Calculating photonic band structure

Photonic materials structured on the scale of the wavelength of light have become the subject of an active field of research fed by the hope of creating novel properties. Theory plays a central role reinforced by the difficulty of manufacturing photonic materials: unusually we are better able to design a photonic material than to build it. In this review we explore the application of scattering theory to Maxwell's equations that has enabled theory to make such a central contribution: implementation of Maxwell's equations on a discrete mesh, development of the electromagnetic transfer matrix, order-N methods, and adaptive meshes. At the same time we present applications to a few key problems by way of illustration, and discuss the special circumstances of metallic photonic structures and their unique properties.