Metamaterials for cloaking: The implications of causality

Causality and passivity constraints appear in constitutive equations of any material or meta-material and characterise the interaction electromagnetic radiation with matter. These constraints result in the well-known physical limitations which affect the design of a cloak. However, there are items in the realm of partial differential equations, namely transmission eigenvalue problems in electromagnetics, which have ignored such physical limitations and are nonetheless believed to play a role in cloaking theory. Herewith, some properties of elliptic partial differential equations are recalled; the main properties of Maxwell-Herglotz pairs are listed; transmission eigenvalue problems are stated, their connections with the properties of the "far-field scattering" operator and to "nearly non-scattering" solutions are discussed. Finally, results coming from transmission eigenvalue problems of electromagnetics, where material models ignore causality and passivity, are shown to be of limited application to cloaking.