The cauchy integral should be performed from zero to infinity. The measured interval is limited. How to complete the interval related to cauchy integral?
The Kramers Kronig inversion is based on the Cauchy integral. This integral is performed from Zero to infinity, but the experimental measure is generally made in a limited interval. Thus , we should complete the missing data in order to use this Cauchy integral. My problem is how to complete these data.
My problem rest until the present date without any response, I am waiting your precious help.
I think you can find some help in the following paper:
Johnson, D.W. (1975) A Fourier method for numerical Kramers-Kronig analysis. J. Phys. A (Math. Gen. Phys.) 8, 490–495.
The book "Optical Properties of Solids" by Wooten is an excellent textbook in this regard. See Appendix G of the book for Kramers-Kronig analysis. The routine taken by most researchers is to interpolate and extrapolate the reflectance data for missing data. For interpolation you need reflectance data from other research groups on your sample. For extrapolation, it depends on the type of your sample. For example, for metals, the extrapolation to zero frequency is made by Hagen-Rubens approximation and the extrapolation to high frequencies is made by using free-electron approximation. Now you have a set of reflectance data from zero to high frequencies and you can apply the Kramers-Kronig integrals to compute optical functions such as complex optical conductivity, epsilon and refractive index. In your program, it is easy to write codes for extrapolation, but you need to patch your data with data from others as much as you can before feeding into your program. This increases the accuracy of the outcome. Hopefully, this helps.
You need to extrapolate your data both to 0 and to infinity. Then you can apply the KK transformation. Please have a look in the following paper that I wrote some years ago
“Determination of the complex refractive index of materials via infrared measurements”, C.P.E. Varsamis, Appl. Spectroscopy 56, 1107 (2002)
If you have further questions do not hesitate to contact me.
Best regards
Christos
Yes, that is what I need. Please send me your paper.
Best regards.
Zoubir
Please go to my profile in publications and you can download the full text
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