Review of Finite Element Methods for Microwave and Optical Waveguides zy B.M. AZIZUR RAHMAN, MEMBER, IEEE, F. ANIBAL FERNANDEZ, MEMBER, IEEE, AND zyxwvutsrq J. BRIAN DAVIES, MEMBER, IEEE zyxwvutsr This paper reviews the application of the finite element method to analysis of microwave and optical waveguide problems. It also discusses the significance zyxwvutsrq of different variational formula- tions, the modeling of the infinite domain of open-boundary wave- guides, techniques to avoid spurious solutions, and matrix solution techniques. We will also briefly refer to the application of these techniques to waveguides containing nonlinear materials and to three dimensional problems. I. INTRODUCTION The optimization of the performance of microwave and optical waveguide devices requires the knowledge of the propagation characteristics and field distributions and their dependence on the fabrication parameters. Therefore, there is a great interest in theoretical methods of waveguide analysis. It is appropriate to refer to two early review pa- pers, by Davies [l] and Ng [2] on microwave waveguides. More recent review papers by Saad [3] and Sorrentino [4] discuss some of the techniques for microwave and optical waveguides. This paper discusses two-dimensional finite element tech- niques applied to microwave and optical waveguide prob- lems. The waveguide is assumed to be uniform along its longitudinal zyxwvutsrqpon z axis. The electromagnetic fields at the frequency w have the form: where zyxwvutsrqpo P is the propagation constant in the positive z- direction. The general geometry of the guide can be quite complicated, with an arbitrary permittivity profile zyxwvutsr E(Z, y) in the transverse directions. As the range of guiding structures becomes more intri- cate, so the need for computer analysis becomes greater and more demanding. The finite element method has become a powerful tool throughout engineering for its flexibil- ity and versatility, being used in complicated structural, thermal, fluid flow, semiconductor, and electromagnetic problems. This method is particularly advantageous for electromagnetic field problems, because of its applicability to waveguides with arbitrary shape, arbitrary refractive index profile, and even anisotropy. zyxw 11. FINITE ELEMENT FORMULATION Finite element formulations are usually established via a variational or a Galerkin (method of moments [5] or weighted residuals) approach. The latter is more flexible but when it is possible, it is advantageous to take a variational approach, especially when one global parameter (like propagation constant) is needed. We will only refer to this form of derivation. A. Scalar Approximation Several different variational formulations have been pro- posed for use with the finite element method. The simplest is a scalar one in terms of the longitudinal components of TE or TM modes. This can be used in situations where the fields can be described as predominantly TE or TM. It has been used for solving homogeneous waveguide problems [6]-(91, for approximate analysis of lossy guides [lo], [ll], for open-boundary problems (121, [13], and for analysis of anisotropic waveguides [ll], [14]-[16]. This formulation can be written as [3], [14], [16]: Manuscript received May 16, 1990; revised January 8, 1991. B. M. A. Rahman is with the Department of Electrical, Electronic, and Information Engineering, The City University, Northampton Square, London EClV OHB, U.K. F. A. Fernandez and J. B. Davies are with the Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WClE 7JE, U.K. S S IEEE Log Number 9102642. where, 4 is either H, or E,. 0018-9219/91/$01.00 0 1991 IEEE 1442 PROCEEDINGS OF THE IEEE, VOL. 79, NO. 10, OCTOBER 1991