Glossary entry

Kramers-Kronig relations

Integral relations, following from causality, that connect the real and imaginary parts of any linear response function. In optics they tie the refractive index at every wavelength to the absorption spectrum at all others, so a change in absorption, from carriers, excitons or a resonance, always comes with a change in index.

Optics fundamentalsUpdated September 2026

A material cannot respond to a light field before the field arrives. That requirement of causality, applied to any linear response, forces the real and imaginary parts of the response function to be Hilbert transforms of each other. For the complex refractive index $\tilde n(\omega) = n(\omega) + i\kappa(\omega)$ the Kramers-Kronig relation for the real part reads

n(\omega) - 1 = \frac{2}{\pi}\,\mathcal{P}\!\int_0^\infty \frac{\omega'\,\kappa(\omega')}{\omega'^2 - \omega^2}\,d\omega',

where $\mathcal{P}$ denotes the principal value, and a companion relation gives $\kappa$ from $n$ . The same pair holds for the real and imaginary parts of the permittivity and of the susceptibility.

What it means in practice

The refractive index at one wavelength is set by absorption at all others. Glass is transparent in the visible but absorbs strongly in the ultraviolet and the infrared, and its visible index, and the normal dispersion in which $n$ falls with wavelength, comes from those distant bands. Near an absorption line the index swings up on one side and down on the other, the anomalous dispersion around a resonance. The relations also work for changes: if an effect alters the absorption spectrum by $\Delta\alpha(\omega)$ , the index changes by an amount fixed by the same integral. This is how the plasma dispersion effect in silicon was quantified, by Soref and Bennett from measured free-carrier absorption; how the electroabsorption of quantum wells comes with electrorefraction; and why a semiconductor laser's gain changes carry index changes, giving the linewidth enhancement factor.

Using the relations

Because the integral runs over all frequencies, a measured spectrum must be extended beyond its range with physical models, and the result is most accurate near the middle of the measured band. The relations are used to extract $n$ and $\kappa$ from reflectance spectra of absorbing materials, to check the consistency of optical-constant data, to compute phase from measured amplitude in minimum-phase systems such as some filters and pulses, and, in their sum-rule forms, to relate integrated absorption to the density of electrons. Any model of a material's optical response, such as a Lorentz or Drude oscillator, satisfies them automatically if it is causal.

References: J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Ch. 7; R. A. Soref, B. R. Bennett, IEEE J. Quantum Electron. 23, 123 (1987); V. Lucarini, J. J. Saarinen, K.-E. Peiponen, E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).