A generalized finite difference approach to the computation of modes

This thesis deals with numerical techniques for the computation of modes in electromagnetic structures with arbitrary geometry. The approach proposed in this work is based on the Finite Difference (FD) and Vector Finite Difference (VFD), which are applied to rectangular, circular, elliptical geometries, and to combination of them. The FD is applied using a 2D cartesian, polar and elliptical grid in the waveguide section. A suitable Taylor expansion of the mode function allows, either for scalar and for vector FD, to take exactly into account the boundary condition. To prevent the raising of spurious modes, the VFD approximation results in a constrained eigenvalue problem, that has been solved using a decomposition method. All approaches presented have been validated comparing the results to the analytical modes of rectangular and circular waveguide, and to known data for the elliptic case. The standard calculation of the waveguide modes using FD requires the use of two different grids, namely one for TE modes and the other for TM modes, due to the different boundary condition. It has been shown that a single grid can be used for all modes, thus allowing an effective mode-matching solution. The FD approach has been extended to waveguides (and apertures) with irregular boundaries, and therefore non-regular discretization grids. It has been shown that a suitable FD approximation of the Laplace operator is still possible. A ridged-waveguide, with trapezoidal ridges, and a rounded-ended waveguide have been considered in detail.