Categories: Complex analysis, Mathematics, Optics, Physics.

Kramers-Kronig relations

Let χ(t)\chi(t) be the response function of a system to an external impulse f(t)f(t), which starts at t=0t = 0. Assuming initial equilibrium, the principle of causality states that there is no response before the impulse, so χ(t)=0\chi(t) = 0 for t<0t < 0. To enforce this, we demand that χ(t)\chi(t) satisfies a causality test, where Θ(t)\Theta(t) is the Heaviside step function:

χ(t)=χ(t) Θ(t)\begin{aligned} \chi(t) = \chi(t) \: \Theta(t) \end{aligned}

If we take the Fourier transform (FT) χ(t) ⁣→ ⁣χ~(ω)\chi(t) \!\to\! \tilde{\chi}(\omega) of this equation, the right-hand side becomes a convolution in the frequency domain thanks to the convolution theorem, where AA, BB and ss are constants determined by how we choose to define our FT:

χ~(ω)=(χ~∗Θ~)(ω)=B∫−∞∞χ~(ω′) Θ~(ω−ω′)dω′\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 full expression for Θ~(ω)\tilde{\Theta}(\omega), which involves the signum function sgn(t)\mathrm{sgn}(t), the Dirac delta function δ\delta, and the Cauchy principal value  P ⁣\pv{}. Inserting that, we arrive at:

χ~(ω)=AB∣s∣ P ⁣∫−∞∞χ~(ω′)(πδ(ω−ω′)+isgn(s)ω−ω′)dω′=(22πAB∣s∣)χ~(ω)+i sgn(s)(2π2πAB∣s∣) P ⁣∫−∞∞χ~(ω′)ω−ω′dω′\begin{aligned} \tilde{\chi}(\omega) &= \frac{A B}{|s|} \pv{\int_{-\infty}^\infty \tilde{\chi}(\omega') \bigg( \pi \delta(\omega - \omega') + i \frac{\mathrm{sgn}(s)}{\omega - \omega'} \bigg) \dd{\omega'}} \\ &= \bigg( \frac{2}{2} \frac{\pi A B}{|s|} \bigg) \tilde{\chi}(\omega) + i \: \mathrm{sgn}(s) \bigg( \frac{2 \pi}{2 \pi} \frac{A B}{|s|} \bigg) \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}

From the definition of the FT we know that 2πAB/∣s∣=12 \pi A B / |s| = 1, so this reduces to:

χ~(ω)=12χ~(ω)+i sgn(s)12π P ⁣∫−∞∞χ~(ω′)ω−ω′dω′\begin{aligned} \tilde{\chi}(\omega) &= \frac{1}{2} \tilde{\chi}(\omega) + i \: \mathrm{sgn}(s) \frac{1}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}

We rearrange this equation a bit to get the final version of the causality test:

χ~(ω)=i sgn(s)1π P ⁣∫−∞∞χ~(ω′)ω−ω′dω′\begin{aligned} \boxed{ \tilde{\chi}(\omega) = i \: \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}} } \end{aligned}

Next, we split χ~(ω)\tilde{\chi}(\omega) into its real and imaginary parts, i.e. χ~(ω)=χ~r(ω)+iχ~i(ω)\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega):

χ~r(ω)+iχ~i(ω)=i sgn(s)1π P ⁣∫−∞∞χ~r(ω′)ω−ω′dω′−sgn(s)1π P ⁣∫−∞∞χ~i(ω′)ω−ω′dω′\begin{aligned} \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega) = i \: \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega - \omega'} \dd{\omega'}} - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega - \omega'} \dd{\omega'}} \end{aligned}

This equation can likewise be split into real and imaginary parts, leading to the Kramers-Kronig relations, which enable us to reconstruct χ~r(ω)\tilde{\chi}_r(\omega) from χ~i(ω)\tilde{\chi}_i(\omega) and vice versa:

χ~r(ω)=−sgn(s)1π P ⁣∫−∞∞χ~i(ω′)ω−ω′dω′χ~i(ω)=sgn(s)1π P ⁣∫−∞∞χ~r(ω′)ω−ω′dω′\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}

The sign of these expressions deserves special attention: it depends on an author’s choice of FT definition via sgn(s)\mathrm{sgn}(s), and, to make matters even more confusing, many also choose to use the opposite sign in the denominator, i.e. they write ω′−ω\omega' - \omega instead of ω−ω′\omega - \omega'.

In the special case where χ(t)\chi(t) is real, we can take advantage of the property that the FT of a real function always satisfies χ~(−ω)=χ~∗(ω)\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega). Here, this means that χ~r(ω)\tilde{\chi}_r(\omega) is even and χ~i(ω)\tilde{\chi}_i(\omega) is odd. To use this fact, we simultaneously multiply and divide the integrands by ω+ω′\omega + \omega':

χ~r(ω)=−sgn(s)1π( ⁣ P ⁣∫−∞∞ωχ~i(ω′)ω2−ω′2dω′+ P ⁣∫−∞∞ω′χ~i(ω′)ω2−ω′2dω′)χ~i(ω)=sgn(s)1π( ⁣ P ⁣∫−∞∞ωχ~r(ω′)ω2−ω′2dω′+ P ⁣∫−∞∞ω′χ~r(ω′)ω2−ω′2dω′)\begin{aligned} \tilde{\chi}_r(\omega) &= - \mathrm{sgn}(s) \frac{1}{\pi} \bigg( \!\pv{\int_{-\infty}^\infty \frac{\omega \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} + \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \bigg) \\ \tilde{\chi}_i(\omega) &= \mathrm{sgn}(s) \frac{1}{\pi} \bigg( \!\pv{\int_{-\infty}^\infty \frac{\omega \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} + \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \bigg) \end{aligned}

In χ~r(ω)\tilde{\chi}_r(\omega)’s equation, the first integrand is odd, so the integral’s value is zero. Similarly, for χ~i(ω)\tilde{\chi}_i(\omega), the second integrand is odd, so we drop it too. We thus arrive at the following common variant of the Kramers-Kronig relations, only valid for real χ(t)\chi(t):

χ~r(ω)=−sgn(s)2π P ⁣∫0∞ω′χ~i(ω′)ω2−ω′2dω′χ~i(ω)=sgn(s)2π P ⁣∫0∞ωχ~r(ω′)ω2−ω′2dω′\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}{\pi} \pv{\int_0^\infty \frac{\omega \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \end{aligned} } \end{aligned}

Note that we have modified the integration limits using the fact that the integrands are even, leading to an extra factor of 22.

References

  1. M. Wubs, Optical properties of solids: Kramers-Kronig relations, 2013, unpublished.