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diff --git a/source/know/concept/kramers-kronig-relations/index.md b/source/know/concept/kramers-kronig-relations/index.md index bb9b08c..3880113 100644 --- a/source/know/concept/kramers-kronig-relations/index.md +++ b/source/know/concept/kramers-kronig-relations/index.md @@ -10,22 +10,22 @@ categories: 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 $\chi(t)$, +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 $\chi_i(t)$: +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, +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 $\chi(t)$: +$$\Theta(t)$$ to create a **causality test** for $$\chi(t)$$: $$\begin{aligned} \chi(t) = \chi(t) \: \Theta(t) @@ -34,7 +34,7 @@ $$\begin{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: +where $$A$$, $$B$$ and $$s$$ are constants from the FT definition: $$\begin{aligned} \tilde{\chi}(\omega) @@ -42,10 +42,10 @@ $$\begin{aligned} = B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'} \end{aligned}$$ -We look up the FT of the step function $\tilde{\Theta}(\omega)$, -which involves the signum function $\mathrm{sgn}(t)$, -the [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$, -and the Cauchy principal value $\pv{}$. +We look up the FT of the step function $$\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} @@ -58,7 +58,7 @@ $$\begin{aligned} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}$$ -From the definition of the Fourier transform we know that $2 \pi A B / |s| = 1$: +From the definition of the Fourier transform we know that $$2 \pi A B / |s| = 1$$: $$\begin{aligned} \tilde{\chi}(\omega) @@ -66,7 +66,7 @@ $$\begin{aligned} + \mathrm{sgn}(s) \frac{i}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}$$ -We isolate this equation for $\tilde{\chi}(\omega)$ +We isolate this equation for $$\tilde{\chi}(\omega)$$ to get the final version of the causality test: $$\begin{aligned} @@ -76,7 +76,7 @@ $$\begin{aligned} } \end{aligned}$$ -By inserting $\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$ +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: @@ -92,13 +92,13 @@ $$\begin{aligned} } \end{aligned}$$ -If the time-domain response function $\chi(t)$ is real +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: +$$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$$, 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: $$\begin{aligned} \tilde{\chi}_r(\omega) @@ -110,8 +110,8 @@ $$\begin{aligned} + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg) \end{aligned}$$ -For $\tilde{\chi}_r(\omega)$, the second integrand is odd, so we can drop it. -Similarly, for $\tilde{\chi}_i(\omega)$, the first integrand is odd. +For $$\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} @@ -126,7 +126,7 @@ $$\begin{aligned} } \end{aligned}$$ -To reiterate: this version is only valid if $\chi(t)$ is real in the time domain. +To reiterate: this version is only valid if $$\chi(t)$$ is real in the time domain. |