Kramers–Kronig relation
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The '''Kramers–Kronig relations''' are [[mathematics|mathematical]] properties, connecting the [[real number|real]] and [[imaginary number|imaginary]] parts of any [[complex analysis|complex function]] which is [[analytic function|analytic]] in the [[upper half plane]]. These relations are often used to relate the real and imaginary parts of [[linear response function|response functions]] in [[physical system]]s because [[causality (physics)|causality]] implies the analyticity condition is satisfied and conversely.<ref>John S. Toll, ''Causality and the Dispersion Relation: Logical Foundations'', Physical Review, vol. ''104'', pp. 1760 - 1770 (1956).</ref> The relation is named in honor of [[Ralph Kronig]]<ref>R. de L. Kronig, ''On the theory of the dispersion of X-rays,'' J. Opt. Soc. Am., vol. '''12''', pp. 547-557 (1926).</ref> and [[Hendrik Anthony Kramers]].<ref>H.A. Kramers, ''La diffusion de la lumiere par les atomes,'' Atti Cong. Intern. Fisica, (Transactions of Volta Centenary Congress) Como, vol. '''2''', p. 545-557 (1927) .</ref>
==Definition==
For a complex function <math>\chi(\omega) = \chi_1(\omega) + i \chi_2(\omega)</math> of the complex variable <math>\omega </math>, analytic in the upper half plane of <math>\omega </math> and which vanishes as <math>|\omega| \rightarrow \infty</math>, the Kramers–Kronig relations are given by
:<math>\chi_1(\omega) = {1 \over \pi} \mathcal{P} \int \limits_{-\infty}^{\infty} {\chi_2(\omega') \over \omega' - \omega}\,d\omega'</math>
and
:<math>\chi_2(\omega) = -{1 \over \pi} \mathcal{P} \int \limits_{-\infty}^{\infty} {\chi_1(\omega') \over \omega' - \omega}\,d\omega',</math>
where <math>\mathcal{P}</math> denotes the [[Cauchy principal value]]. We see that the real and imaginary parts of such a function are not independent, so that the full function can be reconstructed given just one of its parts.
==Derivation==
The proof begins with an application of the [[residue theorem]] for complex integration. Given any analytic function <math>\chi(\omega)</math> in the upper half plane, consider the integral
:<math> \oint {\chi(\omega') \over \omega'-\omega}\,d\omega' = 0. </math>
The [[methods of contour integration|contour]] encloses the upper half plane at [[infinity (mathematics)|infinity]], the real axis and a hump over the [[pole (complex analysis)]] at <math>\omega = \omega'</math> leaving no poles inside, and so the integral vanishes. We decompose the integral into its contributions along each of these three contour segments. The segment at infinity vanishes since we assume <math>\chi(\omega)</math> vanishes as we take <math>|\omega| \rightarrow \infty</math>. We are left with the segment along the real axis and the half-circle:
:<math>\mathcal{P} \int \limits_{-\infty}^\infty {\chi(\omega') \over \omega'-\omega}\,d\omega' - i \pi \chi(\omega) = 0.</math>
Rearranging, we arrive at the compact form of the Kramers–Kronig relations,
:<math>\chi(\omega) = {1 \over i \pi} \mathcal{P} \int \limits_{-\infty}^\infty {\chi(\omega') \over \omega'-\omega}\,d\omega'. </math>
The single <math>i</math> in the [[denominator]] hints at the connection between the real and imaginary components. Finally, split <math>\chi(\omega)</math> and the equation into their real and imaginary parts to obtain the forms quoted above.
==Physical interpretation and alternate form==
We can apply the Kramers–Kronig formalism to [[linear response function|response functions]]. In [[physics]], the response function <math>\chi(t-t')</math> describes how some property <math>P(t)</math> of a physical system responds to an applied [[force (physics)|force]] <math>F(t')</math>. For example, <math>P(t)</math> could be the [[angle|angle]] of a [[pendulum]] and <math>F(t)</math> the applied force of a [[actuator|motor]] driving the pendulum motion. The response <math>\chi(t-t')</math> must be zero for <math>t<t'</math> since a system cannot respond to a force before it is applied. It can be shown that this causality condition implies the [[Fourier transform]] <math>\chi(\omega)</math> is analytic in the upper half plane. Additionally, if we subject a system to high frequency oscillatory forcing, there will be no time for the system to respond before the forcing has switched direction, and so <math>\chi(\omega)</math> vanishes as <math>\omega</math> becomes large. From these physical considerations, we see that <math>\chi(\omega)</math> satisfies conditions needed for the Kramers–Kronig relations to apply.
The Kramers–Kronig relations have a physical interpretation. The imaginary part of a response function describes how a system [[dissipation|dissipates energy]], since it is out of [[phase]] with the [[driving force]]. The Kramers–Kronig relations imply that observing the dissipative response of a system is sufficient to determine its in-phase (reactive) response, and vice versa.
The formulas above are not useful for reconstructing physical responses, as the integrals run from <math>-\infty</math> to <math>\infty</math>, implying we know the response at negative frequencies. Fortunately, in most systems, the positive frequency-response determines the negative-frequency response because <math>\chi(\omega)</math> is the Fourier transform of a real quantity <math>\chi(t-t')</math>, so <math>\chi(-\omega) = \chi^*(\omega)</math>. This means <math>\chi_1(\omega)</math> is an [[even and odd functions|even function]] of frequency and <math>\chi_2(\omega)</math> is [[even and odd functions|odd]].
Using these properties, we can collapse the integration ranges to <math>[0,\infty)</math>. Consider the first relation giving the real part <math>\chi_1(\omega)</math>. Transform the integral into one of definite parity by multiplying the numerator and denominator of the [[integrand]] by <math>\omega' + \omega</math> and separating:
:<math> \chi_1(\omega) = {1 \over \pi} \mathcal{P} \int \limits_{-\infty}^\infty {\omega' \chi_2(\omega') \over \omega'^2 - \omega^2}\, d\omega' + {\omega \over \pi} \mathcal{P} \int \limits_{-\infty}^\infty {\chi_2(\omega') \over \omega'^2 - \omega^2}\,d\omega'. </math>
Since <math>\chi_2(\omega)</math> is odd, the second integral vanishes, and we are left with
:<math>\chi_1(\omega) = {2 \over \pi} \mathcal{P} \int \limits_{0}^{\infty} {\omega' \chi_2(\omega') \over \omega'^2 - \omega^2}\,d\omega'.</math>
The same derivation for the imaginary part gives
:<math>\chi_2(\omega) = -{2 \over \pi} \mathcal{P} \int \limits_{0}^{\infty} {\omega \chi_1(\omega') \over \omega'^2 - \omega^2}\,d\omega' = -{2 \omega \over \pi} \mathcal{P} \int \limits_{0}^{\infty} {\chi_1(\omega') \over \omega'^2 - \omega^2}\,d\omega'.</math>
These are the Kramers–Kronig relations useful for physical response functions.
==See also==
* [[Hilbert transform]]
* [[Linear response function]]
* [[Dispersion (optics)]]
==References==
===Inline===
{{reflist}}
===General===
* Mansoor Sheik-Bahae: ''Nonlinear Optics Basics. Kramers–Kronig Relations in Nonlinear Optics'', in: Robert D. Guenther (Ed.): ''Encyclopedia of Modern Optics'', Academic Press, Amsterdam 2005, ISBN 0-12-227600-0
* J. D. Jackson, ''Classical Electrodynamics'', 2nd edition, Wiley, New York (1975), Sec. 7.10, ISBN 0-471-43132-X.
[[Category:Complex analysis]]
[[Category:Electric and magnetic fields in matter]]
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