Novel finite element analysis of optical waveguide discontinuity problems

In this paper, a novel finite-element method (FEM) to rigorously and efficiently solve the optical waveguide discontinuity problems is presented. Instead of performing the time-consuming modal solutions on both discontinuity sides, the square root of the characteristic matrix is efficiently approximated using Taylor's series expansion, and then the interface boundary conditions are enforced at the discontinuity plane to solve for the reflected and transmitted fields. The high numerical precision and effectiveness of the proposed method is demonstrated through the analysis of various discontinuity problems, and the excellent agreement of the results obtained using the present finite element method and those obtained using other rigorous approaches in the literature.