New methods for computing transmission and reflection spectra of photonic crystals and diffraction gratings

Abstract

Due to their unusual optical properties and significant potentials in applications, photonic crystals (PhCs) have been extensively studied both theoretically and experimentally. Many PhCs devices, such as waveguide bends, branches, frequency filters and waveguide couplers, are important building blocks of integrated optical circuits. Threedimensional (3D) PhCs with a complete bandgap in the optical wavelength region have potential applications, such as ultrahigh quality-factor cavities and zero-threshold lasers. The woodpile structures composed of alternating layers of rods have attracted much attention due to their relatively simple fabrication process compared with other 3D PhCs. Diffraction gratings have been extensively studied for many years, and they are important in practical applications, such as monochromators, spectrometers, lasers, wavelength division multiplexing devices, optical pulse compressing devices, etc. Numerical methods are essential in the design, analysis and optimization of photonic crystals and diffraction gratings. Some of these methods, such as the finitedifference time-domain (FDTD) method, are general methods that can be used to study various aspects of PhCs and diffraction gratings, but their accuracy and efficiency are often limited. FDTD requires a small mesh size to resolve curved material interfaces and often has difficulties truncating periodic structures that extend to infinity. The Fourier modal method (FMM) is suitable for diffraction gratings with uniform layers, but they are not so efficient and may have convergence problems when a general grating with sloping interfaces or a photonic crystal composed of cylinders must be approximated. The boundary integral equation (BIE) method is somewhat complicated to implement, since the integral operators are related to the quasiperiodic Green’s function which requires sophisticated lattice sums techniques to evaluate. The finite element method (FEM) is very general, but it gives rise to large, complex and indefinite linear systems that are expensive to solve. In this thesis, efficient numerical methods based on the Dirichlet-to-Neumann (DtN) or Neumann-to-Dirichlet (NtD) maps (of unit cells or homogeneous sub-domains) are developed for accurate simulations of two- and three-dimensional photonic crystals and diffraction gratings. For photonic crystals composed of interpenetrating cylinders, i.e. when the radius of the cylinders is larger than √3/4 of the lattice constant, an efficient DtN map method is developed for computing the reflection and transmission spectra. Our method manipulates a pair of operators defined on a set of curves. It is efficient since the wave field in the interiors of the unit cells are never calculated. This is achieved by using the DtN maps which map the wave field on the boundaries of the unit cells to its normal derivative. The DtN map method is also developed for two-dimensional photonic crystals with oblique incident waves. In that case, the DtN operator maps the two longitudinal field components to their normal derivatives on the boundary of the unit cell. For three-dimensional photonic crystals composed of crossed arrays of circular cylinders, including woodpile structures as special cases, we develop an efficient and accurate computational method. The method relies on marching a few operators from one side of the structure to another. The marching step makes use of the DtN maps for two-dimensional unit cells in each layer where the structure is invariant in the direction of the cylinder axes. A further simplification is developed based on the so-called Tangential-to-Tangential (T2T) operator which maps two transverse field components to two different transverse field components on the boundary of a 2D unit cell. These operators are approximated by matrices based on expansions in cylindrical waves. For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous sub-domains and computing the NtD maps for these sub-domains by boundary integral equations. For a sub-domain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The method retains the advantages of existing boundary integral equation methods for diffraction gratings, but avoids the quasi-periodic Green's functions that are expensive to evaluate. For diffraction problems in conical mounting, the NtD operator maps the normal derivatives of two longitudinal components of the electromagnetic field to these two components on the boundary of the sub-domain. A differentiation operator along the boundary is also needed to impose proper interface conditions. The method performs equally well for dielectric or metallic gratings. Keywords: Photonic crystal, Diffraction grating, Dirichlet-to-Neumann map, Operator marching, Neumann-to-Dirichlet map, Boundary integral equation

Date of Award	15 Jul 2010
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Ya Yan LU (Supervisor)