# Kramers–Kronig Relations

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: mathematical relations between absorption coefficient and refractive index of media

(also sometimes used, but incorrect: “Kramers–Krönig relations”)

Within the theory of analytic complex functions, general relations have been developed which relate the real part of such a function to an integral containing the imaginary part, and vice versa. Such relations have found widespread application in the area of linear optics and also nonlinear optics. Applied to the frequency-dependent dielectric function <$\epsilon (\omega )$>, they lead to the relation

$${\mathop{\rm Re}} \: \varepsilon (\omega ) = 1 + \frac{2}{\pi }\;\wp \int\limits_0^{ + \infty } {\frac{{\Omega \;{\mathop{\rm Im}} \: \varepsilon (\Omega )}}{{{\Omega ^2} - {\omega ^2}}}{\rm{d}}\Omega } $$which is named after Ralph Kronig and Hendrik Anthony Kramers. <${\rm Re} \: \epsilon (\omega )$> is related to the refractive index (see below), and <$\rm{Im} \: \epsilon (\omega )$> is related to the absorption (or gain) coefficient. The symbol <$\Omega$> is the angular optical frequency variable, running through the whole integration range. The P-like symbol in front of the integral denotes the Cauchy principal value, which requires some care e.g. when calculating such an integral numerically. (Note the pole of the integrand!)

There is a second equation for the imaginary part of <$\epsilon (\omega )$> (not shown here), calculating absorption at one wavelength from the refractive index at all wavelengths. That equation is much less relevant for practical applications. Both equations combined are called the *Kramers–Kronig dispersion relations*.

There is another form of Kramers–Kronig relations, relating the refractive index <$n$> to the intensity absorption coefficient <$\alpha$>:

$$n(\omega ) = 1 + \frac{c}{\pi }\;\wp \int\limits_0^{ + \infty } {\frac{{\alpha (\Omega )}}{{{\Omega ^2} - {\omega ^2}}}{\rm{d}}\Omega } $$These two forms are not directly related; note that in the first, but not in the second form there is a factor <$\Omega$> in the numerator of the integrand.

Another form of the equation involves a wavelength integral:

$$n(\lambda ) = 1 + \frac{1}{{2{\pi ^2}}}\;\wp \int\limits_0^{ + \infty } {\frac{{\alpha (\lambda ')}}{{1 - {{\left( {\frac{{\lambda '}}{\lambda }} \right)}^2}}}{\rm{d}}\lambda {\rm{'}}} $$## Applications of Kramers–Kronig Relations

The Kramers–Kronig relations allow one to calculate the refractive index profile and thus also the chromatic dispersion of an optical material solely from its frequency-dependent absorption losses, which can be measured over a large spectral range. One may, for example, apply this to an optical glass, exhibiting substantial absorption both in the ultraviolet and in the infrared spectral range. Both influence the refractive index in the visible spectral region, where the glass is transparent.

Note that a similar relation, allowing the calculation of the absorption from the refractive index, is far less useful because it is much more difficult to measure the refractive index in a wide frequency range.

Modified Kramers–Kronig relations are also very useful in nonlinear optics [3]. The underlying basic idea is that the *change* in the refractive index caused by some excitation of a medium (e.g. generation of carriers in a semiconductor) is related to the change in the absorption. As the change in the absorption is normally significant only in a limited range of optical frequencies, it is relatively easily measured. Such methods can also be applied to laser gain media, e.g. for calculating phase changes in fiber amplifiers associated with changes of the excitation level of the laser-active ions [4, 5]. Note that in the case of rare-earth-doped laser gain media, for example, it is not sufficient to consider only the changes in gain and loss around a certain laser transition because changes in strong absorption lines in the ultraviolet spectral region are also important. For the effect on the chromatic dispersion, however, nearby absorption and emission lines are more relevant.

## More to Learn

Encyclopedia articles:

### Bibliography

[1] | R. de L. Kronig, “On the theory of the dispersion of X-rays”, J. Opt. Soc. Am. 12 (6), 547 (1926); https://doi.org/10.1364/JOSA.12.000547 |

[2] | M. Beck et al., “Group delay measurements of optical components near 800 nm”, IEEE J. Quantum Electron. 27 (8), 2074 (1991); https://doi.org/10.1109/3.83423 |

[3] | D. C. Hutchings et al., “Kramers–Kronig relations in nonlinear optics”, Opt. Quantum Electron. 24, 1 (1992) |

[4] | M. Montagna et al., “Nonlinear refractive index in erbium-doped optical amplifiers”, Opt. Quantum Electron. 27, 871 (1995) |

[5] | J. W. Arkwright et al., “Experimental and theoretical analysis of the resonant nonlinearity in ytterbium-doped fiber”, J. Lightwave Technol. 16 (5), 798 (1998) |

[6] | M. Sheik-Bahae, “Nonlinear optics basics: Kramers–Kronig relations in nonlinear optics”, in Encyclopedia of Modern Optics (eds. B. Guenter and D. Steel), Academic Press, London (2004) |

[7] | J. D. Jackson, Classical Electrodynamics, 2nd edn., John Wiley & Sons, Inc., New York (1975) |

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2020-07-05

What happens when <$\omega = \Omega$>?

The author's answer:

The integrand has a pole at that location, i.e., it diverges for <$\omega \rightarrow \Omega$>. This problem is handled by taking the so-called Cauchy principal value. Basically, that means removing a tiny interval around the pole from the integration and taking the limit for the width of that interval going towards zero.