Eigenmode Solver For Optical Waveguide With Varying Coefficients And It's Application In The Propagation Computation

Spectral element method (SEM), like the spectral method, expands the so-lution by a combination of basis functions, which uses high - order polynomials,and like the finite element method, decomposes the computational domain into a number of simpler domains where local basis functions are defined. The mesh adaptation algorithm, which is vital in the adaptive finite element method (FEM)and is introduced with the convergence study, "smoothness" and corresponding error estimate, "residues". In an optical waveguide, eigenmode analysis palys an important role. The spectral element method with mesh adaptation (viz, mod-ified spectral element method, MSEM), is studied in this thesis and is further applied in the eigenmode computation of optical waveguide with a varying re-fractive index profile. In the other side, a three dimensional operator marching method is proposed to solve the propagation problem in the three dimensional unbounded optical waveguide, where a three dimensional scalar Helmholtz equa-tion is focused and its efficiency has proved. The MSEM is also involved to solve the eigenmode problem. To be specific:The MSEM is utilized to solve the eigenmode problem for varying refractive index profile's unbounded waveguides, where the refractive index profile (RIP)of the waveguide is a two dimensional function that is not confine to one di-rection. Firstly, a one dimensional unbounded waveguide eignemode problem is studied where the transverse refractive index profile is a continuous function.The perfectly matched layers are used to truncate the unbounded space, so an unbounded eigenmode problem is transformed to a bounded problem. Through the MSEM, the eignemode distribution (including the propagation mode, the leaky mode and the Berenger mode) of this one dimensional problem is present-ed efficiently. Then the MSEM is further applied to the eigenmode problem of two dimensional refractive index profile's waveguides. Indeed, the MSEM is ap-plicable to waveguides with sharply varying RIP, not only be confined to those with slowly varying RIP. Through the compare with Chebyshev spectral colloca-tion method (CSCM), FEM and SEM, which are the classic numerical methods for the solutions of partial differential equation, the efficiency of MSEM has been proved, where uses less interpolation points to achieve the similar accuracy.The improved three dimensional (3D) operator marching method (3DOM-M) that extends the original operator marching method of two dimensions to three dimensions, is proposed to simulate the wave propagation in 3D infinite waveguides. Firstly, the scalar Helmholtz equation with the radiation boundary conditions is considered. The 3DOMM is based on the OMM for two dimensions,where supposes the wave in the propagation direction is slowly varying, mean-while, utilizes the MSEM to study the eigenvalue and eigenvector problem of two dimensional Helmholtz operator and marches piecewise. The modified 3DOMM,in one side, keeps the good merits of the MSEM in eigenmode computation, and in the other side, maintains the advantages of OMM, such as large range step and fast computing.