Algorithms

The Kramers-Kronig Transform is useful for obtaining absorbance and refractive index information from reflectance data. In particular, recent interest in the KK transform has stemmed from the use of FT-IR microscopy.

When the reflectance spectrum of an optically thick sample is measured, it consists of two components. One component is the actual absorbance spectrum, and the other is the refractive index spectrum. In reflectance spectroscopy the refractive index spectrum tends to dominate and make qualitative interpretation of the data difficult if not impossible. The refractive index of a typical mid-IR absorber tends to change rapidly in regions of strong absorbance. This causes the major absorbance peaks to appear as strong first derivative shaped features in the measured reflectance data.

The Kramers-Kronig Transform will decompose this complex reflectance spectrum into its separate extinction coefficient and refractive index spectra. These are also called the K and N spectra. This information can than be used for qualitative evaluation of the sample. The extinction coefficient spectrum can be used to produce the absorbance spectrum.

The KKTrans algorithm assumes reflectance angles near zero which is good for qualitative measurements at most angles not approaching grazing incidence. It is based on a double FFT approach as described by Peterson and Knight. Its main advantage is its speed.

The real (n = refractive index) and imaginary (k = extinction) parts of the complex index of refraction are calculated from the reflectance spectrum via the following formulas:

where

R = the Reflectance spectrum

n = the wave number

q = the phase shift angle of the sample

For a give wave number, the phase shift is calculated using:

Evaluating this integral presents two problems. First, in practice, the spectrum is obtained over a finite range, so that approximations are required at either end. Second, there is a pole at n = nm, which also requires an approximation. The method used to get around this by using a Fourier transform to perform the integral:

1. Take the Fourier transform of the argument of the phase integral.

2. Set the points for t<0 to 0.

3. Take the inverse Fourier transform.

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