| Summary: | 碩士 === 國立中山大學 === 光電工程學系研究所 === 104 === For passive dielectric waveguide devices, we are primary concerned with their steady-state behaviors or narrow-band characteristics. Hence the frequency-domain (FD) methods, such FD finite-difference (FD-FD) methods are often used for solving the Helmholtz equation. Traditional FD-FD methods cannot effectively process points near a dielectric interface with large index contrast. In recent years, we proposed the semi-analytical method of Connected Local Fields (CLF) to solve Helmholtz equation with complex dielectric structures.
In this thesis we will address several key mathematical techniques in order to demonstrate that CLF method can be a highly-accurate method for analysis of two-dimensional complex waveguide structures. First, we review several FD-FD methods in handling dielectric media with interfaces: These includes a straight forward implementation of the interface conditions (such as continuity of normal derivative), a “material averaging” approach for obtaining compact coefficients stencils (CS) of the Helmholtz equation near a dielectric interface and our CLF’s approach of obtaining CS for cells with an inclined linear interface. In chapter 4 we will develop CLF-worthy compact stencils for implementing the transparent boundary condition (TBC) based on local plane wave extrapolation. We have also developed a second-order accurate compact stencils for cell with dielectric corners. Finally we will apply piece-wise linear approximation (PWLA) for dielectric curved interfaces. Combining these techniques we study local corner errors in the 2D rectangular structure as well as PWLA errors in a 2D circular dielectric structure. We report here that compared with exact Green’s functions of the circular dielectric disk we found the PWLA errors are well under two percent for large core-cladding index contrast (3.5:1.0) and much less for small contrasts at low sampling density of 10 points per wavelength.
After two-D dielectric structures, the future work will be focused on the two and a half D dielectric waveguide problems where we encounter three-D wave fields inside a two-D structure. Here we exam a formulation based on coupled longitudinal EM components, namely and When combined with the CLF method, this formulation has the advantage of being able to choose an appropriate local coordinate system whose axes are parallel or perpendicular to the dielectric interface. Thus the complexity of coupled formulation may be greatly reduced.
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