Selected Numerical Approaches

Although the principal emphasis of this book is on the analytical solutions to canonical dielectric waveguide problems, the capability of numerical approaches to yield useful data for problems that had no analytical solutions must be recognized. The steady increase of computing power since 1960 has provided the impetus in increasing the use of numerical means to solve electromagnetic problems. In 1966, Yee [1] demonstrated the use of the finite difference numerical method (FD) to solve boundary value problems involving the Maxwell equations in isotropic media. Goell [2] in 1969 presented his circular-harmonic computer analysis to treat dielectric shapes that are close to a circle. The radially inhomogeneous circular dielectric waveguide problem was solved in 1973 by Dil and Blok [3] using the numerical integration technique, and in 1977 by Yeh and Lindgren [4] using the matrix multiplication technique. The finite element method (FEM) was first used by Yeh et al. [5] in 1975 to solve an arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguide. An improved version was given by Yeh et al. [6] in 1979. Meanwhile, a forward marching, split-step fast Fourier transform technique, also called the beam propagation method (BPM1), was used successfully to treat the problem of wave propagation in a fiber with radially inhomogeneous index variation by Yeh et al. [7] in 1977. A few months later, Feit and Fleck [8] popularized this BPM approach. Subsequently, Yeh et al. [9–11], also showed how this BPM may be used to solve the single mode or multi-mode inhomogeneous fiber coupler problems, the fiber branches, tapers, or horns problem and fibers with longitudinal index variations. The BPM provides good results provided that any longitudinal reflections due to longitudinal variations may be ignored.