A powerful algorithm based on a finite element method (FEM) is newly formulated for the analysis of waveguide discontinuities. In an earlier approach, FEM was applied to the finite region with discontinuites, and a mode expansion technique was applied to the uniform waveguides that are connected to the input and output ports of finite region. Although, in the present approach,the uniform waveguides are replaced by perfectly matched layer (PML) boundary conditions, it is possible to treat periodically varying waveguide structures such as photonic crystal (PC) waveguides. A combined method of beam propagation method (BPM) and FEM is also shown in such a form that a mode expansion technique is not required. To show the validity and usefulness of the present approach,numerical examples for optical gratings, circuit components based on PC waveguides and optical directional couplers are presented.


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