Proceedings of Progress in Electromagnetics Research Symposium (PIERS ’97), Kowloon, Hong Kong, vol. 2, p. 531, 1997 Galerkin's Method Applied to Mapped Infinite Domains: Vector Solutions for Optical Waveguide modes Kai Ming Lo and Alex Tak Ho Li Abstract: We present a numerical method to solve the vector wave equation for optical waveguide devices with an arbitrary two-dimensional refractive index profile. The method used is Galerkin method in a mapped domain. By mapping the whole x-y space onto an unit square eliminate the need of a computational window. This mapping technique have been successfully applied to solve the scalar wave equation for optical waveguides down to modal cut-off. The same technique is applied here to study the vector modes of optical guiding devices.
The transverse electric field of guiding structure is represented by two doubly sine series expansions. Using these field expansions, Galerkin's method is applied to convert the vector wave equation into an equivalent matrix eigenvalue equation. The eigenvalues (propagation constants of guiding modes) and eigenvectors (coefficients of the modes) are then found by solving the matrix eigenvalue problem. The size of matrix can be as large as 5000 × 5000 when 50 terms of sine basis functions are used in each direction. An parallel supercomputer (IBM 9076 SP2) is used for such intensive computation.
We check the implementation of present method and compare results with the exact analytical solution and modal cut-off of a step-index circular core optical fibre with high refractive index contrast. We also study guiding structures such as rectangular waveguide, rib waveguide, and channel waveguide and present their modal propagation constants and field profiles as a function of number of sine basis functions. It is demonstrated that the present method is simple in implementation and useful for the analysis, modeling and design of optical guiding devices down to modal cut-off.
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