IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 9, NO. 10, OCTOBER 1999 389 Efficient Finite-Element Analysis of Tilted Open Anisotropic Optical Channel Waveguides Giovanni Tartarini and Hagen Renner Abstract— An iterative method to analyze open anisotropic dielectric waveguides with arbitrary dielectric tensor such as tilted channel waveguides is presented. The proposed formula- tion approach reduces significantly the computational resources compared to previously proposed other choices. Index Terms—Anisotropic waveguides, finite-element method, integrated optics, open boundary conditions. I. INTRODUCTION T HE FINITE-ELEMENT method (FEM) is a particularly versatile numerical tool which allows an accurate analysis of the electromagnetic behavior of open dielectric waveguides. The most common implementations of this technique rely on the so-called formulation [1], which requires the propagation constant as an input parameter and gives the values of the angular frequency of the modes of the structure as solutions of a generalized eigenvalue problem. Since the typical problem to be solved considers the angular frequency (or the wavelength in case of optical waveguides) as the given value, the program should then be run several times, until the desired value of is obtained through interpolation. The so-called formulation has then been proposed, [2]–[5] where solution is found in a single run as now the input parameter is and is the eigenvalue. This formulation works also when the field is not well confined, if proper absorbers are placed at the computational domain boundaries [6]. However, these boundary conditions (BC’s) do not allow a direct determination of the complex propagation constant of the leaky modes, and this parameter is often of great importance in the design of integrated-optics (IO) devices [7] and in the study of waveguide fabrication processes [8]. To overcome this problem, Fernandez and Lu proposed BC’s which are function of the propagation constant [4] and lead to the determination of the complex propagation constant itself. Despite its iterative nature, this formulation is still superior, as the formulation in this case would require a double cycle of nested iterations. Unfortunately, the generality of this approach is limited by the hypothesis that the permittivity tensor of the dielectric material exhibits . In practice, there is a variety of interesting cases Manuscript received February 24, 1999; revised July 30, 1999. This work was supported by the Consortium CINECA of Italy in the framework of the ICARUS Project of the European Union, the Italian MURST, and CNR. G. Tartarini is with Dipartimento di Elettronica Informatica e Sistemistica, University of Bologna, Italy. H. Renner is with AB Optik und Messtechnik, Technische Universit¨ at Hamburg-Harburg, Germany. Publisher Item Identifier S 1051-8207(99)08532-3. for which this assumption does not apply. For example, we cite important devices made with Titanium Indiffused Lithium Niobate (Ti : LiNbO ) like mode convertors [14] and polarization controllers [10] or frequency shifters [11] and LiNbO devices with bent or branching waveguide sections [12]. To solve general open anisotropic waveguides, one could then seek a similar iterative solution exploiting previously published approaches. For example, Hayata et al. proposed a formulation with completely general permittivity tensor [2], but in this case the size of the algebraic problem to be solved is 8 , where is the number of nodes of the mesh and the matrices involved in the calculations are only partially sparse. Therefore, the iterative solution would be extremely time consuming. Some improvements can be obtained with the formulation proposed by Svedin [3] where matrix sparsity is preserved. However, the size of the matrices themselves is , which is still high and makes also this approach heavy for iterative solutions. In this work, we present a formulation of the FEM which is applicable to open dielectric waveguides where the permittivity tensor can be of completely general form. Moreover, this method reaches iteratively the solution of an algebraic system of size 3 only and also preserves matrix sparsity, leading to a low amount of computational resources required to reach the desired solution. In the following, we will present how this approach can be developed. Subsequently, an example of application of the approach will be illustrated. Finally, conclusions will be drawn. II. THEORY We consider a -invariant dielectric waveguide where the tensor has the most general form The elements of can also be complex, namely the material can exhibit loss or gain, which is important in the analysis of optical amplifiers. Assuming a dependence on time and on the direction of the form , we derive from Maxwell’s Equations the reduced wave equation (1) where is the free-space wavenumber. We apply the usual finite-element discretization to the cross- section of the waveguide using the Galerkin approach [2], which can be applied when the tensor is of the most general 1051–8207/99$10.00  1999 IEEE