Kramers-Kronig Relations

The Kramers-Kronig relations are a special case of the Hilbert transforms. They are those dispersion relations that relate the complex parts of the susceptibility of a material. Thus, they relate the complex parts of the dielectric function and the complex parts of the propagation vectors, such as the index of refraction and the loss factor or absorption. The Kramers-Kronig relations hold if the system is stationary and if one can use the convolution theorem. There is a breakdown in metals because of a pole at ω = 0 due to the DC conduction. In such a case, the pole is separated into an extra term in the imaginary part of the dielectric constant.