Using the R_to_DF software the dielectric function of a material can be computed from its half space intensity reflectance R(w). This is done in several steps which are explained briefly in the following:

1.
The half space intensity reflectance R(w) which is the measured quantity must be known ideally from w=0 to w=¥. Since these conditions cannot be fulfilled (obviously) some kind of approximations must be used. In spectral regions where the sample is transparent and the measured reflectance is not the half space reflectance, or in regions where no experimental data are available the input spectrum of the KKR analysis must be artificially constructed. In most cases this means to do extrapolations of the measured data for low and for high wavenumbers. How you can do these extrapolations in R_to_DF is explained below. Of course, only a finite wavenumber range can be used for doing a digital KKR analysis. How you decide where to set 'infinity' is discussed in the step-by-step example above.

2.   
Since the dielectric function e(w) can be computed easily from the complex amplitude reflection coefficient r(w) this quantity is calculated in a KKR analysis. To do this it is useful to write

.

Now assume that for high frequencies r(w) becomes a constant (actually this must be ensured by the high frequency extrapolation). Then one has .
Now consider the logarithm

.

It is an analytical function and its real and imaginary part are related to each other by a Kramers-Kronig relation. Therefor, the imaginary part f(w)-p can be reconstructed from the real part (which is known from measurement) by a Hilbert transformation.
It has been shown that a two-fold Fourier transformation is equivalent to the Hilbert transformation but has some advantages on digital computers. So it is used here. Using the Fourier transform method one obtains after the first Fourier transform the so-called response function a(t). This function is the key of the KKR analysis. It is a real function, being 0 for t<0. If it is written as a sum of an odd and an even function in time, i.e. a(t)=a_odd(t)+a_even(t) the Fourier transform of the odd part gives the imaginary part of ln r(w) whereas the even one is related to the real part.
Knowing the real part one can calculate the even function a_even(t) by Fourier transformation. Since a(t)=0 for t<0, it holds a_odd(t)=-a_even(t) for t<0. Due to this condition, if a_even(t) is known a_odd(t) is readily known also.
Now since a(t) is known completely the wanted imaginary part f(w)-p can be obtained by inverse Fourier transforming a(t) back to the frequency domain.

3.
Having reconstructed the phase f(w) the complex function r(w) is determined. Now in the last step the dielectric function e' + i e'' must be calculated from the amplitude reflection coefficient which is quite easy.
The generalized dynamic conductivity can be computed from the dielectric function according to

s' = we₀e'' and s'' = -we₀e' .