In mathematics and physics, a Kramers-Kronig relation connects the real part of an analytic complex function to an integral containing the imaginary part of the function and vice versa. In optics, especially nonlinear optics, these relations can be used to calculate the refractive index of a material by the measurement of the absorbance, which is better accessible. The relation is named in honour of Ralph Kronig and Hendrik Anthony Kramers.

Definition

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Assuming a monochromatic electromagnetic radiation whose dependence upon time can be expressed in the form $e^\{-i\; \backslash omega\; t\}$ using a complex representation, then the following relations describe absorption as an effect of the permittivity $\backslash epsilon\; (\; \backslash omega)$:

$\backslash operatorname\{Re\}\; \backslash \{\; \backslash epsilon(\backslash omega)\; \backslash \}\; =\; \backslash epsilon\_0\; +\; \backslash frac\{2\}\{\backslash pi\}\backslash cdot\; \backslash mathcal\{P\}\; \backslash int\; \backslash limits\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash Omega\; \backslash operatorname\{Im\}\; \backslash \{\; \backslash epsilon(\backslash Omega)\; \backslash \}\}\{\backslash Omega^2-\backslash omega^2\}\; \backslash ,\backslash mathrm\{d\}\backslash Omega$

$\backslash operatorname\{Im\}\; \backslash \{\; \backslash epsilon(\backslash omega)\; \backslash \}\; =\; \backslash frac\{2\; \backslash omega\}\{\backslash pi\}\; \backslash cdot\; \backslash mathcal\{P\}\; \backslash int\; \backslash limits\_\{0\}^\{\backslash infty\}\; \backslash frac\{\; \backslash operatorname\{Re\}\; \backslash \{\; \backslash epsilon(\backslash Omega)\; \backslash \}\; -\; \backslash epsilon\_0\; \}\{\backslash Omega^2-\backslash omega^2\}\; \backslash ,\backslash mathrm\{d\}\backslash Omega$

where the above integrals are Cauchy integrals and $\backslash mathcal\{P\}$ denotes the Cauchy principal value.

Reformulated to the intensity absorption coefficient α, the refractive index n and c as the speed of light in vacuum:

$n(\backslash omega)=1+\{c\; \backslash over\; \backslash pi\}\; \backslash cdot\; \backslash mathcal\{P\}\; \backslash int\; \backslash limits\_\{0\}^\{\backslash infty\}\; \backslash ,\backslash mathrm\{d\}\backslash Omega$

The requirements for a function $f(\backslash omega)$ to which Kramers-Kronig relations apply can be interpreted as that the function must represent the Fourier transform of a linear and causal physical process. If we write

$f(\backslash omega)\; =\; f\_1(\backslash omega)\; +\; i\; f\_2(\backslash omega)$,

where $f\_1$ and $f\_2$ are real-valued "well-behaving" functions, then the Kramers-Kronig relations are

$$

f_1(\omega) = \frac{2}{\pi} P\int_0^{\infty} \frac{\omega' f_2(\omega') d\omega'}{\omega'^2 - \omega^2}

$f\_2(\backslash omega)\; =\; -\backslash frac\{2\; \backslash omega\}\{\backslash pi\}\; P\backslash int\_0^\{\backslash infty\}$

\frac{f_1(\omega') d\omega'}{\omega'^2 - \omega^2} ,

The Kramers-Kronig relations are related to the Hilbert transform, and are most often applied on the permittivity $\backslash epsilon(\backslash omega)$ of materials. However, it must be noticed that in this case,

$f(\backslash omega)\; =\; \backslash chi(\backslash omega)\; =\; \backslash epsilon(\backslash omega)/\backslash epsilon\_0\; -\; 1$,

where $\backslash chi(\backslash omega)$ is the electric susceptibility of the material. The susceptibility can be interpreted as the Fourier transform of the time-dependent polarization in the material after an infinitely short pulsed electric field, in other words the impulse response of the polarization.

Derivation

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In physical systems two quantities are generally related by following way

$B(t)\; =\; \backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash infty\}\; K(t,\backslash tau)\; A(\backslash tau)\; \backslash mathrm\{d\}\backslash tau$

, where $A$ and $B$ may stand for example for electrical fields $E$ and $D$. Homogenity in time requires $K(t,\backslash tau)\; =\; K(t-\backslash tau)$, causality requires $K(t)=0$, whenever $t>0$. Using the theta function we can write down the identity $K(t)\; =\; K(t)\backslash cdot\backslash Theta(-t)$.

We are interested in properties of the Fourier transform $f(\backslash omega)$ of the kernel $K(t)$. In electromagnetic field situation this is the permitivity $\backslash epsilon(\backslash omega)$. Fourier transforming former identity and using convolution theorem we get

$$

f = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} f(k) \tilde\Theta(\omega-k) \mathrm{d} k, where $\backslash tilde\backslash Theta(k)$ denotes the Fourier transform of theta function. It can be derived it equals to $\backslash sqrt\{2\backslash pi\}\backslash tilde\backslash Theta(k)=P\; \backslash frac\{i\}\{k\}-\backslash pi\; \backslash delta(k)$. Inserting this formula into convolution and comparing real and imaginary parts, we arrive to the Kramers-Kronig relations.

Reference

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

Complex analysis | Electric and magnetic fields in matter

علاقة كراميرس-كرونيج | Kramers-Kronig-Relation | Relations de Kramers-Kronig | Liên hệ Kramers-Kronig