Stefano,

For this Kramers-Kronig analysis you must choose a model (mathematical approximation) for the parts of the frequency spectrum that you have not measured, from zero frequency to infinity.  In practice, for example with Matlab, you will choose a discrete representation of the spectrum and your maximum frequency will not be infinite, it will be high enough to get the accuracy you need.  

How high is that frequency?  You can figure it out empirically by doing the Matlab computation repeatedly with different maximum model frequencies; but you can also calculate how high it should be analytically using the Kramers-Kronic theory and your chosen model of the high-frequency behavior of your material.

Most materials for example should approach 100% transparency at the extreme high and low frequency ends.  Your difficulty will be finding the most accurate model for the shape of the curve where your material transitions from low but finite absorption to practically zero absorption. 

The book Vladimir R. Tuz mentioned above appears to be one of the definitive sources for this:  V. Lucarini, J.J. Saarinen, K.-E. Peiponen, E.M. Vartiainen, "Kramers–Kronig Relations in Optical Materials Research."

Are you measuring a thin film or bulk material (thousands of times thicker than the wavelengths you are using to probe)?  A brief search on Google Scholar shows there is a separate body of literature on Kramers-Kronig methods for thin films vs bulk materials.

Thin film publications:

K. F. Palmer and M. Z. Williams, “Optical constant determination of thin films condensed on transmitting and reflecting surfaces,” Technical Report AEDC-TR-83-64 (AD-A140845) (Defense Technical Information Center, Fort Belvoir, Va., 1984).

 K. F. Palmer and M. Z. Williams, “Determination of the optical constants of a thin film from transmittance measurements of a single film thickness,” Appl. Opt. 24, 1788–1798 (1985).

More general Kramers-Kronig publications that may be helpful for bulk material transmission measurements:

P.-O. Nilsson, “Determination of optical constants from intensity measurements at normal incidence,” Appl. Opt. 7, 435–442 (1968).

K.-E. Peiponen and E. M. Vartiainen, “Kramers–Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).

Dear Stefano,

I never did that myself, but I know that in the end of the book V. Lucarini, J.J. Saarinen, K.-E. Peiponen, E.M. Vartiainen, "Kramers–Kronig Relations in Optical Materials Research" there is appendix "MATLAB Programs for Data Analysis". If you're lucky, it is possible you can find algorithm which you need in this book. 

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19910004272.pdf

Best regards,

Vladimir 

Dear Stefano,

May be you need to fit by using a model the transmittance spectrum and extract from it the n and k values for your sample. I have used this approach to obtain the refractive index of Al2O3 in nanoporous membranes and it works perfectly.

Best wishes,

Aurelio

Stefano,

For this Kramers-Kronig analysis you must choose a model (mathematical approximation) for the parts of the frequency spectrum that you have not measured, from zero frequency to infinity.  In practice, for example with Matlab, you will choose a discrete representation of the spectrum and your maximum frequency will not be infinite, it will be high enough to get the accuracy you need.  

How high is that frequency?  You can figure it out empirically by doing the Matlab computation repeatedly with different maximum model frequencies; but you can also calculate how high it should be analytically using the Kramers-Kronic theory and your chosen model of the high-frequency behavior of your material.

Most materials for example should approach 100% transparency at the extreme high and low frequency ends.  Your difficulty will be finding the most accurate model for the shape of the curve where your material transitions from low but finite absorption to practically zero absorption. 

The book Vladimir R. Tuz mentioned above appears to be one of the definitive sources for this:  V. Lucarini, J.J. Saarinen, K.-E. Peiponen, E.M. Vartiainen, "Kramers–Kronig Relations in Optical Materials Research."

Are you measuring a thin film or bulk material (thousands of times thicker than the wavelengths you are using to probe)?  A brief search on Google Scholar shows there is a separate body of literature on Kramers-Kronig methods for thin films vs bulk materials.

Thin film publications:

K. F. Palmer and M. Z. Williams, “Optical constant determination of thin films condensed on transmitting and reflecting surfaces,” Technical Report AEDC-TR-83-64 (AD-A140845) (Defense Technical Information Center, Fort Belvoir, Va., 1984).

 K. F. Palmer and M. Z. Williams, “Determination of the optical constants of a thin film from transmittance measurements of a single film thickness,” Appl. Opt. 24, 1788–1798 (1985).

More general Kramers-Kronig publications that may be helpful for bulk material transmission measurements:

P.-O. Nilsson, “Determination of optical constants from intensity measurements at normal incidence,” Appl. Opt. 7, 435–442 (1968).

K.-E. Peiponen and E. M. Vartiainen, “Kramers–Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).

Dear Stefano:

Others have answered your question about using KK relations. I'll ignore this bit and answer your main question more directly.

If you aren't interested in an expression for the imaginary part as a fuction of wavelength, only its value at each wavelength, you can use the relationship Im(n)=ln(Io/I)/kz, where ln is the natural logarithm, Io is the original (incident) intensity, I is the detected intensity, k=2pi/lambda is the wave number in vacuum and z is the sample thickness. Your experimental absorption curve will typically be an I/Io curve, so if you know z you can work out Im(n) for each wavelength.

You can't obtain the real part of n directly from an absorption spectrum (but see answers to KK stuff); if you want to obtain Re(n) experimentally, you need to do it by interferometry, refractometry, etc.

Hope this helps.

No. You are not right and you can have a Re(n) on a longer wavelength range, say from 200 to 1000 nm

See attached file

Best

I deposited a drop of porphyrin in acid enviroment on top of a gold nanostructured surface, and I would say that is not a bulk but it is a film. However I don't know anything about the thickness of this film, but it can be easily much more than the wavelength range of the light probe (400 to 1000 nm). Hence, according to the words of Kenneth Ricci, my porphyrin layer can be a bulk.

Stefano,

As you know the thickness of the film can have a significant effect on the absorption and reflection spectrum: if both the front and back interfaces of the film are relatively smooth (smoother than 400 nm) and mostly parallel, you will probably see some fringe interference between front and back reflections.  Therefore when you search for Matlab codes or algorithms to estimate your Re(n) and Im(n) from your extinction spectrum, you should probably examine BOTH thin-film Kramers-Kronig algorithms and bulk measurement Kramers-Kronig algorithms.  You may have to try each kind of algorithm to see which gives a better result.

If you are lucky enough that your film is MUCH thicker than your longest wavelength (for example 1000 nm is your longest wavelength, perhaps your film could be as much as 100x thicker or 0.1 mm?) and if the two interfaces are not very parallel, a quick plane-wave calculation will show that the front and back interference patterns will be negligible when averaged over your photodetector area (because the film faces are not parallel) and you can just use bulk Kramers-Kronig approximation methods.  But since you do not yet know your film thickness, you should be aware of the possibility of front/back interference.  The thin film Kramers-Kronig models will take this interference pattern into account.

-Ken

How does one handle situations where nanoparticles are involved? Bulk or thin film approaches may not be adequate.

Krishnamurthy, if the particles are much (say, at least 10 times) smaller than the wavelength, you can use the Maxwell-Garnett equation. There are generalisations to nonspherical particles (see, for example, the work of Christian Mätzler). Sorry for the late reply.