20 November 2013

The Kramers-Kronig relations - part 2

In part 1, we had stopped before going to the frequency domain  because we needed the Fourier transform of the sign function. This is where the technical difficulty appears, because we cannot simply write:
\operatorname{sgn}(\omega) = \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t) \operatorname{sgn}(t) \tag{5}
\end{equation} as the integral does not converge. One can however define\begin{align}
&\operatorname{sgn}(\omega) = \lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t - \epsilon |t|) \operatorname{sgn}(t) = \nonumber \\
&- \lim_{\epsilon \to 0} \left [ \frac{1}{i \omega + \epsilon} + \frac{1}{i \omega - \epsilon} \right ]= \lim_{\epsilon \to 0} \frac{2i \omega}{\omega^2 + \epsilon ^2}= \mathcal{P} \left ( \frac{2i}{\omega}\right ) \tag{6}
where the \(\mathcal{P}\) symbol (Cauchy's principal value) is a reminder that we should actually use the limiting process defined on the second line of \eqref{eq:sgnvp}. The sign and prefactor of the result depend on the convention used for the Fourier transform, so you might encounter different forms in the literature.
Using now this expression for \(\operatorname{sgn}(\omega)\) yields:\begin{equation}
\chi '(\omega) =  \mathcal{F}[\chi _{e}(t)] = \mathcal{F}[\operatorname{sgn}(t) \cdot \chi _{o}(t)] = \frac{1}{2 \pi} \operatorname{sgn}(\omega) \ast i \, \chi ''(\omega) \tag{7}
\end{equation}where the first equality is from (4), the second one from (3) and the third one simply states that the Fourier transform of a product is the convolution (denoted by the star) of the Fourier transforms of the factors. Inserting the result for \(\operatorname{sgn}(\omega)\) (6) and recalling the definition of the convolution:
\[ [f \ast g] (\omega) = \int_{-\infty}^{\infty} \text{d} \omega ' \, f(\omega - \omega ') \, g(\omega ')\]we have:\begin{equation}
\chi '(\omega) =  \frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi ''(\omega ')}{\omega ' - \omega} \tag{8}
\end{equation}which is the usual form of the Kramers-Kronig relation. Of course, one can obtain the equation for \(\chi ''(\omega)\) in exactly the same way.

A few final comments are in order concerning the principal value:
  • Although we introduced \(\mathcal{P}\) as a limiting process on regular functions, a more rigorous derivation requires using the concept of distribution.
  • The principal value is usually given a different definition, as \[\lim_{\epsilon \to 0} \left ( \int_{-\infty}^{-\epsilon} + \int_{\epsilon}^{\infty} \right ) \text{d} \omega \]By Sokhotsky's formula, this is however equivalent (in the sense of distributions) to our definition in the second line of \eqref{eq:sgnvp}.

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