Analytical solutions of the wave equation exist for only a few waveguide structures. Direct numeric solution of the wave equation is possible for many structures, although this usually involves iteration to find the approximate eigen-value. In this chapter we will discuss two methods used to study modes of structures which are not amenable to analytic or approximate solution. These methods are the Beam Propagation Method (BPM) and the Finite-Difference Time Domain method (FDTD). Numerical simulations are needed to evaluate special structures such as waveguides with bends or reflective mirrors, split Y couplers, and coupled adjacent waveguides [1, 2, 3], or structures which have reflections, such as a grating. A Y-coupler, shown in Fig. 9.1, is a simple device that connects one waveguide to two waveguides. Since the mode dynamically changes as it enters the structure, it is very difficult to calculate the answers to questions such as “How much loss will the mode encounter?”, “What is the optimum angle to split the waveguide?”, and so forth. The numerical simulation methods allow us to determine these answers.
KeywordsPlane Wave Fast Fourier Transfonn Spatial Mode Local Index Index Profile
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