the PO integrals using quasi-analytical expressions. The complex problem of obtaining the shadowed areas in curved bodies for multiple reflections is avoided, thanks to the electric and magnetic PO currents chosen. These currents extend over the complete area of the body, including lit or shadowed parts. In the lit parts, the currents provide the reflected field, while in the shadow parts, they give a scattered field that, together with the incident field, causes a total null field in the shadows. The currents do not radiate on their back directions. This approach is useful for the analysis of bodies that can be characterized electrically by an impedance boundary condition (IBC). A combination of these currents with the angular Z-buffer (AZB) ray tracing technique makes it possible to analyze simple or complex cases efficiently.     

Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body

The method of moments technique for analyzing electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body is presented based on the combined field integral equations. The body is assumed to be illuminated by a plane wave. The surface equivalence principle is used to replace the body by equivalent electric and magnetic surface currents. These currents radiating in unbounded free space produce the correct scattered field outside. The negatives of these currents produce the correct total internal field, when radiating in an unbounded chiral medium. By enforcing the continuity of the tangential components of the total electric and magnetic fields on the surface of the body, a set of coupled integral equations is obtained for the equivalent surface currents. The surface of the body is modeled using triangular patches. The triangular rooftop vector expansion functions are used for both equivalent surface currents. The coefficients of these expansion functions are obtained using the method of moments. The mixed potential formulation for a chiral medium is developed and used to obtain explicit expressions for the electric and magnetic fields produced by surface currents. Numerical results for bistatic radar cross sections are presented for three chiral scatterers - a sphere, a finite circular cylinder, and a cube.     

On the numerical computation of scattering from inhomogeneous penetrable objects

A formulation of electromagnetic scattering problems involving a penetrable inhomogeneous scatterer is presented. The formulation combines the concepts of invariant imbedding and surface equivalent currents with the method of moments. It effectively circumvents the problem of intractable matrix size which has been a major deterrent in the past.     

A numerical solution to full-vector electromagnetic scattering bythree-dimensional nonlinear bounded dielectrics

This paper deals with electromagnetic scattering by nonlinear dielectric objects. In particular, a numerical approach is developed that is aimed at determining the distributions of the electromagnetic field vector inside a three-dimensional nonlinear, inhomogeneous, isotropic scatterer illuminated by a time-periodic incident electric field vector. An integral-equation formulation for the full-vector scattering problem is considered, and the nonlinear effect is taken into account by introducing equivalent sources and a Fourier-series representation. A system of integral equations (for each harmonic vector component and for the static term) is obtained that includes the internal electric field distribution as the unknown. After discretization, the solution is reduced to solving an algebraic system of nonlinear equations. Some preliminary numerical results are reported concerning scatterers that exhibit a specific (quadratic) dependence of the dielectric permittivity on the total electric field. The harmonic components of the scattered electric field outside the objects are also computed     

Coupling of finite element and moment methods for electromagneticscattering from inhomogeneous objects

A hybrid formulation which combines the method of moments (MM) with the finite element method (FEM) to solve electromagnetic scattering and/or absorption problems involving inhomogeneous media is discussed. The basic technique is to apply the equivalence principle and transform the original problem into interior and exterior problems, which are coupled on the exterior dielectric body surface through the continuities of the tangential electric field and magnetic field. The interior problem involving inhomogeneous medium is solved by the FEM, and the exterior problem is solved by the MM. The coupling of the interior and exterior problems on their common surface results in a matrix equation for the equivalent current sources for the interior and exterior problems. Combining advantages of both methods allows complicated inhomogeneous problems with arbitrary geometry to be treated in a straightforward manner. The validity and accuracy of the formulation are checked by two-dimensional numerical results, which are compared with the exact eigenfunction solution, the unimoment solution, and Richmond's pure moment solution     

Computation of TM and TE modes in waveguides based on a surfaceintegral formulation

The surface integral formulation is used for the computation of TM and TE modes propagating in dielectric loaded waveguides. This formulation makes use of the surface equivalence principle whereby the field at any point internal or external to the waveguide can be expressed in terms of equivalent surface currents. This procedure reduces the original problem into a set of integro-differential equations which is then reduced to a matrix equation using the method of moments. The solution of this matrix equation provides the propagation characteristics of the waveguide and the equivalent surface currents existing on the waveguide walls. The equivalent surface currents can be used to compute the fields at all points, both inside and outside the waveguide. The surface integral method has been used to compute the propagation characteristics of waves propagating in dielectric loaded waveguides. The computed results agree very well with analytical and published data. A method that can be used to remove spurious modes is illustrated     

Scattering by a chiral cylinder of arbitrary cross section

An integral equation and method-of-moments (MM) solution to the problem of scattering by an inhomogeneous chiral cylinder of arbitrary cross section is presented. The volume equivalence theorem for chiral media is developed and used to formulate a set of coupled integral equations for the electric and magnetic volume polarization currents representing the chiral cylinder. These coupled integral equations are solved using a standard pulse basis and point-matching MM solution. Numerical results, including echo width and internal fields, are presented for the scattering by chiral slabs and circular cylinders. These results are compared to exact solutions when available     

Electromagnetic scattering from an inhomogeneous object by raytracing

A shooting and bouncing ray (SBR) formulation is presented for treating the electromagnetic scattering from electrically large, inhomogeneous objects. A dense grid of rays representing the incident plane wave is shot toward the inhomogeneous objects. At the scatterer boundary, reflected rays and refracted rays are generated due to the discontinuity of the medium parameters. The trajectory, amplitude, phase and polarization of the rays inside the inhomogeneous object are traced based on geometrical optics. Whenever the rays cross the scatterer surface, additional reflected/refracted rays are generated and are tracked. This is repeated until the intensities of the refracted/reflected rays become negligible. The contributions of the existing rays to the total scattered field are calculated using the equivalence principle in conjunction with a ray-tube integration scheme. The ray formulation is applied to calculate the backscattering from cylinders and spheres and good agreement with the exact series solutions is observed in the high-frequency range. In addition, the backscattering mechanisms in penetrable objects are interpreted in terms of simple ray pictures     

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     

Electromagnetic diffraction from a dielectric-filled slit-cylinderenclosing a cylinder of arbitrary cross section: TM case

A simple moment solution to the problem of the diffraction of a TM plane wave from an infinite, perfectly conducting slotted cylinder of an arbitrary cross section is summarized. The slit cylinder encloses a smaller perfectly conducting cylinder of an arbitrary cross section, and the space between the cylinders is filled with a dielectric material. The equivalence principle is used to obtain a set of coupled integral equations for the induced/equivalent surface currents on the cylinders, and the method of moments is used to solve numerically the integral equations. The electric field integral equation formulation is used. The advantages and the limitations of the method are discussed. Sample results for the induced current, aperture field, internal field, and scattering cross sections are given. These are in good agreement with some of the available published data     

Shielding and scattering analysis of lossy cylindrical shells usingan extended multifilament current approach

An investigation of the multifilament current method's (MFCM) ability to solve electromagnetic scattering and/or absorption problems involving inhomogeneous cylindrical structures is presented. Dielectric cylinders of arbitrary cross section covered by multiple layers of lossy dielectrics are considered. Both cases of wave incidence, TM and TE, are treated. Like some other moment-method solutions, the MFCM experiences numerical difficulties when dealing with a medium characterized by a high loss tangent or large electrical conductivity. To overcome this problem, a new boundary condition based on an impedance matrix equivalent circuit approach that accounts for the curvature of the surfaces has been developed. This new impedance matrix boundary condition (IMBC) extends the MFCM capability to the analysis of two-dimensional (2-D) structures involving conducting layers with ohmic losses. The usefulness of this extended method is confirmed by the study of metallic shells for which a strong energy coupling with the incident electromagnetic (EM) field is demonstrated at their structural resonance frequencies     

Combined field solution for TM scattering from multiple conducting and dielectric cylinders of arbitrary cross section

A simple moment solution is given for the problem of electromagnetic scattering from multiple conducting and dielectric cylinders of arbitrary cross section. The system of conducting and dielectric cylinders is excited by a plane-wave polarized transverse magnetic to the axis of the cylinders. The equivalence principle is used to obtain three coupled integral equations for the induced electric current on the conducting cylinders and the equivalent electric and magnetic currents on the surface of dielectric cylinders. The combined field integral equation (CFIE) formulation is used. Sample numerical results are presented. The agreement with available published data is excellent.     

Transmission through dielectric-filled slots in a conductingcylindrical shell of arbitrary cross section: TE case

A simple moment solution is summarized for the problem of electromagnetic transmission through dielectric-filled slots in a conducting cylindrical shell of arbitrary cross section. The system is excited by a plane-wave polarized transverse electric (TE) to the axis of the shell. The equivalence principle is used to replace the shell and the dielectric by equivalent electric and magnetic surface currents radiating into an unbounded medium. Two different sets of coupled integral equations involving the surface currents are obtained by enforcing the boundary conditions on the tangential components of the total electric and magnetic fields. The method of moments is used to solve the integral equations. Pulses are used for both expansion and testing functions. Special attention is paid to circular and rectangular shells. Results for shell surface current, the internal field, and the aperture field are presented. For the case of air dielectric filling, the results computed using the electric field and/or the magnetic field formulation are in very good agreement with published data. In general, it is observed that the effect of filling a slot with a dielectric is not predictable from a simple theory     

A new finite element formulation for RF scattering by complexbodies of revolution

A formulation for and results of solving electromagnetic scattering from complex inhomogeneous axisymmetric bodies are presented. The approach presented uses the finite element method in the frequency domain. A node-based approach is used to solve for the three components of the electric or magnetic field. The finite element mesh is truncated using a three-dimensional vector absorbing boundary condition based on the Wilcox expansion theorem. A harmonic expansion of the near-field solution obtained from the finite element solution is used to compute the far fields and radar cross section     

Analysis of multilayer planar circuits by a hybrid method

This letter proposes a hybrid method for the modeling of multilayer planar structures with locally inhomogeneous penetrable objects. By using the equivalence principle, the original problem can be decomposed into external and internal problems, which are formulated by the mixed-potential integral equation (MPIE) method and finite-difference time-domain (FDTD) method, respectively. Instead of directly implementing the continuity boundary conditions, an iterative approach is used to solve the hybrid FDTD-MPIE problem. Numerical results are presented to validate the hybrid method.     

Reflection and transmission characteristics of lossy periodiccomposite structures

A periodic surface integral formulation is proposed to analyze the reflection and transmission properties of a lossy periodic composite structure which has circular conducting fibers embedded in a dielectric matrix. This formulation is based on the equivalence principle which represents the unknown electric and magnetic currents over the material discontinuity interfaces, and uses the structure periodicity and Poisson summation formula to reduce the problem to a periodic cell. These surface integral equations are then solved numerically, using the method of moments with pulse bases and point matching. Only the transverse magnetic (TM) case is analyzed and the numerical results such as reflected, transmitted, and dissipated powers for a single-layer fiber-reinforced composite structure are presented, in detail, to discuss the effects of frequency, incident angle, fiber radius, fiber conductivity, embedding dielectric, etc. A convergence study and a comparison with the previous published results are also included to confirm the accuracy of the new formulation     

Surface integral formulation for calculating conductor anddielectric losses of various transmission structures

The power-loss method, along with a surface integral formulation, has been used to compute the attenuation constant in microstrip and coplanar structures. This method can be used for the analysis of both open and closed structures. Using the surface equivalence principle, the waveguide walls are replaced by equivalent electric surface currents and dielectric surfaces are replaced by equivalent electric and magnetic surface currents. Enforcing the appropriate boundary condition, and E-field integral equation (EFIE) is developed for these currents. Method of moments with pulse expansion and point matching testing procedure is used to transform the integral equation into a matrix one. The relationship between the propagation constant and frequency is found from the minimum eigenvalue of the moment matrix. The eigenvector pertaining to the minimum eigenvalue gives the unknown electric and magnetic surface currents     

Current and charge Integral equation formulation

A new stable frequency domain surface integral equation formulation is proposed for the three dimensional electromagnetic scattering of composite metallic and dielectric objects. The developed formulation does not suffer from the low frequency breakdown and leads to a well balanced and stable system on a wide frequency band. Surface charge densities are used as unknowns in addition to the traditional surface current densities. The balance of the system is achieved by using normalized field quantities and by enforcing the continuity of the fields across the boundaries with carefully chosen scaling factors. The linear dependence between the currents and charges is taken into account with an integral operator, and the linear dependence in charges is removed with the deflation method. A combined field integral equation form of the formulation is proposed to remove the internal resonance problem associated to the closed metallic objects. Due to the good balance in the new formulation, fast converging iterative solutions on a very wide frequency band can be obtained. The new formulation and its convergence is verified with numerical examples.