Diffraction by Perfectly Conducting Capacitive Grids: Photonic Band Structures and Circuit Models

Abstract

This paper outlines a theoretical formulation for the diffraction of plane waves by capacitive grids consisting of perfectly conducting cylinders, and focuses on the importance of the acoustic (lowest frequency) mode as the mechanism for long wavelength energy transmission. Particular attention is paid to boundary conditions and the form of the modes in the quasistatic limit as the wavenumber k approaches O. We develop a scattering matrix formulation and elucidate its properties in the long wavelength limit (for which there is only a single propagating order) using the Sherman-Woodbury formula. With this, we demonstrate a circuit model for grids of infinitesimal thickness, and a thin film model for thick grids. Questions of homogenisation are considered and results applicable to finitely conducting grids are discussed.