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+---
+title: "Kramers-Kronig relations"
+date: 2021-02-25
+categories:
+- Mathematics
+- Complex analysis
+- Physics
+- Optics
+layout: "concept"
+---
+
+Let $\chi(t)$ be a complex function describing
+the response of a system to an impulse $f(t)$ starting at $t = 0$.
+The **Kramers-Kronig relations** connect the real and imaginary parts of $\chi(t)$,
+such that one can be reconstructed from the other.
+Suppose we can only measure $\chi_r(t)$ or $\chi_i(t)$:
+
+$$\begin{aligned}
+    \chi(t) = \chi_r(t) + i \chi_i(t)
+\end{aligned}$$
+
+Assuming that the system was at rest until $t = 0$,
+the response $\chi(t)$ cannot depend on anything from $t < 0$,
+since the known impulse $f(t)$ had not started yet,
+This principle is called **causality**, and to enforce it,
+we use the [Heaviside step function](/know/concept/heaviside-step-function/)
+$\Theta(t)$ to create a **causality test** for $\chi(t)$:
+
+$$\begin{aligned}
+    \chi(t) = \chi(t) \: \Theta(t)
+\end{aligned}$$
+
+If we [Fourier transform](/know/concept/fourier-transform/) this equation,
+then it will become a convolution in the frequency domain
+thanks to the [convolution theorem](/know/concept/convolution-theorem/),
+where $A$, $B$ and $s$ are constants from the FT definition:
+
+$$\begin{aligned}
+    \tilde{\chi}(\omega)
+    = (\tilde{\chi} * \tilde{\Theta})(\omega)
+    = B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'}
+\end{aligned}$$
+
+We look up the FT of the step function $\tilde{\Theta}(\omega)$,
+which involves the signum function $\mathrm{sgn}(t)$,
+the [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$,
+and the Cauchy principal value $\pv{}$.
+We arrive at:
+
+$$\begin{aligned}
+    \tilde{\chi}(\omega)
+    &= \frac{A B}{|s|} \pv{\int_{-\infty}^\infty \tilde{\chi}(\omega')
+    \Big( \pi \delta(\omega - \omega') + i \:\mathrm{sgn} \frac{1}{\omega - \omega'} \Big) \dd{\omega'}}
+    \\
+    &= \Big( \frac{1}{2} \frac{2 \pi A B}{|s|} \Big) \tilde{\chi}(\omega)
+    + i \Big( \frac{\mathrm{sgn}(s)}{2 \pi} \frac{2 \pi A B}{|s|} \Big)
+    \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
+\end{aligned}$$
+
+From the definition of the Fourier transform we know that $2 \pi A B / |s| = 1$:
+
+$$\begin{aligned}
+    \tilde{\chi}(\omega)
+    &= \frac{1}{2} \tilde{\chi}(\omega)
+    + \mathrm{sgn}(s) \frac{i}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
+\end{aligned}$$
+
+We isolate this equation for $\tilde{\chi}(\omega)$
+to get the final version of the causality test:
+
+$$\begin{aligned}
+    \boxed{
+        \tilde{\chi}(\omega)
+        = - \mathrm{sgn}(s) \frac{i}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
+    }
+\end{aligned}$$
+
+By inserting $\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$
+and splitting the equation into real and imaginary parts,
+we get the Kramers-Kronig relations:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \tilde{\chi}_r(\omega)
+            &= \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega' - \omega} \dd{\omega'}}
+            \\
+            \tilde{\chi}_i(\omega)
+            &= - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega' - \omega} \dd{\omega'}}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+If the time-domain response function $\chi(t)$ is real
+(so far we have assumed it to be complex),
+then we can take advantage of the fact that
+the FT of a real function satisfies
+$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$, i.e. $\tilde{\chi}_r(\omega)$
+is even and $\tilde{\chi}_i(\omega)$ is odd. We multiply the fractions by
+$(\omega' + \omega)$ above and below:
+
+$$\begin{aligned}
+    \tilde{\chi}_r(\omega)
+    &= \mathrm{sgn}(s) \bigg( \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
+    + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg)
+    \\
+    \tilde{\chi}_i(\omega)
+    &= - \mathrm{sgn}(s) \bigg( \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
+    + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg)
+\end{aligned}$$
+
+For $\tilde{\chi}_r(\omega)$, the second integrand is odd, so we can drop it.
+Similarly, for $\tilde{\chi}_i(\omega)$, the first integrand is odd.
+We therefore find the following variant of the Kramers-Kronig relations:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \tilde{\chi}_r(\omega)
+            &= \mathrm{sgn}(s) \frac{2}{\pi} \pv{\int_0^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
+            \\
+            \tilde{\chi}_i(\omega)
+            &= - \mathrm{sgn}(s) \frac{2 \omega}{\pi} \pv{\int_0^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+To reiterate: this version is only valid if $\chi(t)$ is real in the time domain.
+
+
+
+## References
+1.  M. Wubs,
+    *Optical properties of solids: Kramers-Kronig relations*, 2013,
+    unpublished.