Permittivity - Dispersion and Causality

Dispersion and Causality

In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is

That is, the polarization is a convolution of the electric field at previous times with time-dependent susceptibility given by χ(Δt). The upper limit of this integral can be extended to infinity as well if one defines χ(Δt) = 0 for Δt < 0. An instantaneous response corresponds to Dirac delta function susceptibility χ(Δt) = χ δ(Δt).

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Because of the convolution theorem, the integral becomes a simple product,

This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. χ(Δt) = 0 for Δt < 0), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility χ(0).