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).
Read more about this topic: Permittivity
Famous quotes containing the words dispersion and/or causality:
“The slogan offers a counterweight to the general dispersion of thought by holding it fast to a single, utterly succinct and unforgettable expression, one which usually inspires men to immediate action. It abolishes reflection: the slogan does not argue, it asserts and commands.”
—Johan Huizinga (1872–1945)
“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
—Jorge Luis Borges (1899–1986)