Volume integral equations for analysis of dielectric branchingwaveguides

Novel forms of volume integral equations are developed for the exact treatment of wave propagation in two-dimensional dielectric branching waveguides. The integral equations can be obtained by considering the condition at a point far away from the junction section. An approximate solution by the Born approximation and a numerical solution by the moment method establish the validity of the new volume integral equations. The numerical results are discussed from the viewpoint of energy conservation and reciprocity. The solution is exact if sufficiently large computer memory and computational time are used. The method can be extended to problems of a more general nature (i.e. the incident TM mode), and complex configurations of branching waveguides. The basic idea is also applicable to techniques using boundary (surface) integral equations which are applicable to three-dimensional problems     

A Modified Residue-Calculus Technique for Solving a Class of Boundary Value Problems Part I: Waveguide Discontinuities

The modified residue-calculus technique, which is a generalization of the conventional function theoretic procedure for solving certain infinite sets of equations, permits solution of waveguide discontinuity problems which include dielectric, diaphragm, and step modifications of a basic discontinuity problem exactly solvable by the Wiener-Hopf techniqne. Solutions in scattering matrix form, including both propagating and nonpropagating modes, are found by a rapidly convergent and very accurate numerical procedure which eliminates many of the computational difficulties associated with integral equation or matrix equation solutions of the same problems.     

Regularized integral equations and curvilinear boundary elementsfor electromagnetic wave scattering in three dimensions

The boundary integral equations (BIEs), in their original forms, which govern the electromagnetic (EM) wave scattering in three-dimensional space contain at least a hypersingularity (1/R³ ) or a Cauchy-singularity (1/R²), usually both. Thus, obtaining reliable numerical solutions using such equations requires considerable care, especially when developing systematic numerical integration procedures for realistic problems. Regularized BIEs for the numerical computation of time-harmonic EM scattering fields due to arbitrarily-shaped scatterers are introduced. Two regularization approaches utilizing an isolation method plus a mapping are presented to remove all singularities prior to numerical integration. Both approaches differ from all existing approaches to EM scattering problems. Both work for integral equations initially containing either hypersingularities or Cauchy-singularities, without the need to introduce surface divergences or other derivatives of the EM fields on the boundary. Also, neither approach is limited to flat surfaces nor flat-element models of curved surfaces. The Muller linear combination of the electric- and magnetic-field integral equations (EFIE) and (MFIE) is used to avoid the resonance difficulty that is usually associated with integral equation-based formulations. Some preliminary numerical results for EM scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions     

On volume integral equations

The focus of this paper is on the volume integral representations to be used in constructing integral equations for composite volume media. The major thrust of the paper is to identify where derivatives of a discontinuous function arise in the derivation of the volume representation. Three different derivation methods are presented, resulting in identical representation independent of the derivation method. These representations agree with some in the existing literature and disagree with others. When an electric field formulation is considered, the source of disagreement manifests itself only when magnetic materials are present. Likewise, for the dual situation, the inconsistency appears for a magnetic field formulation of dielectric materials. This paper identifies the sources of error in the incorrect representations and its major contribution is the rigorously correct derivation of the representations to be used in volume integral equations. We also present numerical results for an integral equation derived from our representation. The numerical results employ only the E-field as the unknown and the singularity is handled in a manner analogous to a standard numerical treatment of the electric field integral equation.     

A new approach to the thin scatterer problem using the hybrid equations

A new set of integral equations for electromagnetic scattering problems, the "hybrid" equations, is presented. The advantages of these equations for thin perfect conductors are discussed in comparison to the magnetic and electric field integral equations. Specific comparisons are made with the solution of the electric field integral equation for a finite hollow cylinder. It is demonstrated that the primary advantage of these equations is obtained by minimizing the coupling between component equations for the two surface currents.     

Analysis of a Longitudinal Gyroelectric Discontinuity Inside a Fiber Waveguide

The scattering of guided electromagnetic waves from a finite-length longitudinal gyroelectic discontinuity inside a fiber waveguide is treated analytically. An integral equation approach is employed to formulate the corresponding boundary-value problem. The induced field inside the gyroelectic discontinuity region is expanded into a Fourier-type series in terms of the well-known cylindrical waves M and N plus a purely longitudinal wave Q. Then the method of moments is applied to decouple the basic integral equation. The resulting infinite coupled system of equations is truncated and solved numerically. After determining the field inside the discontinuity, the scattered far field inside the dielectric-rod waveguide is computed by employing a steepest descent integration technique. Numerical results for the scattering coefficients of an incident HE/sub 11/ dominant mode are obtained. Finally, design principles are discussed for practical components based on the treated longitudinal gyroelectric discontinuity.     

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.     

Rank Reduction of Ill-Conditioned Matrices in Waveguide Junction Problems

A new low-rank spectral expansion technique for solving the ordinarily intractable matrix equations obtained from waveguide field equivalence theorem decompositions is described. The method facilitates the analysis of waveguide discontinuity problems that resist ordinary methods of solution. The technique is illustrated for the problem of scattering at a slant interface in a rectangular waveguide.     

A comparison of integral equations with unique solution in theresonance region for scattering by conducting bodies

A comparison of integral equations, for problems involving scattering by arbitrary-shape conducting bodies, having a unique solution in the resonance region is presented. The augmented electric and magnetic field integral equations and the combined field integral equation, in their exact and approximate versions, are considered. The integral equations and the basis and test functions used in the method of moments to solve them are reviewed. Their implementation in a computer code is analyzed, mainly the relation between the matrix properties and the CPU time and memory. Numerical results (condition number and backscattering cross section) are presented for the cube. It is shown that the combined field integral equation, and the approximate (symmetric) combined field integral equation, are the most efficient equations to use in the neighborhood of resonant frequencies, because the overdetermined augmented integral equations require an extra matrix multiplication     

IMPROVED GENERALIZED ADMITTANCE MATRIX TECHNIQUE AND ITS APPLICATIONS TO RIGOROUS ANALYSIS OF MILLIMETER-WAVE DEVICES IN RECTANGULAR WAVEGUIDE

An improved generalized admittance (GAM) matrix technique is presented in this paper. Matrix transformation eliminates the singularity factor of GAM, denominator (1+Γ), because of new presentations of GAM. The relationship equations between II-port current and I-port incidence wave is computed by mode matching method. The generalized scattering matrix (GSM) of waveguide structure and its discontinuity problems is obtained with relationship equations and reflection coefficients. The GSM’s of millimeter-wave multistepped bend and T-junction in rectangular waveguide are computed by the improved GAM technique. The results comparisons between the proposed method and commercial software HFSS10.0 show the validity of the proposed method, which improves the validity of the GAM technique and reduces mathematical efforts. It is general, very efficient and can be used to solve other complicated and multiport network problems.     

Rigorous Analysis of the Scattering of Surface Waves in an Abruptly Ended Slab Dielectric Waveguide

The reflection and the scattering properties of even TE and TM surface waves incident in an abruptly ended dielectric slab waveguide are analyzed. The discontinuity is regarded as a junction between two open waveguides namely the dielectric slab waveguide and the free space waveguide. The boundary conditions acting together with the orthogonality provide singular coupled integral equations on the discrete and the continuous wave amplitudes at the discontinuity. These singular coupled intergral equations with Cauchy kernels and infinite limits of integration are solved by iteration via the Neuman series. Numerical results are presented for the reflectivity of the even TE/sub 0/ and TM/sub 0/ fundamental modes, together with their mode conversion on even TE/sub 2/ and TM/sub 2/ in a slab where two guided modes can propagate. Reflectivity and mode conversion of higher order excitations are also investigated     

An integral transform technique for analysis of planar dielectricstructures

This paper presents a novel efficient technique for the study of planar dielectric waveguides for submillimeter-wave and optical applications. In an appropriate integral transform domain, which is determined by the Green's function of the substrate structure, higher-order boundary conditions are enforced in conjunction with Taylor expansions of the fields to derive an equivalent one-dimensional integral equation for the corresponding two-dimensional waveguide geometry. This reduction in the dimensionality of the boundary-value problem can easily be extended to three-dimensional planar structures, with equivalent two-dimensional integral equations being formulated. The reduced integral equations are solved numerically by invoking the method of moments, in which the transform-domain unknowns are expanded in a smooth localized entire-domain basis. It is demonstrated that using orthogonal Hermite-Gauss functions as an expansion basis provides very satisfactory results with only a few expansion terms. For the validation of the technique, single and coupled dielectric slab waveguides are treated     

SOLUTION OF ARBITRARY CROSS-SECTION WAVEGUIDE USING METHOD OF EIGEN WEIGHTED BOUNDARY INTEGRAL EQUATION

Based on the second kind of Green's identity,a boundary integral equation forarbitrary cross-section waveguide is transformed to a system of linear homogeneous algebraicequations by means of expansion of boundary bases and by using the eigenfunctions of a fictitiousregular boundary as weighting functions,which corresponds to less algebraic equations than BEMand simpler coefficients than the modified BEM.The numerical results for some typical metallicwaveguides are given by using the method of eigen-weighted boundary integral equation,and theyare accurate enough with fast convergence.     

Note on an E-Plane Waveguide Step with Simultaneous Change of Media (Short Papers)

A simple quasi-static formula for the discontinuity susceptance of a 2:1 E-plane waveguide step with a simultaneous change of media is found using the theory of singular integral equations. An important difference from previous solutions by this method comes from the determination of the constants in the solution.     

Analyse de discontinuitiés rayonnantes sur ligne microruban au moyen d’équations intégrates

An integral equations technique solved by the moment method associated with the simple multiport model is used to analyse radiating open discontinuities in microwave circuits. Results obtained on a microstrip gap discontinuity are compared with those given by Hammerstad’s which are utilised in almost commercial cad software.     

Integral equations with reduced unknowns for the simulation oftwo-dimensional composite structures

A set of integral equations with reduced unknowns is derived for modeling two-dimensional inhomogeneous composite scatterers. The scatterer is first simulated in terms of thin curvilinear material layers of constant thickness. The traditional integral equations corresponding to each inhomogeneous layer are then manipulated in a manner allowing the identification of a new equivalent current component to replace two of the traditional ones across the layer. The given integral equations require approximately 2N current-component unknowns for their numerical implementation instead of the 3N unknowns generally required with traditional formulations. The implied computational efficiency though, was obtained at the expense of some complexity in the resulting pair of integral equations. To test the validity of the derived integral equations, special attention is given to a moment-method implementation of the authors' compact set of integral equations, with emphasis on the analytical evaluation of the diagonal and near-diagonal elements of the impedance matrix. Scattering patterns are presented as computed with the compact set of integral equations. These are further compared with measured data and computations using alternate analytical techniques. In all cases, these were in excellent agreement with corresponding results achieved by alternate methods     

Mathematical foundations for error estimation in numericalsolutions of integral equations in electromagnetics

The problem of error estimation in the numerical solution of integral equations that arise in electromagnetics is addressed. The direct method (Green's theorem or field approach) and the indirect method (layer ansatz or source approach) lead to well-known integral equations both of the first kind [electric field integral equations (EFIE)] and the second kind [magnetic field integral equations (MFIE)]. These equations are analyzed systematically in terms of the mapping properties of the integral operators. It is shown how the assumption that field quantities have finite energy leads naturally to describing the mapping properties in appropriate Sobolev spaces. These function spaces are demystified through simple examples which also are used to demonstrate the importance of knowing in which space the given data lives and in which space the solution should be sought. It is further shown how the method of moments (or Galerkin method) is formulated in these function spaces and how residual error can be used to estimate actual error in these spaces. The condition number of all of the impedance matrices that result from discretizing the integral equations, including first kind equations, is shown to be bounded when the elements are computed appropriately. Finally, the consequences of carrying out all computations in the space of square integrable functions, a particularly friendly Sobolev space, are explained     

On energy conservation and the method of moments in scattering problems

Electromagnetic scattering problems, including waveguide discontinuity, phased array, and scattering (exterior type) problems, are frequently described by integral equations that can be solved by the Ritz-Galerkin or generalized method of moments. Under appropriate conditions, it has been shown that reciprocity and variational properties are, in fact, preserved in the approximate solutions. It is shown here that in the Ritz-Galerkin method, energy is also conserved under certain conditions, even in those scattering problems where reciprocity does not exist. Hence energy conservation cannot serve as a check for accuracy of a numerical solution obtained by the Ritz method or other related methods.