Computing photonic crystal waveguide and cavity modes using dirichlet-to-neumann maps

Abstract

In this thesis, efficient numerical methods are developed to calculate differ ent photonic crystal modes such as propagating modes, cavity modes and the corresponding leaky modes based on the Dirichlet-to-Neumann(DtN) maps of the defect and normal unit cells. A DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative. Due to the periodicity of the photonic crystals, it is advantageous to find the DtN maps so that further computation can be reduced to the edges of the unit cells. For a unit cell containing a circular cylinder, the DtN map can be easily approximated by a small size matrix based on the cylindrical wave expansions. For a perfect photonic crystal waveguide where the PhC cladding is infinite and the propagation constant is real, we develop two approaches to find the propagating modes. The first approach is to solve a linear eigenvalue problem with the DtN map of a supercell. The second approach is a non-linear eigenvalue problem using boundary conditions that terminate infinite PhC claddings. For a leaky waveguide where the PhC cladding is finite and the propagation constant is complex, we find the leaky modes by establishing exact boundary conditions terminating the surrounding homogeneous media. For perfect cavities in an infinite PhC, we solve the cavity modes by reducing the problem to a condition on edges of the defect cell. For leaky cavities surrounded by a finite PhC, the multipole method is used. For leaky cavities coupled to PhC waveguides or homogeneous media, boundary conditions are established to terminate semi-infinite waveguides or the free space. Our DtN method reduces original eigenvalue problems to conditions that involves small matrices. It is proved to be efficient and accurate by a number of numerical examples.

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