@Sufal, Waranatha is talking about the spectral range of 300...1000nm, which covers the visible and adjacent (NIR & NUV) spectral ranges. I would suspect that this range of data (besides all what others mentioned above) is not sufficient to reliably apply the Kramers Kronig relations, simply because the integrals will not nearly have converged.

@ Waranatha, the KK relations formulate that real and imaginary part of a response function are not independent. If one is known **for all frequencies**, then you can calculate the other using formulae involving integrals over the entire frequency axis. If you have (which is most common) only a finite frequency range available then you can 1) estimate errors or 2) apply models (see also comments of Tsvetanka above) in order to get approximate results. In the x-ray regime, the "normal" optical processes are so far away in terms of frequency that their contribution to the integral is small and can be quite well controlled. this is what Sufal refers to, I think. (Sufal, you're invited to correct me in case I do get something wrong here)

If you know the basic parameters (density, molecular wt. etc) you should be able to obtain the energy dependent complex refractive index values. You first need to calculate the absorptive component (imaginary part) of the refractive index from the absorption spectra. Then apply the Kramers-kronigs transformation to obtain the diffractive component (real part) of the refractive index. This works very well for organic materials in the soft x-ray regime.

Are you certain the RI information for these materials is not available elsewhere, or by the manufactuer?  If not, you could calculate it by optical methods if you can model the material as a flat, composite dielectric slab.  Check our fresnel's equations, and dielectric slab calculations:

http://www.comsol.com/model/dielectric-slab-waveguide-14709

Sufal, can you please elaborate on energy dependent complex refractive index?  Also I'm more interested in the wavelength range of 300-1000 nm.

First you need to calculate the absorption coefficient. Then calculate reflectivity from absorption coefficient values. Since there is a direct relation between reflectivity and refractive index, wavelength dependent refractive index can be calculated if you have wavelength dependent reflectivity spectra.

Wavelength dependent refractive index can also be calculated using ellipsometer directly.

To Shashikant Sharma - If you do not know refractive index it is not possible to calculate reflectance. In the case of thin films it is even worst because R is also a function of film thickness.

If you have only absorbance spectra you can use nonlinear fitting of transmittance curve (A=-log(T)). However you will need a good guess for the dispersion models and the initial values of the model parameters. If you know the thickness of the films you can measure transmittance and reflectance and determine n and k simultaneously. If you do not know thickness you can deposit your film on two types of substrate - for example glass and silicon wafer and determine n, k and d simultaneously using T and R from film/glass system and R from film/Si system. See for example our paper. Also you can measure n and k using ellipsometry.

Some authors has reported that they have calculated reflectance using the values of absorption coefficient, thickness and transmittance.  Then they have derived wavelength dependent refractive index from reflectance for thin films.

To Tsvetanka Babeva :  Please suggest me how wavelength dependent  refractive index for thin films can directly be calculated if value of transmittance, absorbance and film thickness is known.

Dear Shashikant Sharma, direct calculation of n and k from T and R is not possible in the common case because of the strong non-linearity of the dependences T,R = f(n,k,d,...). There are few cases where some approximations can be done leading the equations to be simplified.

Eq. 8 from the paper.pdf is for R of slab with thickness higher than the coherence lenght of the used waves and the sumation is performed over the intensities of multiple reflected waves from top and bottom boundaries of the slab instead of  their amplitudes. This is the reason R to be independent of thickness. I will leave you to decide if this is correct in the case of the studies in the paper.pdf.

Further, I said that n and k can be calculated from T and R spectra if d is known, not T and A. (T-transmittance, R-reflectance, A-absorbance) In this case at each wavelength you have two known (T and R) and two unknown parameters (n and k). Check Newton-Raphson algorithm and Abeles method for more details about solving a system of two equations with two unknowns.

Finally, have in mind that transmittance and absorbance are related with the relation A=-log(T), so they are not independent.

Warantha,

here are two files that might explain what I meant by energy dependent n. Also attached is instructions to calculate the optical constants from X-ray absorption spectra using KK integration (link to igor module is in the file). I have never tried it outside the soft x-ray regime though. I do not see why it shouldn't work.

@Sufal, Waranatha is talking about the spectral range of 300...1000nm, which covers the visible and adjacent (NIR & NUV) spectral ranges. I would suspect that this range of data (besides all what others mentioned above) is not sufficient to reliably apply the Kramers Kronig relations, simply because the integrals will not nearly have converged.

@ Waranatha, the KK relations formulate that real and imaginary part of a response function are not independent. If one is known **for all frequencies**, then you can calculate the other using formulae involving integrals over the entire frequency axis. If you have (which is most common) only a finite frequency range available then you can 1) estimate errors or 2) apply models (see also comments of Tsvetanka above) in order to get approximate results. In the x-ray regime, the "normal" optical processes are so far away in terms of frequency that their contribution to the integral is small and can be quite well controlled. this is what Sufal refers to, I think. (Sufal, you're invited to correct me in case I do get something wrong here)

@ Sufal, Tsvetanka & Kai,

Thank you very much for your detailed responses. My task is more complicated than I initially presumed. The thickness of the films I'm talking about is around 100-200nm.

So I guess my options are to determine the absorption spectra over a larger range of wavelengths, and use KK integration or measure R,T and thickness simultaneously as Tsvetanka suggested. I do not have the facility to measure these parameters right now, and it was difficult to locate published data on complex refractive index of the organic semiconductor thin films I'm modelling. But the absorption spectra for most of these polymers are available.  I'm not sure when I can get the R,T, and thickness measurements , so would like to know more about dispersion models. 

Can you please direct me to some literature on dispersion models, and how to select the correct dispersion model for a particular material? 

Dear Waranatha,

it is not so easy to give you detailed advice. Much could also depend on the level of accuracy you wish to achieve. Whenever you measure absorption by doing a transmission experiment, you will have reflexion "losses" at both ends of your sample (in the simple case where you would not bother about interference). That adds up to a geometric series of single reflexion terms.

Each time, the reflected amplitude will be governed by Fresnel's equations. At normal incidence at an interface to air, you have for the amplitude reflection coefficient

r = (n-1) / (n+1)

where n is the complex index of refraction. So, already these terms depend on both real and imaginary part of n. Therefore these measurements cannot immediately tell you one apart from the other.

In practice, much shall depend on how large your absorption is and what the (surface) quality of your specimens is.

Dispersion models I have come across include those carrying  the names of Cauchy and Sellmeier, respectively. (See wikipedia and consider the link below for a starter). These are more phenomenological parametrizations of dispersion rather than really models. Their range of application is restricted to spectral ranges with no absorption.

So, if your polymer has significant absorption in the spectral range under consideration these dispersion laws will rapidly lose their validity and significance. You should then really look into working with the KK relations. This will not be an easy task though, unless your absorption features within a relatively narrow band somewhere in the middle of your experimentally accessible wavelength range.

Books I found useful when I studied optical properties years ago include those of Abeles and - in particular - Wooten ("optical properties of solids" or some similar title). These are old books, but should still be useful. I recall Wooten nicely discussing real and imaginary parts of optical response functions (and hope I am remembering right). There are certainly other very good and more contemporary books around. An extremely good general textbook on optics is the one by Eugene Hecht.

Kai, Thank you for your detailed response. I definitely need to read some literature and get familiar with the basics, before selecting what method is more suitable for me. I will go through the textbooks you have pointed out.

Hi Waranatha,

You can calculate the refractive index from the absorption spectra through the calculations or theoretical references but not directly. Ellipsometry technique can measure directly .

0) first convert your absorbance data in to transmittance % data by Beer- Lamberts law

http://teaching.shu.ac.uk/hwb/chemistry/tutorials/molspec/beers1.htm

1) try PARAV software http://www.chalcogenide.org/parav-v2-0-2/

2) try this reference paper to understand the theoretical relationship between the different physical (optical) quantities.  

I hope you can get some idea. good luck

Dear All,

I have reflectance spectra of cell culture in 400- 1000 nm wavelength range.

I would like to do the analysis of keramers-kroning. I tried to write a MATLAB based code for calcultions. However, I an interrupted for phse shift integral part.

this is the most common formula : -w/pi (integral (ln (R(w)-ln(R(w0))/ w^2-w0^2) dw

I only have R(w) and am really confused which w0 is????

in the case that anybody have done this analysis , would you please clarify the variables.

I don't know how to read R(W0)