Kramers-Kronig relations on a finite data set

  1. Hi

    Say I have a finite data set (frequency, absorption) and I would like to find the corresponding dispersion. For this I could use the Kramers-Kronig (KK) relation on the absorption data. What I would do is to make a qubic spline and then perform the KK-transformation.

    However, the absorption data naturally doesn't run from ±∞, but what I would do is simply to use the extremes of my frequency-data instead - this will naturally introduce some numerical error. What do professional people do in this case, do they quantify the error? Or is there not a way to extract the dispersion from the absorption data?

    Thanks in advance.
     
  2. jcsd
  3. Hmm, if you have a data set, you probaby want to calculate the Kramers-Kroning integrals numerically. I wouldn't recommend first building a spline because they are terribly inaccurate outside of the range where you have data points, and using that could lead to very uncontrolled errors.
     
  4. DrDu

    DrDu 4,421
    Science Advisor

    I have no idea what professional packages do, but have some general information on how the dielectric constant should behave asymptotically. Namely ε-1 should fall off like 1/ω at very high frequencies and should go to a constant in the limit ω→0. There are also lots of sum rules which provide further information on the relevant constants as far as you cannot infer them from your data.
     
  5. I only integrate (numerically!) from the first and last frequency data point, so I never go outside the range.


    Thanks. They behave as anticipated, but I'm worried about the precision.
     
  6. DrDu

    DrDu 4,421
    Science Advisor

    I meant that you could integrate over the corresponding asymptotic expressions in the range where you don't have data.
     
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