Two-and Three-Dimensional Periodic Structures
Wave propagating characteristics in periodic structures (gratings) have been intensively studied for several decades; most of the research results were focused on those of 1D periodic structures employed as diffraction gratings or grating waveguides. In the previous chapters the phase and dispersion relations of waves supported in a 1D periodic medium have been studied using the Fourier and modal transmission-line approaches, respectively. Moreover, the scattering of a plane wave by a 1D periodic structure (or grating) and the guiding characteristics of a dielectric waveguide incorporating 1D periodic structures have also been investigated. In this chapter we will extend the 1D periodic structure to two dimensions: one that is periodic in two directions and homogeneous in the third direction. Furthermore, the 3D periodic structure, which can be considered as a stack of 2D periodic layers, is also taken into account for evaluating the eigenwave and scattering characteristics. The rigorous mode-matching method will be employed for solving the electromagnetic boundary-value problem of the two types of periodic structures including the metallic and dielectric cylinders. Specifically, the Brillouin and phase diagrams will be exploited to understand the physical picture of the wave process involved in the structure under consideration.