Kramers-Kronig relations

In mathematics and physics, the Kramers-Kronig relations describe the relation between the real and imaginary part of a certain class of complex-valued functions. The requirements for a function $f (ω)$ to which they apply can be interpreted as that the function must represent the Fourier transform of a linear and causal physical process. If we write

f (ω) = f 1 (ω) + i f 2 (ω)

where $f 1$ and $f 2$ are real-valued "well-behaving" functions, then the Kramers-Kronig relations are

$f_1(\omega) = \frac{2}{\pi} \int_0^{\infty} \frac{\omega' f_2(\omega') d\omega'}{\omega^2 - \omega'^2}$

$f_2(\omega) = -\frac{2 \omega}{\pi} \int_0^{\infty} \frac{f_1(\omega') d\omega'}{\omega^2 - \omega'^2}$ .

The Kramers-Kronig relations are related to the Hilbert transform, and are most often applied on the permittivity $ε(ω)$ of materials. However, it must be noticed that in this case,

f (ω) = χ(ω) = ε(ω) / ε 0 - 1

where $χ(ω)$ is the electric susceptibility of the material. The susceptibility can be interpreted as the Fourier transform of the time-dependent polarization in the material after an infinitely short pulsed electric field, in other words the impulse response of the polarization.

Categories: Complex analysis

08-19-2006 13:07:39
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