Kramers-Kronig relations on a finite data set

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Discussion Overview

The discussion revolves around the application of Kramers-Kronig relations to a finite data set of frequency and absorption data, specifically focusing on the challenges of extracting dispersion from this data. Participants explore numerical methods, potential errors, and the behavior of dielectric constants in relation to the Kramers-Kronig transformation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests using cubic spline interpolation on the absorption data before applying the Kramers-Kronig transformation, while expressing concern about the introduction of numerical error due to the finite nature of the data set.
  • Another participant advises against using splines, arguing that they can lead to significant inaccuracies outside the range of available data and recommends calculating the Kramers-Kronig integrals numerically instead.
  • A different participant provides information on the expected asymptotic behavior of the dielectric constant, noting that ε-1 should decrease like 1/ω at high frequencies and approach a constant as ω approaches zero, while also mentioning sum rules that could inform the analysis.
  • One participant reiterates the importance of integrating numerically only within the bounds of the data set to avoid extrapolation errors.
  • A later reply suggests that integrating over asymptotic expressions could be a viable approach in regions lacking data.

Areas of Agreement / Disagreement

Participants express differing views on the use of cubic splines versus numerical integration for Kramers-Kronig transformations, indicating a lack of consensus on the best method to handle finite data sets and the associated errors.

Contextual Notes

Participants highlight potential limitations related to numerical errors introduced by extrapolating outside the data range and the dependence on asymptotic behavior, but do not resolve these issues.

Niles
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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.
 
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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.
 
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.
 
Zarqon said:
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.

I only integrate (numerically!) from the first and last frequency data point, so I never go outside the range.


DrDu said:
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.

Thanks. They behave as anticipated, but I'm worried about the precision.
 
I meant that you could integrate over the corresponding asymptotic expressions in the range where you don't have data.
 

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