Computation of light waves in photonic crystal slabs and crossed gratings

Abstract

In the last two decades, photonic crystals (PhCs) have been extensively studied due to their unusual ability to control and manipulate light. The most important property of a PhC is the existence of bandgaps, i.e. frequency intervals in which light cannot propagate. Since the fabrication of three-dimensional (3D) PhCs with sub-micro periods is a very challenging and expensive task, much attention has been turned to PhC slabs which are 3D structures with a two-dimensional (2D) periodicity. PhC slabs can confine light by the bandgap effect in the two periodic directions and index guiding (or total internal reflection) in the third direction. In this thesis, we consider two problems for PhC slabs: a basic boundary value problem for transmission and reflection spectra of finite number of hole arrays in a slab, and an eigenvalue of problem for waveguide modes of line defect waveguides in PhC slabs. These two problems are very important, since PhC slab waveguides are the basic building blocks in future photonic integrated circuits and they are usually separated by a finite number of unit cells. Since a PhC slab is a 3D structure, these two mathematical problems are computationally expensive. Existing numerical and semi-analytic methods for analyzing PhC slabs include the effective index method, the finitedifference time-domain method, the plane-wave expansion method, the multipole method, etc.. Among these methods, the multipole method is relatively efficient, but it is very complicated for slabs containing infinite number of holes. We develop a Dirichlet-to-Neumann (DtN) map method to solve these two problems. The DtN map of a unit cell is an operator that provides the relation between the wave field and its normal derivative on the cell boundary. It can be used to avoid further computation in the interiors of the unit cells and repeated calculations in identical unit cells. Unlike the multipole method, our method does not require lattice sums, it is much simpler. In this thesis, we also study crossed gratings involving circular inclusions. These are very important optical structures with applications in solar-selective elements, antireflection devices, etc.. Diffraction gratings with one dimensional periodicity have been extensively studied, but there are no efficient numerical methods for crossed gratings which are 3D structures with 2D periodicity. A widely used numerical method for crossed gratings is the Fourier modal method which unfortunately suffers from poor convergence. We develop a least squares semi-analytical modal method for crossed gratings with circular inclusions. Our method is accurate and efficient for calculating the eigenmodes of the grating layer. Diffraction efficiencies are obtained based on the computed eigenmodes and a least squares method.

Date of Award	4 Oct 2010
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Ya Yan LU (Supervisor)