A vector H -field finite-element method has been used for the solution of optical waveguide problems. The permittivity of the guiding structures can be an arbitrarily tensor, only limited to being lossless. To extend the domain of the field representation, infinite elements have been introduced. To eliminate spurious solutions and to improve eigenvectors, a penalty function method has been introduced. To show the validity and usefulness of this formulation, computed results are illustrated for step channel waveguide, diffused channel waveguide, anisotropic channel waveguide, and channel waveguide directional couplers.

Finite-element solution of integrated optical waveguides

A vector H-field formulation is developed for electromagnetic wave propagation for a wide range of guided-wave problems. It is capable of solving microwave or optical waveguide problems with arbitrarily anisotropic materials. We have introduced infinite elements to extend the region of explicit field representation to infirdly, to consider open-type waveguides more accurately. Computed results are given for a variety of optical planar guides, image lines, and waveguides containing skew anisotropic dielectic.

Finite-Element Analysis of Optical and Microwave Waveguide Problems

The finite element method is a well-established method for the solution of a wide range of guided wave problems. One drawback associated with the powerful vector formulation is the appearance of spurious or nonphysical solutions. A penalty function method has been introduced to the finite element formulation, to reduce or eliminate spurious solutions. It also improves the quality of the physical field solutions. The method has been applied for the solution of metallic homogeneous and inhomogeneous guides, and integrated optics guides.

Penalty Function Improvement of Waveguide Solution by Finite Elements

The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current interest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary interest is with the computational methods, there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in themselves. These results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some of the more powerful techniques available are illustrated by applying them to a model problem.

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Eigenvalues of the Laplacian in Two Dimensions

It is pointed out that for a closed cavity should not be solved by the 'vectorial' analog of the classical finite-element method (that is, with node-based finite elements). It is suggested that there are good reasons to use edge-elements instead. Such elements (whose degrees of freedom correspond to the edges of the mesh, not its nodes) not only guarantee the right kind of continuity for fields like h and e, but also prevent the appearance of unwanted, unphysical divergent solutions in the problem of eigenmodes, the so-called spurious modes. >

Solving Maxwell equations in a closed cavity, and the question of 'spurious modes'

Dispersion characteristics for open-typed dielectric waveguide structures operated at millimeter-and submillimeter-wave frequencies are calculated by a finite-element iterative procedure with a given criterion on the maximum field strength at the virtual boundary. Numerical results for a rectangular dielectric image guide are presented and compared with results from other methods. The strip dielectric guide and the insulated image guide with finite- or infinite-width substrates are also analyzed.

Analysis of Open-Type Dielectric Waveguides by the Finite-Element Iterative Method

An iterative test for H -matrix

We propose a preconditioned iterative method with the preconditioning matrixPsm =I +S +Sm forAx =b, whereA is an irreducibly diagonal dominantZ-matrix with unit diagonal. The convergence property and the comparison theorem of the proposed method are discussed. Moreover, some numerical examples are reported to confirm the theoretical analysis.

Hideo Sawami

Papers

Analysis of Open-Type Dielectric Waveguides by the Finite-Element Iterative Method

An iterative test for H -matrix

The Gauss-Seidel iterative method with the preconditioning matrix (I +S +Sm)

Spurious solutions in the finite-element analysis of microstrip lines

The alpha interpolation method for the solution of an eigenvalue problem