Vertical mode expansion method for scattering of light by multiply layered photonic structures

Abstract

A layered structure is a one-dimensional (1D) structure for which the material properties depend only on one spatial variable z. A multiply layered structure is a three-dimensional (3D) structure consisting of different cylindrical 1D layered structures in different regions. In photonics, the study of multiply layered structures is of great practical importance due to the existing fabrication techniques. Some important examples of multiply layered structures include a metallic film with cylindrical apertures, cylindrical metallic nanoparticles on a substrate, and a photonic crystal slab with cylindrical holes. A fundamental problem is to analyze the scattering of light by 3D multiply layered structures. Numerical methods, such as the finite-difference time-domain method, the finite element method, the volume and surface integral equation methods, can be used to solve these scattering problems. However, it is desirable and often possible to develop special numerical or semi-analytic methods that are more efficient and accurate than the general methods. The usual mode matching method (also called modal method or mode expansion method) is applicable to piecewise z-invariant structures, and it expands the electromagnetic field in each z-invariant segment using the eigenmodes of that segment. These eigenmodes are functions of two transverse variables. The method is not very efficient since a large number of eigenmodes are required, and they are full vectorial and expensive to calculate. In this thesis, we develop a vertical mode expansion method (VMEM) for analyzing the scattering of light by multiply layered structures. Our starting point is a mode expansion technique in general 1D layered structures. The electromagnetic field is expanded in 1D vertical modes which depend on z, where the “expansion coefficients” are functions of the two transverse variables and satisfy two-dimensional (2D) Helmholtz equations. Our VMEM requires the so-called Dirichlet-to-Neumann (DtN) or Neumann-to-Dirichlet (NtD) maps for the related 2D Helmholtz equations. These operators provide relations between the solutions and their normal derivatives on the boundaries of 2D domains in the xy plane. With the help of DtN or NtD maps, VMEM establishes a linear system by matching the tangential components of the electromagnetic filed on the vertical boundaries of the different regions. The VMEM gives a 2D formulation for the original 3D problem. It is relatively simple to implement and relatively efficient. In Chapter 3, we present a VMEM for multiply layered structures with an elliptic cylindrical region. The method is developed based on a numerical separation of variables in the elliptic coordinates. The key step is to calculate the DtN maps for 2D Helmholtz equations inside or outside an ellipse. For numerical stability reasons, we avoid the analytic solutions of the Helmholtz equations in terms of the angular and radial Mathieu functions, and construct the DtN maps by a fully numerical method. The method is used to analyze the transmission of light through an elliptic aperture in a metallic film, and the scattering of light by elliptic gold cylinders on a substrate. In Chapter 4, we develop a more general VMEM for layered cylindrical structures with arbitrary cross sections in a layered background. A boundary integral equation (BIE) is used to construct the NtD maps for 2D Helmholtz equations that appear in the mode expansion process. The method is applied to analyze subwavelength apertures in metallic films and nanoparticles on substrates. In Chapter 5, we further extend the VMEM to multiply layered periodic structures, such as a photonic crystal slab with a square lattice of holes, and a periodic array of metallic nanoparticles on a substrate. For a multiply layered periodic structure, a unit cell consists of different 1D layered cylindrical regions. A BIE is again used to construct the NtD maps, but the quasi-periodic boundary conditions must be incorporated in the process, and a graded mesh technique is used to handle the corner singularities. Using the VMEM for periodic structures, we calculate the transmission and extinction spectra for photonic crystal slabs and other plasmonic structures

Date of Award	2 Oct 2015
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Ya Yan LU (Supervisor)