Kramers-Kronig relationships

The Kramers-Kronig relations , also Kramers-Kronig relation (after their discoverers Hendrik Anthony Kramers and Ralph Kronig ), relate the real and imaginary parts of certain meromorphic functions in the form of an integral equation . They therefore represent a special case of the Hilbert transformation .

An important application is the connection between the absorption and the dispersion of a “ray of light” propagating in a medium . There are other applications in high energy physics and engineering .

Mathematical formulation

Let be a meromorphic function whose poles lie in the lower half-plane . The causality postulate corresponds physically to this requirement for the position of the poles . Furthermore, let or be the real and imaginary part of the function . It is assumed that these two functions are even and odd, respectively. This means that by means of Fourier integration, it is not possible to form an arbitrary complex function, but a real one. ${\displaystyle F:\mathbb {C} \rightarrow \mathbb {C} }$ ${\displaystyle \operatorname {Re} \,F|_{\mathbb {R} }}$ ${\displaystyle \operatorname {Im} \,F|_{\mathbb {R} }}$ ${\displaystyle F}$ ${\displaystyle F}$

In physics, instead of looking at the function , one often looks at the assumptions regarding even and odd. ${\displaystyle F}$ ${\displaystyle F/\mathrm {i} }$

Finally be . Then the following equations, called Kramers-Kronig relationships, hold : ${\displaystyle \lim _{|z|\rightarrow \infty }|F(z)|=0}$ ${\displaystyle x\in \mathbb {R} }$

${\displaystyle \operatorname {Im} \,F(x)=-{\frac {2}{\pi }}\cdot \;\mathrm {CH} \,\int _{0}^{+\infty }{\frac {x\cdot \operatorname {Re} \,F(t)}{t^{2}-x^{2}}}\mathrm {d} t}$

${\displaystyle \operatorname {Re} \,F(x)={\frac {2}{\pi }}\cdot \;\mathrm {CH} \,\int _{0}^{+\infty }{\frac {t\cdot \operatorname {Im} \,F(t)}{t^{2}-x^{2}}}\mathrm {d} t}$

${\displaystyle \mathrm {CH} }$denotes the main Cauchy value of the occurring integral.

The real and imaginary parts of the function are mutually dependent on integration. This finds applications in optics and in systems theory when the susceptibility of a system indicates, see causality . Applications can also be found in high-energy physics for the dispersion relations of the S matrix . ${\displaystyle F}$ ${\displaystyle F}$

Motivation (a boundary value problem)

A continuous real function is given on the real axis , which, analogously to , should be assumed to be straight. For this purpose, a complex (!) Function holomorphic in the entire upper half-plane should be constructed in such a way that applies. ${\displaystyle \mathbb {R} }$ ${\displaystyle \,f}$ ${\displaystyle \operatorname {Re} \,F}$ ${\displaystyle \,F}$ ${\displaystyle \operatorname {Re} \,F|_{\mathbb {R} }{\stackrel {!}{=}}f}$

So a boundary value problem is to be solved, whereby inside the considered area , i. H. above , because of the holomorphism condition, the Cauchy-Riemann differential equations must be fulfilled and on the boundary, a continuous real function , is given, which is to be assumed there. ${\displaystyle \,G}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \partial G=\mathbb {R} \,,}$ ${\displaystyle f}$

A holomorphic function can be represented according to the residual theorem as:

${\displaystyle F(z)={\frac {1}{2\pi \mathrm {i} }}\left(\int _{HK_{r}}{\frac {F(t)}{t-z}}\mathrm {d} t+\int _{-r}^{r}{\frac {F(t)}{t-z}}\mathrm {d} t\right)}$,

where the (positively oriented) semicircle in the upper half plane denotes the center and radius . If it drops fast enough at infinity, the representation is reduced to an integral over the real axis at the limit crossing , i.e.: ${\displaystyle HK_{r}(0)}$ ${\displaystyle 0}$ ${\displaystyle r>0}$ ${\displaystyle F}$ ${\displaystyle r\rightarrow \infty }$

${\displaystyle F(z)={\frac {1}{2\pi \mathrm {i} }}\int _{-\infty }^{+\infty }{\frac {F(t)}{t-z}}\mathrm {d} t}$

In the case of and because or should be an even function , this finally results ${\displaystyle \operatorname {Im} \,z=0}$ ${\displaystyle \,f}$ ${\displaystyle \operatorname {Re} \,F(t)}$

${\displaystyle \operatorname {Im} \,F(z)=-{\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {\operatorname {Re} \,F(t)}{t-z}}\mathrm {d} t=-{\frac {2}{\pi }}\int _{0}^{+\infty }{\frac {z\cdot \operatorname {Re} \,F(t)}{t^{2}-z^{2}}}\mathrm {d} t}$,

whereby the occurring integral is to be interpreted as Cauchy's main value (singularity for ) and agrees with the Hilbert transformation of . The residual theorem is applied to the integration path. This equation corresponds to the one Kramers-Kronig relationship. ${\displaystyle t=z}$ ${\displaystyle f}$ ${\displaystyle [-r,z-\varepsilon ]\cdot HK_{\varepsilon }(z)\cdot [z+\varepsilon ,r]\cdot HK_{r}(0)}$

To solve the boundary value problem one only needs to use the relation . ${\displaystyle \operatorname {Re} \,F|_{\mathbb {R} }{=}f}$

The same procedure is used for odd functions and the other Kramers-Kronig relationship is obtained. Any function can always be the rule , with , be broken down into an even or odd part. The simplest case of a meromorphic function with the assumed properties is the linear response function of the damped harmonic oscillator, with a positive damping constant and a positive characteristic oscillator angular frequency ${\displaystyle f}$ ${\displaystyle \,f=f_{+}+f_{-}}$ ${\displaystyle f_{\pm }(t)={\frac {1}{2}}\left(f(t)\pm f(-t)\right)}$ ${\displaystyle F(z)}$ ${\displaystyle F(z):=1/(z^{2}-\omega _{0}^{2}+\mathrm {i} \gamma z),}$ ${\displaystyle \gamma }$ ${\displaystyle \omega _{0}.}$

Applications

The Kramers-Kronig relationships are used where a real even function - possibly made odd by an additional factor - is to be supplemented to a holomorphic function . This mostly serves to simplify the calculations that occur, especially with wave functions , i.e. mainly in signal processing and optics , but also in 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 . It is sufficient to know the dependence of one of the two quantities on the frequency in order to be able to calculate the other. ${\displaystyle F(\omega )}$ ${\displaystyle 1/\mathrm {i} }$ ${\displaystyle F(z)}$

The permittivity depending on the angular frequency can be expressed as the integral of the absorption depending on the angular frequency : ${\displaystyle \omega }$ ${\displaystyle \varepsilon (\omega )}$

${\displaystyle \operatorname {Re} (\varepsilon (\omega ))=1+{\frac {2}{\pi }}\cdot \;\mathrm {CH} \,\int \limits _{0}^{\infty }{\frac {\Omega \cdot \operatorname {Im} (\varepsilon (\Omega ))}{\Omega ^{2}-\omega ^{2}}}\,\mathrm {d} \Omega }$

in which

• the real angular frequency , the variable of integration is ${\displaystyle \Omega }$
• the variable, which is also real, represents a characteristic system angular frequency ${\displaystyle \omega }$
• the abbreviation  for the Cauchy principal value of the integral stands ( is even, odd). ${\displaystyle \mathrm {CH} \int _{0}^{\infty }\dots }$ ${\displaystyle \operatorname {Re} \varepsilon (\omega )}$ ${\displaystyle \operatorname {Im} \varepsilon (\omega )}$

An alternative approach arises with the absorption coefficient , the refractive index and the speed of light : ${\displaystyle \alpha }$ ${\displaystyle n}$ ${\displaystyle c}$

${\displaystyle n(\omega )=1+{\frac {c}{\pi }}\cdot \;\mathrm {CH} \,\int \limits _{0}^{\infty }{{\alpha (\Omega )} \over {\Omega ^{2}-\omega ^{2}}}\,\mathrm {d} \Omega }$

This enables the complex shape of the refractive index to be derived from a simple absorption measurement , especially in non-linear optics . The name of the dispersion relations in high energy physics also refers to this example.

In the engineering sciences, the Kramers-Kronig relationships are mainly used in the context of impedance spectroscopic measurements, where non-fulfillment leads to the conclusion that the frequency response is incorrectly measured .

The restriction of the generally valid Kramers-Kronig relationships to two-pole systems leads to the ZHIT algorithm, which can be used to validate the impedance spectra of electrochemical systems ( electrochemical impedance spectroscopy ).

literature

Original works:

• R. de L. Kronig: On the theory of dispersion of X-rays . In: Journal of the Optical Society of America . tape 12 , no. 6 , 1926, pp. 547-556 , doi : 10.1364 / JOSA.12.000547 .
• HA Kramers: La diffusion de la lumiere par les atomes . In: Atti Cong. Intern. Fisici, (Transactions of Volta Centenary Congress) Como . Vol. 2, 1927, pp. 545-557.

Further literature:

• Mansoor Sheik-Bahae: Nonlinear Optics Basics. Kramers-Krönig Relations in Nonlinear Optics . In: Robert D. Guenther (Ed.): Encyclopedia of Modern Optics . Academic Press, Amsterdam 2005, ISBN 0-12-227600-0 , pp. 234-240.

Individual evidence

1. ^ Safa Kasap, Peter Capper: Springer Handbook of Electronic and Photonic Materials . Springer, 2006, ISBN 978-0-387-26059-4 , pp. 49 .
2. ↑ BA Boukamp: A Linear Kronig-Kramer's Transform Test for Immittance Data Validation . In: J. Electrochem. Soc . 142, 1995, pp. 1885-1894.
3. ↑ M. Schönleber, D. Klotz and E. Ivers-Tiffée: A Method for Improving the Robustness of linear Kramers-Kronig Validity Tests . In: Electrochimica Acta . 131, 2014, pp. 20-27. doi : 10.1016 / j.electacta.2014.01.034 .