structural:reformatting, quality:corruption, structural:scholar_edition
Dispersion and causality

In general, a material cannot polarise instantaneously in response to an applied field. The more general formulation as a function of time is

P (t) = ε 0 ∫ − ∞ t χ e (t − t ′) E (t ′) d t ′. {\displaystyle \mathbf {P} (t)=\varepsilon _{0}\int _{-\infty }^{t}\chi _{e}\left(t-t'\right)\mathbf {E} (t')\,dt'.}

That is, the polarisation is a convolution of the electric field at previous times with time-dependent susceptibility given by χ e (Δ t) {\displaystyle \chi _{e}(\Delta t)}. The upper limit of this integral can be extended to infinity as well if one defines χ e (Δ t) = 0 {\displaystyle \chi _{e}(\Delta t)=0} for Δ t < 0 {\displaystyle \Delta t<0}. An instantaneous response corresponds to Dirac delta function susceptibility χ e (Δ t) = χ e δ (Δ t) {\displaystyle \chi _{e}(\Delta t)=\chi _{e}\delta (\Delta t)}.

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product, P (ω) = ε 0 χ e (ω) E (ω). {\displaystyle \mathbf {P} (\omega)=\varepsilon _{0}\chi _{e}(\omega)\mathbf {E} (\omega).}

The susceptibility (or equivalently the permittivity) is frequency dependent. The change of susceptibility with respect to frequency characterises the dispersion properties of the material.

Moreover, the fact that the polarisation can only depend on the electric field at previous times (i.e., χ e (Δ t) = 0 {\displaystyle \chi _{e}(\Delta t)=0} for Δ t < 0 {\displaystyle \Delta t<0}), a consequence of causality, imposes Kramers–Kronig constraints on the real and imaginary parts of the susceptibility χ e (ω) {\displaystyle \chi _{e}(\omega)}.