Waveguides of Arbitrary Cross Section by Solution of a Nonlinear Integral Eigenvalue Equation

The problem of electromagnetic wave propagation in hollow conducting waveguides of arbitrary cross section is formulated as an integro-differential equation in terms of fields at the waveguide boundary. Cutoff wave numbers and wall currents appear as eigenvalues and eigenfunctions of a nonlinear eigenvalue problem involving an integro-differential operator. A variational solution is effected by reducing the problem to matrix form using the method of moments. A specific solution of the problem is developed using triangle expansion functions in the method of moments. The solution is simplified by symmetry considerations and is implemented by two digital computer programs. Listings and full documentation of these programs are available. This solution yields accurate determinations of cutoff wave numbers, wall currents, and distributions of both longitudinal and transverse modal field components for the first several modes. Illustrative computations are presented for the single-ridge waveguide, which has a complicated boundary shape that does not lend itself to exact solution.     

On the Variational Reaction Theory for Dielectric Waveguides

By the reaction concept, a variational theory is established for treating the scattering and propagation problem associated with a dielectric waveguide which is illuminated by an obliquely incident plane wave. The theory is characterized by properly absorbing radiation and continuity conditions into the variational equation. This equation is then solved by the finite-element method together with the frontal solution technique. In this paper, the propagation constants of the guide are obtained by the procedure of searching for the poles of the scattering coefficients when an inhomogeneous wave is incident. Two most attractive features of this approach are the avoidance of the spurious modes and the accuracy of the results even for the modes near cutoff. Although the proposed theory may be applied to dielectric waveguides of arbitrary cross section, only the one with rectanguIar shape is investigated in detail. Also included in this study are numericul results for the propagation constants in discussing the effects due to differences in refractive indices, aspect ratios, and index profiles.     

On Guided Waves in Moving Anisotropic Media

Based upon the Maxwell-Minkowski theory, the equations governing the propagation of electromagnetic waves in a cylindrical waveguide of an arbitrary cross section filled with a moving anisotropic medium are derived. The governing equations are reducible to a pair of coupled wave equations in the axial components of the electric and magnetic fields, which in turn can be solved through the solution of a single second order scalar homogeneous Hehnholtz equation. For a general anisotropic medium no pure TM or TE modes can exist in the waveguide. However, if the moving medium is uniaxially anisotropic, TM and TE modes are possible. It is interesting to note that the cutoff frequencies are always lowered by a factor which depends upon the velocity of the medium and is independent of the guide geometry. The formulas for the characteristic wave impedance and power flow in a waveguide for a moving uniaxial medium, if expressed in terms of the new cutoff frequency, have the same forms as those for a moving isotropic medium. The propagation characteristics of waveguides of rectangular and circular cross sections filled with a moving uniaxial gyroelectric medium are discussed.     

Waveguide Modes Via an Integral Equation Leading to a Linear Matrix Eigenvalue Problem

A numerical method for determining the modes of a rectanguIar or a circular waveguide strongly perturbed by axial cylindrical conducting objects is presented. The method is based upon an integral equation which leads to a matrix eigenvalue problem by using the Galerkin procedure. Cutoff wavenumbers are simultaneously calculated with very good precision for a number of modes near to the order of the matrix eigenvalue problem. Excellent results are obtained also when the perturbed waveguide section exhibits reentrant parts or edges. Computing time is short and storage requirements are moderate. The method is also applicable for waveguides of arbitrary cross section.     

基于多分辨分析计算波导的TE和TM模

本文根据多分辨分析理论,给出了计算具有任意横截面波导的TE模和TM模的一种方法。在该方法中,用尺度函数把场分量(Ez或Hz)近似为级数,通过变分公式把波导问题化为矩阵方程。并且把波导截面分为许多矩形区域,用三阶B-样条函数,推出了矩阵元素的具体表达式。为了验证该方法,以三种不同的波导为实例进行了计算。     

Complex and Backward-Wave Modes in Inhomogeneously and Anisotropically Filled Waveguides

A rigorous analysis of lossless inhomogeneously and anisotropically filled waveguides of arbitrarily shaped cross section is presented. The mode propagation constants squared appear as eigenvalues of a real infinite-dimensional characteristic matrix, which is, in the general case, nonsymmetric. Complex conjugate pairs of eigenvalues are then possible, which give rise to complex modes. Modes at cutoff are shown to be either TE or TM with real cutoff frequencies. An investigation of the power flow shows that backward-wave modes may exist as well. Different orthogonality relations are derived from which the power coupling between complex modes is investigated.     

Eigenmodes of sectoral coaxial ridged waveguides

The results of numerical investigation of sectoral coaxial ridged waveguides eigenmodes of two configurations (with a ridge on inner or outer wall) for different cross-section dimensions are presented. In particular, dependences of cutoff wave numbers on geometrical dimensions ratios for first four modes are investigated, electric field components distributions for these modes have been obtained and the optimization of sectoral coaxial ridged waveguides has been carried out to provide maximal single-mode operation frequency band. Two optimal configurations of waveguides with single-mode operation bandwidth ratio 5.6:1 are obtained. It is shown that smaller cross-section dimensions at the fixed single-mode operation frequency band has the waveguide with the ridge at the inner round wall. The size of the gap between the ridge and the round wall of optimal waveguide is identical for both configurations and is determined by the required ratio of cutoff frequencies of two lower TE modes. Calculations are conducted utilizing the mathematical model obtained in [1] by the integral equation technique with the correct account of singular behavior of the field at the ridge.     

Cutoff frequencies of uniform waveguides of regular polygonal cross section

This letter deals with the problem of calculating the cutoff frequencies of hollow piped waveguides of regular polygonal cross section. By conformally transforming the given cross section onto a unit circle, the boundary conditions can be identically satisfied; however, the transformed partial differential equation must be solved by approximate procedures. The results are considered excellent, demonstrating remarkable agreement with the exact values in the case of square and circular cross sections. Only TM modes are considered in the present study; TE modes can be treated in a similar manner.     

Numerical solution of transmission line problems by a network modeldecomposition method based on polygon discretization

A polygon discretization technique for establishing a network model suitable for both TEM (transverse electromagnetic) transmission lines and hollow waveguides is introduced. A network model decomposition algorithm, or diakoptic algorithm, is also presented for solving the transmission-line problems and is shown to be useful when a computer is not sufficiently large to accommodate a problem. The network model decomposition algorithm can be used to calculate the characteristic impedance of an arbitrarily shaped TEM transmission line and the cutoff wavenumbers of a hollow waveguide of arbitrary section. Numerical results are also presented as a demonstration of the method's validity     

Uniform Waveguides with Arbitrary Cross-Section Considered by the Point-Matching Method

The point-matching method applies to the problem of wave propagation in many uniform waveguides of very general cross-sections. The boundary conditions are satisfied at a finite number of points on the guide wall only. This method applies when the contour of the cross section of the guide is a closed curve, the function of which is single-valued. The validity of the point-matching method is demonstrated qualitatively. Examples show that accurate values of cutoff wave numbers can be achieved easily.     

Eigenmodes of coaxial quad-ridged waveguides. Theory

The electrodynamics eigenmodes boundary problems’ solutions for the eigenmodes of coaxial quad-ridged waveguides are presented. These solutions have been obtained by integral equations technique utilizing the proposed system of orthogonal basis functions, which take correctly into account singular behavior of the field at the ridges’ edges. The formulas obtained provide possibilities to calculate cutoff wave numbers and electric and magnetic fields distributions for TEM, TE and TM modes in the presence of the ridges either on the inner or on the outer conducting cylinder. Analysis of the dependence of numerical solutions convergence for cutoff wave numbers on the number of basis functions and partial modes has been carried out. It has been shown that for calculation of cutoff wave numbers with residual error less than 0.1% it is enough to utilize the 7 proposed orthogonal basis functions and the 21 partial modes.     

Application of the Fourier-grid method to guided-wave problems

A method recently developed for solving the Schrodinger equation is applied to dielectric waveguides. The technique, which is extremely simple to implement, involves representing the differential operator in the scalar Helmholtz equation on a grid of discrete points in coordinate space, and then diagonalizing the resulting matrix to reveal the propagation constants and field patterns of the guided modes. The square of the transverse index profile is specified directly as a diagonal matrix in coordinate space, while the matrix for the transverse Laplacian is obtained through the Fourier relationship between its diagonal form in momentum space and the equivalent representation in coordinate space. The accuracy and computational performance of this procedure is assessed for one- and two-dimensional transverse profiles. Modal refractive indices and fields computed by the grid method are found to agree well with those derived by means of other techniques     

任意截面形状同轴线高次模的直线法分析

本文应用直线法理论严格推导了在极坐标系统中分析任意截面形状同轴线高次模的公式。根据这些公式可求出各模式的截止波数和相应的场分布。计算结果与已有的数据的一致性，证明了本文分析的正确性。     

计算复杂形状波导截止频率的高效方法

提出一种使用保角变换结合矩量法,计算复杂形状波导截止频率的新方法,该方法采用解析法或数值法,把复杂截面的波导变换为圆形区域,再采用矩量法求解波导的截止频率。实验结果验证了该方法的有效性。     

A General High-Order Finite-Element Analysis Program Waveguide

A very general computer program for determining sets of propagating modes and cutoff frequencies of arbitrarily shaped waveguides is described. The program uses a new method of analysis based on approximate extremization of a functional whose Euler equation is the scalar Helmholtz equation, subject to homogeneous boundary conditions. Subdividing the guide cross section into triangular regions and assuming the solution to be representable by a polynomial in each region, the variational problem is approximated by a matrix eigenvalue problem, which is solved by Householder tridiagonalization and Sturm sequences. For reasonably simple convex polygonal guide shapes, the dominant eigenfrequencies are obtained to 5-6 significant figures; for nonconvex or complicated shapes, the accuracy may fall to 3 significant figures. Use of the program is illustrated by calculating the propagating modes of a class of degenerate mode guides of current interest, for which experimental data are available. Numerical studies of convergence rate and discretization error are also described. It is believed that the new program produces waveguide analyses of higher accuracy than any general program previously available.     

Dispersion characteristics of asymmetric coupled anisotropic dielectric waveguides using FDFD

The formulation is developed in the frequency domain and the finite difference method is used for the numerical solution of the scalar wave equation, written in terms of the transverse components of the magnetic field. As a result a conventional eigenvalue problem is obtained without the presence of spurious modes due to the implicit inclusion of the divergence of the magnetic field equal to zero. The formulation is developed to include biaxial anisotropic dielectrics with an index profile varying arbitrarily in the cross section of the waveguide under analysis. This formulation is then applied to the analysis of the influence on the dispersion characteristics of the dimensions of asymmetric coupled rectangular uniaxial anisotropic dielectric waveguides. As expected, the reduction of the height or the width of one of the rectangular dielectric waveguides causes the dispersion curves to move towards higher frequencies.     

Cross-shaped and four-ridge waveguide—Horn radiators: Internal and external characteristics

A mathematical model is developed for smooth-wall waveguide-horn radiators with an arbitrary cross section of the aperture. The exterior problem is solved in the Huygens-Kirchhoff approximation and the interior problem is solved with the use of a numerical algorithm constructed on the basis of the finite-element method. Internal (cutoff frequencies of modes, the field structure, attenuation coefficients, the wave impedance, and the maximum transmitted power of the fundamental mode) and external (radiation patterns, polarization characteristics, the aperture efficiency, and matching to free space) characteristics of dual-polarization waveguide-horn radiators with cross-shaped and four-ridge circular and square cross sections are investigated. Optimum relationships for cross-sectional dimensions that maximize the operating bandwidth, minimize the loss level, and maximize the power transmitted through the waveguides are found. It is shown that optimization of cross-sectional parameters can ensure a high degree of the axial symmetry of radiation patterns, a good matching to free space, and a low level of sidelobe radiation in the operating frequency band—characteristics which are better than those of similar radiators with square and circular cross sections. Numerical results are compared with experimental data.     

Wave Scattering in Nonuniform Waveguides with Large Flare Angles and Near Cutoff

A set of coupled first-order differential equations for the wave amplitudes in nonuniform waveguides is developed. The coupling coefficients are regarded as differentiaI transmission and reflection scattering coefficients between two adjacent elementary radial waveguide sections. The analysis is an extension of an earlier quasi-optical solution. This set of coupled equations is compared with the familiar generalized telegraphist's equations which may be derived by considering the nonuniform waveguide to consist of elementary rectangular waveguide sections. The equations for the wave amplitudes derived in this paper are less coupled than the commonly used telegraphist's equations, and they may also be applied to waveguides with large flare angles and in regions at which the waveguide modes are at their respective cutoff cross sections.     

Infrared surface waves in circular hollow waveguides with smallcore diameters

An asymptotic form of the characteristic equation that describes wave propagation at near-infrared wavelengths in small core hollow circular waveguides is developed. Analytic solutions for the transverse and axial propagation constants are obtained. These demonstrate the transition of the TE₁₁ and TM₀₁ modes to surface waves as the guide radius is increased to values much greater than at cutoff. Relative power density distributions illustrating these mode transitions are shown     

Transfer matrix function (TMF) for wave propagation in dielectricwaveguides with arbitrary transverse profiles

A transfer matrix function (TMF) is derived for the analysis of electromagnetic (EM) wave propagation in dielectric waveguides with arbitrary profiles, situated inside rectangular metal tubes. The TMF relates the wave profile at the waveguide output to the (arbitrary profile) input wave in the Laplace space. The TMF consists of the Fourier coefficients of the transverse dielectric profile and those of the input-wave profile. The method is applicable for inhomogeneous dielectric profiles with single or multiple maxima in the transverse plane. The TMF is useful for the analysis of dielectric waveguides in the microwave and the millimeter-wave regimes and for diffused optical waveguides in integrated optics