Kramers-Kronig relation

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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.

[edit] Definition

Assuming a monochromatic electromagnetic radiation whose dependence upon time can be expressed in the form e − iωt using a complex representation, then the following relations describe absorption as an effect of the permittivity ε(ω):

\operatorname{Re} \{ \epsilon(\omega) \} = \epsilon_0 + \frac{2}{\pi}\cdot \mathcal{P} \int \limits_{0}^{\infty} \frac{\Omega \operatorname{Im} \{ \epsilon(\Omega) \}}{\Omega^2-\omega^2} \,\mathrm{d}\Omega
\operatorname{Im} \{ \epsilon(\omega) \} = \frac{2 \omega}{\pi} \cdot \mathcal{P} \int \limits_{0}^{\infty} \frac{ \operatorname{Re} \{ \epsilon(\Omega) \} - \epsilon_0 }{\Omega^2-\omega^2} \,\mathrm{d}\Omega


where the above integrals are Cauchy integrals and \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(\omega)=1+{c \over \pi} \cdot \mathcal{P} \int \limits_{0}^{\infty} {{\alpha(\Omega)} \over {\Omega^2-\omega^2}} \,\mathrm{d}\Omega

The requirements for a function f(ω) 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(ω) = f1(ω) + if2(ω),

where f1 and f2 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(\omega) = -\frac{2 \omega}{\pi} P\int_0^{\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 ε(ω) of materials. However, it must be noticed that in this case,

f(ω) = χ(ω) = ε(ω) / ε0 − 1,

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

[edit] Derivation

In physical systems two quantities are generally related by following way

B(t) = \int\limits_{-\infty}^{\infty} K(t,\tau) A(\tau) \mathrm{d}\tau,

where A and B may stand for example for electrical fields E and D. Homogeneity in time requires K(t,τ) = K(t − τ), causality requires K(t) = 0, whenever t > 0. Using the step function we can write down the identity K(t) = K(t)\cdot\Theta(-t).

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

f(\omega) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} f(k) \tilde\Theta(\omega-k) \mathrm{d} k,

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

[edit] Reference

  • 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
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