Modeling photonic crystal devices by Dirichlet-to-Neumann maps

Abstract

Photonic crystals (PhCs) are important artificial materials. In recent years, they have been extensivly studied both theoretically and experimentally, due to their unusual ability to control and manipulate light. Because of the periodicity of the dielectric constant, PhCs exhibit unusual dispersion properties and frequency gaps in which propagating Bloch waves do not exist. These properties have been widely used to design photonic crystal devices, such as waveguide bends, branches, frequency filters, waveguide couplers, Mach-Zehnder interferometers, etc. They are important building blocks of integrated optical circuits. Numerical methods are essential to analyze basic properties of photonic crystals and to design and optimize photonic crystal devices. Many PhC devices are simulated in time domain, for example, by the finite difference time domain (FDTD) method. For some problems, such as the propagation of a pulse, the time domain methods are essential. Other problems, such as the transmission and reflection spectra, are more naturally formulated in the frequency domain. However, even for two-dimensional (2D) problems, standard numerical methods for frequency domain formulations, such as the finite element method, often give rise to large linear systems that are complex, non-Hermitian, indefinite but sparse. These systems are expensive to solve by direct methods. Iterative methods often have a very slow convergence and even fail to converge, since existing preconditioning techniques for indefinite linear systems are not very effective. In this thesis, an efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the unit cells is developed for accurate simulations of two-dimensional photonic crystal devices in the frequency domain. An important advantage of frequency domain formulations is that they allow us to make use of the geometric features of the structure. We study devices in a 2D PhC composed of a lattice of infinitely long and parallel cylinders in a homogeneous background, such as air-holes in a dielectric medium or dielectric rods in air. For frequencies in a bandgap, microcavities and waveguides can be developed by introducing point and line defects and they can be further combined to produce various components and devices with many different functions. When cavities and waveguides are introduced as point and line defects, the structure loses its periodicity, but it still has many identical unit cells. Very often, there are only two different types of unit cells: the regular unit cell and the defect unit cell. Of course, the wave fields are different on different cells, but it is possible to take advantage of the many identical cells by using their Dirichlet-to-Neumann (DtN) maps. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative, and it can be approximated by a small matrix. Using the DtN maps of the regular and defect unit cells, we can avoid computations in the interiors of the unit cells and calculate the wave field only on the edges. This gives rise to a significant reduction in the total number of unknowns. Reasonably accurate solutions can be obtained using 10 to 15 unknowns for each unit cell. In contrast, standard finite element, finite difference or plane wave expansion methods may require a few hundreds unknowns for each unit cell at the same level of accuracy. For more complicated photonic crystal devices, there can be several thousands of unit cells in the truncated domain. We develop an improved Dirichlet-to-Neumann map method by incorporating an operator marching (OM) method for devices where a main propagation direction can be identified in at least part of the structure and a Bloch mode expansion technique for structures with partial periodicity along the main propagation direction. For simulating such large structures, standard finite element or finite difference method are prohibitively expensive, but our DtN map method with OM and Bloch mode expansion techniques is still efficient and accurate. To realize various applications of photonic crystals, it is important to have efficient coupling of light between a PhC waveguide and a different structure such as free space, a conventional waveguide and a different PhC waveguide. Our Dirichlet-to-Neumann map methods is also efficient to analyze these couplers. Besides bandgaps, photonic crystals also exhibit many unusual refraction properties. In the final part of this thesis, we demonstrate abnormal refraction phenomenon of photonic crystals by our DtN map method. Keywords: Photonic crystal, Numerical method, Dirichlet-to-Neumann map, Operator marching, Bloch mode expansion

Date of Award	2 Oct 2009
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Ya Yan LU (Supervisor)