Chapter 6 - Finite Element Method

Abstract

This chapter focuses on the finite element method (FEM), which is suitable for the mode analysis of optical waveguides having arbitrary refractive-index profiles and complicated waveguide structures. FEM is also applicable to the stress analysis of optical waveguides. The chapter presents the FEM mode analysis of slab waveguides, rectangular waveguides, and optical fibers. Stress analysis of waveguides and the combination of stress analysis with mode analysis are also reviewed. In the finite element method, the domain of the problem is discretized into small elements. Therefore, FEM is applicable to complicated domain structures and to problems in which electromagnetic fields are localized. In FEM, a simple form of function is adopted to approximate the field in each element. The possible error in the solution is alleviated by increasing the number of elements thus reducing the element size. All of the element contributions to the system are assembled to form the functional. Waveguide analysis and stress analysis are totally different problems. However, the discretization procedures and the formulation of the functionals are rather similar in both problems. Such versatility in the mathematical procedures is the great advantage of FEM.