Abstract-A simple and efficient ftnite.element method for the analysis or microwave and optical waveguiding problems is formulated using three components of the electric or magnetic field . In order to eliminate spurious solutions, edge elements are introduced. In the edge element approach the nodal parameters are not limited to the magnetic fi eld as in the conventional three-component formulation ror the dielectric waveguiding problem. An eigenvalue equation derived here involves only the edge variables in the transversal plane and ca n provide a direct solution ror the propagation constant. To show the vailidity and usefulness of this approach, computed results are illustrated fo r microstrlp transmission lines and dielectric waveguides. I. I NTRODUCTIONTO RIGOROUSLY evaluate propagat ion characteris-1. tics of microwave and optical wavegu ides with arbitrarily shaped cross sections, vectorial wave analys is is necessary , and different types o f the vector finit e-element method (FEM) have been developed . Of the various formulations, the FEM using full vector H field is quite suitable for a wide range of practical, complicated problems [I ] -[ I OJ . Thi s approach has been widely used fo r various dielectric waveguiding structures in microwave , millimeter-wave , and optical wavelength regions, and recently has been utilized as the waveguide solver of CAD packages [7] . The most serious problem associated with this approach is the appearance of spurious solutions. The penalty fun ction method [31. [4] , (6) , [7] has been used to cure this problem, but in this technique an arbitrary positi ve constant , ca lled the penalty coeffi cient , is involved and the accuracy of solutions depends on its magnitude . Funhennore, in the full vectorial fonnulation [1]-[10] the propagat ion conslant is fi rsl given as an inpul datum , and subsequently the operating frequency is obtained as a solution. More recently, several methods for solving directly the propagation constant have been developed , but each has its drawback, e.g., a large number of field components (11] -[13) , consideration of the adjoint fi eld which does not correspond to the actual electromagnet ic field [14], o r the need to estimate the line integral in the varialional expression [1 5). IEEE' Log Nu mber9104779.In this paper a simple and efficient FEM fo r the analysis of microwave and optical waveguiding problems is formulated using three components of the electric or magnetic fi eld. [n order to el iminate spurious solut ions and 10 treat arbitrarily shaped waveguidcs, triangu la r edge elements are introduced . An e igenvalue equation derived here invo lves only the edge variables in the transversal plane and can prov ide a direct solution fo r the propagation constant. To show the validity and usefulness of this approach, examples are computed fo r microstrip transmission lines on isolropic or ani sotropic substrates, dielectric rectangular wavegu ides, and equi lateral triangular core waveguides. II . B ASIC EQUATIONSWe consider a dielectric...

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With a view to developing a method enabling derivation of the propagation characteristics of a three‐dimensional (3‐D) periodic waveguide with an arbitrary shape, an analysis technique based on the finite‐element method with tetrahedral elements is proposed and its formulation developed. For the finite‐element analysis of the electromagnetic field in a 3‐D inhomogeneous region including a dielectric material, the variational expression in terms of three components of magnetic field is convenient. However, if the conventional variational expression is used directly for solving eigenvalue problems, spurious solutions can appear. In this paper, to suppress and eliminate these spurious solutions, various countermeasures to spurious solutions, developed for the finite‐element analysis of a waveguide uniform in the propagation direction that can equivalently be treated as a 2‐D problem. These include: the penalty function method, the transverse magnetic‐field component method and the reduction method, all of which are extended to the finite‐element analysis of 3‐D periodic waveguide problems. Specifically, by way of the numerical examples for a rectangular waveguide loaded with dielectric material with a periodic structure, the validity of the proposed method and the adaptability of each method for suppressing and eliminating spurious solutions are investigated.

A numerical approach to determine the propagation characteristics of periodic waveguides with circular symmetry is proposed. This approach is based on the finite‐element method using axisymmetric triangular elements. Although it is convenient to use the functional in terms of three components of the magnetic field in the finite‐element analysis of the electromagnetic field in a three‐dimensional inhomogeneous region including a dielectric, if the eigenvalue problem is solved with the unaltered functional, spurious solutions do appear. Thus, to eliminate spurious solutions, the penalty function method developed for two‐dimensional finite‐element analysis of uniform waveguides is extended to analyze the axisymmetric periodic waveguide problem. Moreover, numerical integration based on the formulas derived by Hammer et al. is used for evaluating an element integral including singularity, which appears in the functional when it is expressed in a cylindrical coordinate system (r, θ, z). The validity and efficiency of the numerical analysis proposed here is verified by taking the example of circular waveguides loaded with a dielectric having periodic structure.