Kramers–Kronig relations

The Kramers -Kronig relations, including Kramers -Kronig relation, set real and imaginary parts of certain meromorphic functions in the form of an integral equation related to each other. They are therefore a special case of the Hilbert transform represents the relations were named after their discoverers, Hendrik Anthony Kramers and Kronig Ralph.

An important application of the Kramers-Kronig relations, the relation between the absorption and dispersion of the spread of light in a medium. Other applications are in high-energy physics.

Mathematical formulation

Is a meromorphic function whose poles are located in the lower half-plane. This call to the location of the poles corresponds physically to the causality postulate. Further are or real and imaginary parts of the function. It is assumed that these two functions are even or odd. This means that can not be formed from any complex, but out of a real function by Fourier integration.

In physics, often considered instead of the function, which reverse the conditions relating odd and even. Finally, it should. Then apply to the following referred to as Kramers -Kronig relations equations:

Denotes the Cauchy principal value of the integral occurring.

Real and imaginary parts of the function that is mutually dependent by integration. This finds applications in optics and in systems theory if the susceptibility of a system indicates see causality. Applications can be found even in high -energy physics in the dispersion relations of the S- matrix.

Motivation ( A boundary value problem)

On the real axis is a continuous real function is defined to be analogous to a straight condition. This is a holomorphic around the upper half-plane complex (!) Function are designed so that the following applies.

It is a boundary value problem to be solved, where inside the considered volume, that is, above, because of the holomorphy condition the Cauchy- Riemann equations must be satisfied and, is given on the boundary, a continuous real function to be adopted there.

A holomorphic function can be represented by the residue theorem as:

Where the ( positively oriented ) semicircle in the upper half-plane with center and radius respectively. If now at infinity sufficiently rapidly so reduced in the limit the presentation to an integral over the real axis, ie:

In the event, and as a straight or should be function is finally obtained

Wherein the integral appearing to be interpreted as Cauchy principal value ( for singularity ) and the Hilbert transform of matches. The residue theorem is thereby applied to the integration path. This equation corresponds to a Kramers -Kronig relationship.

We need now to use only the relationship to the solution of the boundary value problem.

For odd functions, the procedure analogous and receives the other Kramers -Kronig relationship. An arbitrary function can always be the rule, with, be decomposed into an even and odd part.

Applications

The Kramers -Kronig relations are used in applications, where a real even function to a holomorphic function is to be added, which is usually to simplify the calculations occurring Especially recommended for wave functions, ie, mainly in signal processing and optics, but also in the statistical physics in connection with the fluctuation-dissipation theorem. In this way the absorption of electromagnetic waves in a medium is related to the refractive index. So it suffices to know the dependence of the two variables on the frequency in order to calculate the other can.

Which depends on the angular frequency absorption can be expressed as a function of a dependent of the angular frequency permittivity:

Where the angular frequency as the variable of integration and the Cauchy principal value (English Cauchy principal value) are the integral.

An alternative approach is achieved with the absorption coefficient, the refractive index and the speed of light:

This can be especially in the nonlinear optics of a simple absorption measurement, the complex shape of the refractive index derived.