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Why are Kramers-Kroning relations useful?

  1. Oct 8, 2006 #1

    quasar987

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    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?
     
  2. jcsd
  3. Oct 10, 2006 #2

    Claude Bile

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

    Claude.
     
  4. Oct 10, 2006 #3

    quasar987

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

    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 [itex]n(\omega)[/itex]. 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?
     
    Last edited: Oct 10, 2006
  5. Oct 10, 2006 #4
    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
     
  6. Oct 11, 2006 #5

    ZapperZ

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    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.
     
  7. Oct 11, 2006 #6

    Dr Transport

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  8. Oct 11, 2006 #7

    Claude Bile

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    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.
     
    Last edited: Oct 11, 2006
  9. Oct 11, 2006 #8

    quasar987

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