Numerical simulation techniques for optical waveguides and photonic crystals

Abstract

In this thesis, a number of numerical methods are developed for conventional optical waveguides, photonic crystals and two-dimensional scattering prob- lems. For a straight waveguide, the most important problem is to determine the propagating modes. Many di®erent numerical methods have already been de- veloped for this purpose. The ¯nite di®erence method is simple to implement, but it gives rise to eigenvalue problems of large sparse unsymmetric matri- ces. Iterative methods can be used, but the convergence is often very slow. Rayleigh Quotient Iteration (RQI) requires few iterations for convergence, but each iteration is expensive since one has to solve a large linear system. We develop a domain decomposition method for solving the linear system. The overall computation cost of the RQI is signi¯cantly reduced. The scattering problem at a longitudinal discontinuity along the waveguide axis is important in applications. For two dimensional waveguide discontinu- ities, a number of e±cient iterative methods have been developed based on rational approximations of square root operators. For three dimensional waveg- uide discontinuities, an existing method based on the paraxial preconditioner appears to have a slow convergence. In this thesis, we develop an e±cient iterative method for the three-dimensional waveguide discontinuity problem based on a new preconditioner using the semi-vectorial approximations. Photonic Crystals (PhC) have attracted much attention in recent years. In this thesis, we develop a new approach to analyze PhC structures. Our approach is based on the Dirichlet-to-Neumann (DtN) map which maps the wave ¯eld on the boundary of a unit cell to its normal derivative. The DtN approach is ¯rst used to analyze the transmission and re°ection of a PhC of ¯nite size. Compared with existing methods for this problem, our method is simple and e±cient, since it avoids the discretization of the unit cell completely and does not require sophisticated lattice sums techniques. The DtN approach is also applied to compute propagating modes in photonic crystal waveguides. Numerical examples are used to illustrate the e±ciency and accuracy of our methods. Finally, for the classical two-dimensional scattering problem of a perfectly conducting cylinder, we develop two fast approximate methods based on one- way models. The ¯rst method follows the Beam Propagation Method for optical waveguides. The second method uses a rational interpolation of the exact DtN map and a local approximation by circular scatterers. Numerical experiments are used to illustrate the accuracy of these two methods.

Date of Award	15 Feb 2007
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Ya Yan LU (Supervisor)