Recent progress in research on the finite element method (FEM) for optical waveguide design and analysis is reviewed, focusing on the author's works. After briefly reviewing fundamentals of FEM such as a theoretical framework, a conventional nodal element, a newly developed edge element to eliminate nonphysical, spurious solutions, and a perfectly matched layer to avoid undesirable reflections from computational window edges, various FEM techniques for a guided-mode analysis, a beam propagation analysis, and a waveguide discontinuity analysis are described. Some design examples are introduced, including current research activities on multi-core fibers.

A formal necessary and sufficient condition on the general Petri net reachability problem is presented by eliminating all spurious solutions among known nonnegative integer solutions of state equation and unifying all the causes of those spurious solutions into a maximal-strongly-connected and siphon-and-trap subnet Nw. This result is based on the decomposition of a given net (N, Mo) with Md and the concepts of "no immature siphon at the reduced initial marking Mwo" and "no immature trap at the reduced end marking Mwd" on Nw which are both extended from "no token-free siphon at the initial marking Mo" and "no token-free trap at the end marking Md" on N, respectively, which have been both effectively, explicitly or implicitly, used in the well-known fundamental and simple subclasses.