## Kramers–Kronig relations > [!Note] > Let $\chi =\chi_{1}(\omega)+ i \chi_{2}(\omega)$, $\chi_{1}$ and $\chi_{2}$ are real, then > > $\begin{aligned} > \chi_1(\omega) &= \frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \frac{\chi_2(\omega')}{\omega' - \omega} \mathrm{d}\omega'\\ > \chi_2(\omega) &= -\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \frac{\chi_1(\omega')}{\omega' - \omega} \mathrm{d}\omega' > \end{aligned} > $ Kramers–Kronig relations are derived from properties of analytic functions and are a mathematical concept, and valid for any analytic functions $\chi$. But in physics, it is used to determine the real part of a property from the linear response, from its imaginary part, or vice versa. The linear susceptibility is one typical example. To make it clear, this susceptibility $\chi$ does not have to be optical susceptibility. As long as we are under this linear and causality assumption, it is safe to say that Kramers–Kronig relations are valid. ### $\chi$ is an analytic function in the upper half plane In complex analysis, analytic function is equivalent to holomorphic function, and these two terms are interchangeable (but for real analysis, the name "analytic function" is still applied while holomorphic is a specific term in complex analysis. Analytic function has a boarder definition). A complex-valued function is analytic/holomorphic means that - Single-valued, this is assumed implicitly in most of the cases. - Convergent, or being locally expressible as a power series. - Continuously differentiable. - Satisfy Cauchy-Riemann equations, $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$, $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$, $z=x+iy$. We may check them one by one. - **Single-valued**: By the physically meaningfulness, we may say $\chi^{(1)}$ is single-valued. - **Convergence**: This may be deduced from the linear response expression. Because of the **causality**, the integration limit is from zero to infinite. $\chi^{(1)}(\omega) = \int_0^{\infty} \mathrm{d}\tau \, R^{(1)}(\tau) e^{i \omega \tau}$ Write $\omega$ as a complex quantity (no physical meanings). $\omega = \text{Re}(\omega) + i\text{Im}(\omega)$, and insert inside $\chi^{(1)}(\omega)$ expression, we have $\chi^{(1)}(\omega) = \int_0^{\infty} R^{(1)}(\tau) e^{i(\text{Re}(\omega) + i\text{Im}(\omega)) \tau} \, \mathrm{d}\tau$ The exponential part can then be separated into two parts, $e^{i(\text{Re}(\omega) + i\text{Im}(\omega)) \tau} = e^{i \text{Re}(\omega) \tau} e^{-\text{Im}(\omega) \tau}$ Therefore, we have $\chi^{(1)}(\omega) = \int_0^{\infty} R^{(1)}(\tau) e^{i \text{Re}(\omega) \tau} e^{-\text{Im}(\omega) \tau} \, \mathrm{d}\tau$ The causality condition makes $\tau>0$, and for $\text{Im}(\omega) > 0$, $e^{-\text{Im}(\omega) \tau}$ decays to zero for $\tau \rightarrow \infty$. The $e^{i \text{Re}(\omega) \tau}$ part is just a finite periodic quantity (so we don't have to concern about $\text{Re}(\omega)$), and $R^{(1)}(\tau)$ has to be finite to be physically reasonable (no response can be infinite). This means $\chi^{(1)}(\omega)$ is finite for $\text{Im}(\omega) > 0$. For $\text{Im}(\omega) = 0$, $R^{(1)}(\tau)$ is square integrable (or from the fact that $\chi^{(1)}$ is a physically measurable quantity), $\chi^{(1)}$ is finite. Therefore, for $\text{Im}(\omega) \geq 0$, or say in the upper half plane, $\chi^{(1)}$ is converged. - **Continuously differentiable**: This can be directly proved from the expression of linear response. We have $\chi^{(1)}(\omega) = \int_0^{\infty} R^{(1)}(\tau) e^{i(\text{Re}(\omega) + i\text{Im}(\omega)) \tau} \, \mathrm{d}\tau$ Calculate the derivative, $\frac{d}{\mathrm{d}\omega} \chi^{(1)}(\omega) = \frac{d}{\mathrm{d}\omega} \left( \int_0^{\infty} R^{(1)}(\tau) e^{i\omega \tau} \, \mathrm{d}\tau \right) = \int_0^{\infty} R^{(1)}(\tau) \frac{d}{\mathrm{d}\omega} \left( e^{i\omega \tau} \right) \, \mathrm{d}\tau= \int_0^{\infty} R^{(1)}(\tau) i\tau e^{i\omega \tau} \, \mathrm{d}\tau$ This is converged, the reason is the same as above. For higher order derivatives, we have $\frac{d^n}{\mathrm{d}\omega^n} \chi^{(1)}(\omega) = \int_0^{\infty} R^{(1)}(\tau) (i\tau)^n e^{i\omega \tau} \, \mathrm{d}\tau$ also converged, and we show that $\chi^{(1)}$ is continuously differentiable. - **Cauchy-Riemann equations**: Similarly, from $\chi^{(1)}(\omega) = \int_0^{\infty} R^{(1)}(\tau) e^{i(\text{Re}(\omega) + i\text{Im}(\omega)) \tau} \, \mathrm{d}\tau$, we have $\chi^{(1)}(\omega) = \int_0^{\infty} R^{(1)}(\tau) e^{i\text{Re}(\omega) \tau} e^{-\text{Im}(\omega) \tau} \, \mathrm{d}\tau$. Write the real and imaginary parts explicitly, $U(\text{Re}(\omega), \text{Im}(\omega)) = \int_0^{\infty} R^{(1)}(\tau) e^{-\text{Im}(\omega) \tau} \cos(\text{Re}(\omega) \tau) \, \mathrm{d}\tau$ $V(\text{Re}(\omega), \text{Im}(\omega)) = \int_0^{\infty} R^{(1)}(\tau) e^{-\text{Im}(\omega) \tau} \sin(\text{Re}(\omega) \tau) \, \mathrm{d}\tau$ So, $\chi^{(1)}(\omega) = U + iV$, calculate the partial derivatives, $\frac{\partial U}{\partial \text{Re}(\omega)} = -\int_0^{\infty} R^{(1)}(\tau) e^{-\text{Im}(\omega) \tau} \tau \sin(\text{Re}(\omega) \tau) \, \mathrm{d}\tau$ $\frac{\partial V}{\partial \text{Im}(\omega)} = -\int_0^{\infty} R^{(1)}(\tau) e^{-\text{Im}(\omega) \tau} \tau \sin(\text{Re}(\omega) \tau) \, \mathrm{d}\tau$ we have $\frac{\partial U}{\partial \text{Re}(\omega)} = \frac{\partial V}{\partial \text{Im}(\omega)}$ Similarly, $\frac{\partial U}{\partial \text{Im}(\omega)} = -\int_0^{\infty} R^{(1)}(\tau) e^{-\text{Im}(\omega) \tau} \tau \cos(\text{Re}(\omega) \tau) \, \mathrm{d}\tau$ $\frac{\partial V}{\partial \text{Re}(\omega)} = \int_0^{\infty} R^{(1)}(\tau) e^{-\text{Im}(\omega) \tau} \tau \cos(\text{Re}(\omega) \tau) \, \mathrm{d}\tau$ This gives, $\frac{\partial U}{\partial \text{Im}(\omega)} = -\frac{\partial V}{\partial \text{Re}(\omega)}$ In conclusion, the quantity $\chi^{(1)}$ is analytic function. ### Proof of Kramers–Kronig relations The integral we should consider is $\text{Int} = \mathcal{P}\int_{-\infty}^{\infty} \frac{\chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega}$ Take Cauchy principal value of the integral (namely dealing with the pole), $\mathcal{P}\int_{-\infty}^{\infty} \frac{\chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega} \equiv \lim_{\delta \to 0} \left[ \int_{-\infty}^{\omega-\delta} \frac{\chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega} + \int_{\omega + \delta}^{\infty} \frac{\chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega} \right]$ ![[Drawing 2024-09-19 14.11.39.excalidraw.svg]] Notice our desired integration can be separated into three path integrals, $\colorbox{#ffa94d66}{\(\text{Int}\)} = \colorbox{#87D4FF66}{\(\text{Int}(A)\)} - \colorbox{#76EE5366}{\(\text{Int}(B)\)} - \colorbox{#FF878766}{\(\text{Int}(C)\)}$ (contour integration), and we consider $r_{1}\rightarrow \infty$ and $r_{2}\rightarrow 0$. Now calculate each integral. For path $A$, since we have shown that the upper plane is analytic and the only pole is $\omega'=\omega$. By Cauchy's integral theorem, this integral, closed path with no poles inside, gives zero. $\colorbox{#87D4FF66}{\(\text{Int}(A) =0\)}$ For path $B$, for sufficiently large $\omega '$, the integrated quantity $\frac{\chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega}$ scales with $\frac{\chi^{(1)}(\omega')}{\omega'}$, while we have shown that $\chi^{(1)}(\omega')$ converge in the upper half plane. This means for $r_{1}\rightarrow \infty$, we have $\colorbox{#76EE5366}{\(\text{Int}(B)=0\)}$. The $\colorbox{#FF878766}{\(\text{Int}(C)\)}$ is not straightforward. Although in most proof people directly use the residue theorem (namely consider the circle with the pole, $\oint_{C} \frac{\chi(\omega')}{\omega'-\omega}\, \mathrm{d}\omega = 2\pi i \chi(\omega)$, and upper half would be $i\pi\chi(\omega)$, but strictly speaking the analytic property of lower plane is hard to say). One may consider Sokhotski–Plemelj theorem in real axis, $\lim_{\epsilon \to 0^+} \int_a^b \frac{\chi(\omega')}{\omega' - \omega \pm i\epsilon} \mathrm{d}\omega' = \mathcal{P} \int_a^b \frac{\chi(\omega')}{\omega' - \omega} \mathrm{d}\omega' \mp i\pi \chi(\omega)$, take $a=\omega^{-}$ and $b=\omega^{+}$, which will make the principal integral becomes zero (also if set the limit being $\pm \infty$ we may directly give the KK relation). Or even write the parametric expression of $\omega'$, $\omega'=\omega+\epsilon e^{i\theta}$ and calculate the integral. In short, we have $\colorbox{#FF878766}{\(\text{Int}(C)=-\pi i\chi(\omega)\)}$. Combine all these together, we have $\chi^{(1)}(\omega) = \frac{-i}{\pi} \mathcal{P}\int_{-\infty}^{\infty} \frac{\chi^{(1)}(\omega') \, \mathrm{d}\omega'}{\omega' - \omega}$ and if separate the real and imaginary parts, $\chi^{(1)}(\omega) = \text{Re} \, \chi^{(1)}(\omega) + i \, \text{Im} \, \chi^{(1)}(\omega)$, we have $\text{Re} \, \chi^{(1)}(\omega) = \frac{1}{\pi} \mathcal{P}\int_{-\infty}^{\infty} \frac{\text{Im} \, \chi^{(1)}(\omega') \, \mathrm{d}\omega'}{\omega' - \omega}$ $\text{Im} \, \chi^{(1)}(\omega) = -\frac{1}{\pi} \mathcal{P}\int_{-\infty}^{\infty} \frac{\text{Re} \, \chi^{(1)}(\omega') \, \mathrm{d}\omega'}{\omega' - \omega}.$ To make this result actually meaningful and applicable, we should convert the frequency to positive values. With $\chi^{(1)}(-\omega)=\chi^{(1)}(\omega)^{*}$, we can write $\text{Re} \, \chi^{(1)}(-\omega) = \text{Re} \, \chi^{(1)}(\omega)$ $\text{Im} \, \chi^{(1)}(-\omega) = -\text{Im} \, \chi^{(1)}(\omega)$ Combine with the KK relation we obtained, we have $\begin{aligned} \text{Im} \, \chi^{(1)}(\omega) & = -\frac{1}{\pi} \mathcal{P}\int_{-\infty}^{0} \frac{\text{Re} \, \chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega} - \frac{1}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\text{Re} \, \chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega}\\&= \frac{1}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\text{Re} \, \chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' + \omega} - \frac{1}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\text{Re} \, \chi^{(1)}(\omega') \mathrm{d}\omega'}{\omega' - \omega}\\&=\frac{-2\omega}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\text{Re} \, \chi^{(1)}(\omega')}{\omega'^2 - \omega^2} \, \mathrm{d}\omega' \end{aligned}$ and $\begin{aligned} \text{Re} \, \chi^{(1)}(\omega) &=\frac{1}{\pi} \mathcal{P}\int_{-\infty}^{0} \frac{\text{Im} \, \chi^{(1)}(\omega') \, \mathrm{d}\omega'}{\omega' - \omega}+\frac{1}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\text{Im} \, \chi^{(1)}(\omega') \, \mathrm{d}\omega'}{\omega' - \omega}\\ &=-\frac{1}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\text{Im} \, \chi^{(1)}(\omega') \, \mathrm{d}\omega'}{\omega' + \omega}+\frac{1}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\text{Im} \, \chi^{(1)}(\omega') \, \mathrm{d}\omega'}{\omega' - \omega}\\ &=\frac{2}{\pi} \mathcal{P}\int_{0}^{\infty} \frac{\omega' \, \text{Im} \, \chi^{(1)}(\omega')}{\omega'^2 - \omega^2} \, \mathrm{d}\omega' \end{aligned} $ ### Kramers–Kronig relations in nonlinear optics One see that the **linearity** and **causality** are the fundamentals in derivation of Kramers–Kronig relation. The expansion to nonlinear susceptibility is possible, though not straightforward. There are some ways to ensure linearity and causality, and have multiple frequencies involved at the same time. The most intuitive method is letting one frequency being the linear variable, and keeping all others constants. For example, for third order susceptibility, $\chi^{(3)}(\omega_{\sigma}; \omega_1, \omega_2, \omega_3) = \frac{1}{i\pi} \mathcal{P}\int_{-\infty}^{\infty} \frac{\chi^{(3)}(\omega_{\sigma}'; \omega_1', \omega_2, \omega_3)}{\omega_1' - \omega_1} \, \mathrm{d}\omega_1'$ where $\omega_{\sigma}' = \omega_1' + \omega_2 + \omega_3$. The same result holds for $\omega_{2}'$ and $\omega_{3}'$. The proof is similarly to the linear one. And just intuitively, the causality still holds, and $\chi^{(3)}(\omega_{\sigma}; \omega_1, \omega_2, \omega_3) E(\omega_1) E(\omega_2) E(\omega_3)$ is also linear for field $E_{2}$, the KK relation should be valid. A more general form would be $\chi^{(n)}(\omega_{\sigma}; \omega_1 + p_1 \omega, \omega_2 + p_2 \omega, \dots, \omega_n + p_n \omega) = \frac{1}{i \pi}\mathcal{P} \int_{-\infty}^{\infty} \frac{\chi^{(n)}(\omega_{\sigma}'; \omega_1 + p_1 \omega', \omega_2 + p_2 \omega', \dots, \omega_n + p_n \omega')}{\omega' - \omega} \, \mathrm{d}\omega'$ This expression has $\omega'$ being the linear variable and use parameters $p_{i}$ to control the frequencies. Special cases like SHG and THG can be derived from this $\chi^{(2)}(2\omega; \omega, \omega) = \frac{1}{i \pi} \mathcal{P}\int_{-\infty}^{\infty} \frac{\chi^{(2)}(2\omega'; \omega', \omega')}{\omega' - \omega} \, \mathrm{d}\omega'$ $\chi^{(3)}(3\omega; \omega, \omega, \omega) = \frac{1}{i \pi} \mathcal{P}\int_{-\infty}^{\infty} \frac{\chi^{(3)}(3\omega'; \omega', \omega', \omega')}{\omega' - \omega} \, \mathrm{d}\omega'$ Another example is refractive index change by auxiliary beam with frequency $\omega_1$, $\chi^{(3)}(\omega; \omega, \omega_1, -\omega_1) = \frac{1}{i \pi}\mathcal{P}\ \int_{-\infty}^{\infty} \frac{\chi^{(3)}(\omega'; \omega', \omega_1, -\omega_1)}{\omega' - \omega} \, \mathrm{d}\omega'$ >[!Notice] >To have Kramers–Kronig relation established, **linearity** and **causality** must be ensured simultaneously. For example, the self-induced refractive index change **cannot** be described by Kramers–Kronig relation, since $\chi^{(3)}(\omega; \omega, \omega, -\omega)$ could not be written in a linear way. >[!Info] >See https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations. >Such relation works for any response functions being linear and causality. Therefore, it could naturally be extended to Green functions. See [[LRT.pdf#page=19&selection=260,0,262,35|https://eduardo.physics.illinois.edu/phys582/LRT.pdf, page 19]]. >The relation, real and imaginary parts of generalized $\chi$ are the behavior of [[Fluctuation-Dissipation theorem]]. Check the linear response theory. > >For mathematical concepts: [Holomorphic function](https://en.wikipedia.org/wiki/Holomorphic_function), [Cauchy–Goursat theorem/Cauchy's integral theorem](https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem), [Cauchy–Riemann equations](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations), [Residue theorem](https://en.wikipedia.org/wiki/Residue_theorem), [Sokhotski–Plemelj theorem](https://en.wikipedia.org/wiki/Sokhotski%E2%80%93Plemelj_theorem#Version_for_the_real_line).