Exception to Causality of the Effective Refractive Index Function in Heterogeneous Media

Abstract

Optical constants are generally assumed to be causal functions. Mathematically, causality manifest itself in the Kramers–Kronig (KK) relations that relate the real and imaginary part of physical parameters in stable systems. These relations are also widely used to retrieve optical constants. Here, we present a stable physical heterogeneous system that is macroscopically described by a noncausal effective refractive index. We find that elastic scattering in heterogeneous media where particles have a lower refractive index than the surrounding matrix, results in optical constants that are not connected through KK relations. A modification of KK relations for anticausal media is presented and simulations with established scattering theories, numerical full-field electromagnetic analysis and experimental evidence support the agreement of these relations. Therefore, this physical system, although having a causal impulse response, has noncausal optical constants. These results offer a new insight into causality and restrict its range of applications, showing that some stable physical media can be described by noncausal functions. It also opens new paths to shape the electromagnetic response of artificial materials.