TABLE OF CONTENTS Curvilinear directional couplers, branching and combining waveguides, S-shaped bent waveguides and tapered waveguides are indispensable components in constructing integrated optical circuits. Mode-coupling phenomena in the parallel directional couplers have been investigated in Chapter 4. In practical directional couplers, however, light coupling in the S-shaped bent waveguide regions in the front and rear of parallel waveguides should be taken into account so as to evaluate the propagation characteristics precisely. For axially varying waveguides, the FEM stationary mode analysis described in Chapter 6 should be modified by using the paraxial wave approximation and Galerkin s method [1]. The Beam propagation method (BPM) is the most powerful technique to investigate linear and nonlinear lightwave propagation phenomena in axially varying waveguides such as curvilinear directional couplers, branching and combining waveguides, S-shaped bent waveguides, and tapered waveguides. BPM is also quite important for the analysis of ultrashort light pulse propagation in optical fibers. Two kinds of BPM procedures are described in this chapter: one based on the fast Fourier transform (FFT) and one based on the finite difference method (FDM).
The three-dimensional scalar wave equation (Helmholtz equation), which is the basis of BPM, is expressed by
We separate electric field E( x, y, z) into two parts: the axially slowly varying envelope term of 
( x, y, z) and the rapidly varying term of exp ( ? jk n 0 z) . Here,
