Determination of Optical Spectra by a Modified Kramers Kronig Integral

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 λ₀, \[ n(λ_0)=\sum_{i=1}^{\rm n}(-1)^{\rm n+1}\prod_{\stackrel{j=1}{j \not=i}}^{\rm n} \frac{(λ_j^2-λ_0^2)}{(λ_i^2- λ_j^2)}n(λ_i)+\frac{2}{\pi}P\int_{0}^{\infty}(-1)^{\rm n+1} \frac{\prod_{i=1}^{\rm n}(λ_i^2-λ_0^2)}{\prod_{i=0}^{\rm n}(λ^2-λ_i^2)}λ k(λ)dλ \] with relative error given by, \[ R_n(λ_0)=\prod_{i=1}^{\rm n}\frac{λ_i^2- λ_0^2}{λ_Σ^2-λ_i^2} . \] 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.