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Novel boundary conditions based on Padé approximations for the frequency domain two-dimensional/finite element (2-D/FE) simulation of planar optical junctions of arbitrary geometry and number of accessing waveguides are presented and described in detail. This efficient formulation is straightforwardly implemented within the 2-D/FE framework and also can easily be used in finite difference schemes. Three examples show the applicability and reliability of the present method: a waveguide step discontinuity, waveguide transverse displacement and T-shaped beam splitter.

A numerical analysis of periodic segmented waveguides (PSWs) using the 2-D finite element method (2D-FEM) in the frequency domain is presented. This method has significantly lower computational cost when compared with 3-D methods that have been used to model PSWs, and can also model back reflected signals. Unlike photonic crystal waveguides, light confinement in a PSW is due to total internal reflection as in a continuous waveguide (CWG). We show that the dispersion relation of the guided modes in PSW is strongly influenced by the dielectric periodicity along the waveguide. We calculate the mode profile of a PSW in a region far away from the bandgap and we showed that it is comparable to the mode profile of the equivalent CWG even for relatively high values of averaged refractive index contrast.Index Terms-Continuum waveguide, equivalent index, finite element methods, mode profile, segmented waveguide.