Research On Collocation Methods And Their Applications In Optical Waveguide Computation

Collocation methods (CMs) have become a kind of commonly used numeri-cal methods for the solutions of partial differential equations due to their simple forms and high computing efficiency. The CM expands the solution by a combina-tion of basis functions, while the unknown coefficients are determined by forcing the solution to satisfy the equation and boundary conditions on a set of pre-given collocating nodes. Two kinds of typical collocation methods with complementar-ity, the regularized meshless method (RMM) and the Chebyshev pseudo-spectral method (CPM), are studied here respectively and further applied in the com-putation of optical waveguides according to their different characteristics. The main contents include:Some new kinds of RMMs are proposed to solve the Laplace and Helmholtz equations on irregular domains. These new RMMs overcome the shortcomings of the original one, which can be only applied for regular domain problems. This thesis also derives the analytical diagonal elements'formula of RMM, and further analyzes the key factors that affect the RMM's solution accuracy. In addition, based on the new RMMs and the Chebfun package, this thesis introduces a new method to compute the cutoff wavelength of elliptical waveguides, which greatly improves the efficiency and stability for the analysis of waveguides with irregular shapes.A new one-dimensional multi-domain Chebyshev pseudo-spectral method (MCPM) is proposed for solving the full-vectorial modes of circular waveguides. Based on the idea of domain decomposition, this method overcomes the difficulty of discontinuous derivatives of electromagnetics at the interface of different me-dia, and guarantees the spectral accuracy of solution. It also takes advantage of the circular symmetry of the optical waveguide to reduce the dimension, thereby only one-dimensional mode equations are needed to be solved. Particularly, the MCPM truncates the unbounded space by the perfectly matched layer (PML), which makes it possible to calculate both the propagation and leaky modes ac- curately. Compared with the existing numerical methods, the MCPM greatly improves the accuracy and efficiency for the calculation of circular waveguide modes.Improved second-order and fourth-order operator marching methods (OMMs) are proposed respectively, based on the original OMM for lossless waveguide, to simulate the wave propagations in single and multi-layer infinite lossy waveg-uides. In the second-order improved OMM(OMM2), new local base transform techniques are introduced to overcome the difficulty caused by the non-self-adjoint transverse operator in lossy waveguides. In the fourth-order improved OMM(OMM4), the PML is used to truncate the unbounded domain and the MCPM is applied to discretize the transverse operators. It totally avoids the use of the local base transform and greatly reduces the computation cost. In addition, the analytical expression of the wave field can also be reconstructed in OMM4 by the barycentric Chebyshev interpolation. The improved OMMs greatly extend the application scope of the original method, and maintain the advantages such as small memory requirement, fast computing speed and large range step.