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In this review, we take an extensive look at the role that the principles ofcausality and passivity have played in various areas of physics andengineering, including in the modern field of metamaterials. The aim is not toprovide a comprehensive list of references as that number would be in thethousands, but to review the major results and contributions which haveanimated these areas and to provide a unified framework from which tounderstand the developments in different fields. Towards these goals, we chartthe early history of the field through its dual beginnings in the analysis ofthe Sellmeier equation and in Hilbert transforms, giving rise to the farreaching dispersion relations in the early works of Kramers, Kronig, andTitchmarsh. However, these early relations constitute a limited result as theyonly apply to a restricted class of transfer functions. To understand how thisrestriction can be lifted, we take a quick detour into the distributionalanalysis of Schwartz. This approach expands the reach of the dispersionanalysis to distributional transfer functions and also to those functions whichexhibit polynomial growth properties. To generalize the results even further totensorial transfer functions, we consider the concept of passivity - originallystudied in the theory of electrical networks. We clarify why passivity impliescausality. Subsequently, as special cases, we present examples of dispersionrelations from several areas of physics including electromagnetism, acoustics,seismology, reflectance measurements, and scattering theory. We discuss sumrules, derivative analyticity relations, and nearly local approximations.Finally we review the clever applications of ideas from causality and passivityto the recent field of metamaterials. These ideas have provided limits to whatcan be achieved in metamaterial property design and metamaterial deviceperformance.

Causality and Passivity: from Electromagnetism and Network Theory to Metamaterials