Causality and passivity: From electromagnetism and network theory to metamaterials

Abstract

In this review, we take an extensive look at the role that the principles of causality and passivity have played in various areas of physics and engineering, including in the modern field of metamaterials. The aim is not to provide a comprehensive list of references as that number would be in the thousands, but to review the major results and contributions which have animated these areas and to provide a unified framework which could be useful in understanding the developments in different fields. Towards these goals, we chart the early history of the field through its dual beginnings in the analysis of the Sellmeier equation and in Hilbert transforms, giving rise to the far reaching dispersion relations in the early works of Sokhotskii, Plemelj, Kramers, Kronig, and Titchmarsh. However, these early relations constitute a limited result as they only apply to a restricted class of transfer functions. To understand how this restriction can be lifted, we take a quick detour into the distributional analysis of Schwartz, and discuss the dispersion relations in the context of distribution theory. This approach expands the reach of the dispersion analysis to distributional transfer functions and also to those functions which exhibit polynomial growth properties. To generalize the results even further to tensorial transfer functions, we consider the concept of passivity — originally studied in the theory of electrical networks. We clarify why passivity implies causality and present generalized dispersion relations applicable to transfer functions which are distributional, tensorial, and possibly exhibiting polynomial growth. Subsequently, as special cases, we present examples of dispersion relations from several areas of physics including electromagnetism, acoustics, seismology, reflectance measurements, and scattering theory. We discuss sum rules which follow from the infinite integral dispersion relations and also how these integrals may be simplified either by truncating them under appropriate assumptions or by replacing them with derivative relations. These derivative relations, termed derivative analyticity relations, form the basis of the so called nearly-local approximations of the dispersion relations which are extensively employed in many fields including acoustics. Finally, we review the clever applications of ideas from causality and passivity to the recent field of metamaterials. In many ways, these ideas have provided limits to what can be achieved in metamaterial property design and metamaterial device performance.