# Determination of Optical Spectra by a Modified Kramers Kronig Integral

Authors: Wallis, D.H.; Wickramasinghe, N.C.

Source: Astrophysics and Space Science, Volume 262, Number 2, 1998 , pp. 193-213(21)

Publisher: Springer

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Abstract:

A new modified Kramers Kronig Integral is derived and shown to produce excellent results when k data is only known over a limited range. By considering the effect of resonance features simulated using the Dirac-Delta function, the new integral is shown to be more rapidly converging than both the conventional Kramers Kronig integral and a modified (Subtractive Kramers Kronig – SKK) integral introduced by Ahrenkiel (1971). The new integral does not require extensive extrapolation of reflectance data outside the measured region in order to produce reliable results. By extending the above procedure to include n data points, it is shown that at wavelength λ_0, $n(_0)=\sum_{i=1}^{\rm n}(-1)^{\rm n+1}\prod_{\stackrel{j=1}{j \not=i}}^{\rm n} \frac{(_j^2-_0^2)}{(_i^2- _j^2)}n(_i)+\frac{2}{\pi}P\int_{0}^{\infty}(-1)^{\rm n+1} \frac{\prod_{i=1}^{\rm n}(_i^2-_0^2)}{\prod_{i=0}^{\rm n}(^2-_i^2)} k()d$ with relative error given by, $R_n(_0)=\prod_{i=1}^{\rm n}\frac{_i^2- _0^2}{_Σ^2-_i^2} .$ This n^th order expression should prove useful in establishing the internal self-consistency of data sets for which both optical coefficients have been theoretically derived.

Document Type: Regular Paper

Affiliations: School of Mathematics, University of Wales Cardiff, PO Box 926, Senghennydd Road, Cardiff, CF2 4YH, UK; E-mail: wickramasinghe@cf.ac.uk

Publication date: January 1, 1998

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