Kramers-Kronig relations on a finite data set

[Niles]

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.

 

[Zarqon]

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.

 

[DrDu]

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.

 

[Niles]

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.

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.

 

[DrDu]

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