Kramers–Kronig relations
The relations are often used to compute the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the condition of analyticity, and conversely, analyticity implies causality of the corresponding stable physical system.
[1] The relation is named in honor of Ralph Kronig and Hans Kramers.
The proof begins with an application of Cauchy's residue theorem for complex integration.
When the contour is chosen to trace the real axis, a hump over the pole at
This follows decomposition of the integral into its contributions along each of these three contour segments and pass them to limits.
The second term in the last expression is obtained using the theory of residues,[4] more specifically, the Sokhotski–Plemelj theorem.
and the equation into their real and imaginary parts to obtain the forms quoted above.
In certain linear physical systems, or in engineering fields such as signal processing, the response function
It can be shown (for instance, by invoking Titchmarsh's theorem) that this causality condition implies that the Fourier transform
The imaginary part of a response function describes how a system dissipates energy, since it is in phase with the driving force.
Fortunately, in most physical systems, the positive frequency-response determines the negative-frequency response because
We transform the integral into one of definite parity by multiplying the numerator and denominator of the integrand by
Hu[6] and Hall and Heck[7] give a related and possibly more intuitive proof that avoids contour integration.
[8] The conventional form of Kramers–Kronig above relates the real and imaginary part of a complex response function.
[12] Hence, in effect, this also applies for the complex relative permittivity and electric susceptibility.
[13] The Sellmeier equation is directly connected to the Kramer-Kronig relations, and is used to approximate real and complex refractive index of materials far away from any resonances.
Kramers–Kronig relations enable exact solutions of nontrivial scattering problems, which find applications in magneto-optics.
[16] In ellipsometry, Kramer-Kronig relations are applied to verify the measured values for the real and complex parts of the refractive index of thin films.
[17] In electron energy loss spectroscopy, Kramers–Kronig analysis allows one to calculate the energy dependence of both real and imaginary parts of a specimen's light optical permittivity, together with other optical properties such as the absorption coefficient and reflectivity.
Using this data with Kramers–Kronig analysis, one can calculate the real part of permittivity (as a function of energy) as well.
This measurement is made with electrons, rather than with light, and can be done with very high spatial resolution.
Although electron spectroscopy has poorer energy resolution than light spectroscopy, data on properties in visible, ultraviolet and soft x-ray spectral ranges may be recorded in the same experiment.
In angle resolved photoemission spectroscopy the Kramers–Kronig relations can be used to link the real and imaginary parts of the electrons self-energy.
Notable examples are in the high temperature superconductors, where kinks corresponding to the real part of the self-energy are observed in the band dispersion and changes in the MDC width are also observed corresponding to the imaginary part of the self-energy.
Through the use of the optical theorem the imaginary part of the scattering amplitude is then related to the total cross section, which is a physically measurable quantity.
[21] For seismic wave propagation, the Kramer–Kronig relation helps to find right form for the quality factor in an attenuating medium.
Since, it is not possible in practice to obtain data in the whole frequency range, as the Kramers-Kronig formula requires, approximations are necessarily made.
At high frequencies (> 1 MHz) it is usually safe to assume, that the impedance is dominated by ohmic resistance of the electrolyte, although inductance artefacts are often observed.
At low frequencies, the KK test can be used to verify whether experimental data are reliable.
In battery practice, data obtained with experiments of duration less than one minute usually fail the test for frequencies below 10 Hz.
