定义：
吸收与透明介质折射率之间的数学关系

在复解析函数理论中，函数的实部可以表示成虚部的积分形式，反之亦是如此。这一关系在线性和非线性光学中都用途广泛。应用到与频率有关的电介质方程ε(ω)中，得到： 

该方程是以Ralph Kronig和Hendrik Anthony Kramers命名的。其中Re ε(ω)与折射率有关，而Im ε(ω)则与吸收（或增益）有关。积分前面的符号是柯西主值，在进行数值积分时需要认真考虑。 
而ε(ω)的虚部还满足另一个方程（这里未给出），由任意波长的折射率得到对某一波长的吸收。该方程实际中用处不多。两个方程合起来称为Kramers-Kronig色散关系。 
还有另一种形式的Kramer-Kronig关系，指的是折射率n与强度吸收系数α的关系： 

这两种形式并不直接相关，在第一种形式中被积函数的分子中包含一个因子Ω。 

Kramers-Kronig关系的应用
通过Kramers-Kronig关系可以计算折射率分布，因此从与频率相关的损耗就可以得到介质的色散，可以在很大的光谱范围内测量。还有一个类似的关系式，可以由折射率计算吸收，但该关系并不常用，因为它很难在很宽的频率范围内测量折射率。 
改进的Kramers-Kronig关系经常用在非线性光学中。基本原理就是介质受到激发（例如，在半导体中产生载流子）引起的折射率变化与吸收的变化有关。由于只有在有限的频率范围内的吸收变化是比较重要的，因此相对容易测量。这一方法还可以应用于激光增益介质中，例如，计算光纤放大器中与激发能级有关的相位变化。在稀土掺杂增益介质中，得到一定激光跃迁附近增益和损耗的变化是不够的，因为在紫外光谱区域的强吸收线也非常重要。

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 ε(ω), they lead to the relation

which is named after Ralph Kronig and Hendrik Anthony Kramers. Re ε(ω) is related to the refractive index (see below), and Im ε(ω) is related to the absorption (or gain) coefficient. The symbol Ω 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 ε(ω) (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 α:

These two forms are not directly related; note that in the first, but not in the second form there is a factor Ω in the numerator of the integrand.

Another form of the equation involves a wavelength integral:

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.

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