Why are Kramers-Kroning relations useful?

[quasar987]

The Kramers-Kronig relations allows one to calculate the real part of the permitivity knowing the imaginary part or vice-versa:

http://en.wikipedia.org/wiki/Kramers-Kronig_relations

But in what situation will one know either the imginary part but not the real part or the real part but not the imaginary part of the permitivity?

 

[Claude Bile]

KK relations are useful for calculating dispersion (dn/dw) characteristics near an absorption peak.

Claude.

 

[quasar987]

Cool. So, you measure the absorption experimentally and use the KK integral for $n(\omega)$ in terms of $\alpha(\omega)$.

In my EM class, we only saw the KK relations for the permitivity. It would be strange that we stopped there if their only usefulness was to derive the KK relation for $n(\omega)$. I'm putting this on my list of question I have to bug the EM proffessor with.P.S. Why would one want to calculate dn/dw? What does this tell you about what? It gives the "speed" at which the ratio of c to the phase velocity is changind as frequency changes, but why is that important?

 

[lalbatros]

I guess the reason of the KK relation makes its importance: causality.

I may be wrong, but I think that the Landau damping in plasma physics might be a nice example.
The Landau damping is a collisionless damping of plasma waves that needs to take causality into account in its derivation. Therefore it must have the same origin as the Kramers-Kronig relations, and of course it illustrates it anyway.

Michel

 

[ZapperZ]

One of the most common application of Kramers-Kronig transform is in optical conductivity. Often, you cannot obtain the optical conductivity in a material because a particular frequency of light attenuates rather quickly when it enters a material, such as a conductor. Still, one can obtain the optical conductivity from the reflectivity data. One takes the reflectivity data as a function of frequency and do a KK-transformation to obtain the conductivity.

The couple of caveats here are that one has to assume that the sum-rule is obeyed, and that in many instances, the simplest, solvable model requires that the Drude model be valid.

Zz.

 

[Dr Transport]

The KK relations can also be applied to nonlinear optics.

Check this book out,
http://search.barnesandnoble.com/booksearch/isbnInquiry.asp?z=y&EAN=9783540236733&itm=1

I use it all the time and will give you more info about KK relations and optics than you will ever wnat to know.

 

[Claude Bile]

P.S. Why would one want to calculate dn/dw? What does this tell you about what? It gives the "speed" at which the ratio of c to the phase velocity is changind as frequency changes, but why is that important?

Imagine you have two frequencies close together, then knowing dn/dw will tell you how spread out in time and space the two frequencies will be after propagating a certain distance. Essentially, knowing dn/dw will tell you how 'smeared out' your pulse of light will be after propagation. This is particularly important to know in a laser gain medium for instance, where the whole idea is to operate near an absorption band.

And of course, the inverse, dw/dn is related to the group velocity, which is always handy to know.

Claude.

 

[quasar987]

If there's a whole book on them, they must not be completely useless!