A Nonlinear Eigenvalue Hybrid FEM Formulation for Two Dimensional Open Waveguiding Structures

A non linear hybrid finite element formulation, for the two dimensional eigenvalue analysis of open waveguides is proposed. The infinite solution domain of this kind of problems is divided into two region using a fictitious cylindrical surface-C. Inside the surface-C the finite element method is employed. Outside the surface-C the infinite domain is modeled through an infinite sum of cylindrical harmonics. The two solutions are coupled considering the continuity of the tangential field components along the surface-C. The overall procedure ends up to a nonlinear eigenvalue problem of the form A(β) � x = 0 where β is the propagation constant along the axis of the waveguide. For the solution of the nonlinear eigenvalue problem the Regula Falsi method is considered. The solution procedure is based on the initial values provided from a linear approximation of the problem. Finally, the validity of the method is verified by comparison with measurements presented in the bibliography. Introduction During the last years a particular research effort is directed towards the solution of the eigenvalue problem of arbitrary cross-section waveguiding structures in a unified and general way. All of the numerical techniques developed towards this direction try to convert the open-radiating problem to an equivalent closed one and in that way truncating the solution domain. This can be achieved either by making use of an artificial boundary transparent to the solution or by combining the Finite Element Method (FEM) with methods, such as the method of moments, capable of modeling the unbounded region. When the artificial boundary is considered one method to truncate the solution domain is to impose on it either the Absorbing Boundary Conditions (ABCs) or employ the Perfect Matching Layer (PML). An alternative method is to express the field in the unbounded region as an expansion of solution satisfying both the Maxwell equations and the radiation condition. However, for the two dimensional (2D) open waveguides, the performance of the already proposed techniques in the solution of the corresponding eigenvalue problem, is very poor. In particular, while PML is quite efficient in the estimation of the field distribution generated by a specific source, when this is used in the solution of eigenvalue problem leads to spurious (or corrupted) solutions. In the present work a hybrid finite element method capable of handling problems involving open arbitrary shaped waveguides is described. The problem at a first stage is approximated by means of a linear eigenvalue formulation. The formulation is derived by combining the finite element method and an approximate expansion in cylindrical harmonics (1). Namely, the radial wavenumber in the unbounded media is considered approxi- mately equal to that of free space. This is a reasonable approximation for the spectral region only around cut-off. The eigenvalues calculated using this approach were in good agreement with experimental results. However, aiming at a more generally valid method, the present effort considers the accurate radial wavenumber, which unfortunately (as it is already expected) yields a non-linear eigenvalue problem. The final nonlinear algebraic system is formulated employing a full electric field FEM formulation discretized by mixed edge/node triangular elements. The final nonlinear system is solved using the Regula Falsi technique (2) and employing the solution of the first linear approach as an initial guess.