good question

Dear May Ahmed, I would actually question your claim that you have correctly evaluated experimental reflectance and transmittance data to produce the plot you show.

I would rather argue that this is not usually possible without some assumptions or without recurring to specific methods such as e.g. spectroscopic ellipsometry.

Please do tell and elaborate on the procedure you followed, and do give a description of your sample. If you can, do share the data which resulted in your plot.

Observation: εr is fairly large, changes are very small, even where they show up as "strong" in your plot. The energetic position of the change at 3.3eV might hint at Si, for example, but the "true" dielectric function of Si looks different:

https://refractiveindex.info/?shelf=3d&book=crystals&page=silicon

Dear Kai Fauth, Thank you very much for your interest in helping me. My samples are doped metal oxides thin films and to be more specific (ZnO). At first I measured transmittance and reflectance using spectrophotometer at wave length from (185-2500 nm), Then I made the known calculations for alpha, Eg, K, n, εr , εi and optical conductivity. All plots are to some extent logical except εr is different and perplexing.

α= 1/d ln 1/T

αhע = A(hע-Eg)n

K = αλ/4П

n =((1+R)/(1-R))+√(((1+R)/(1-R) )^2-(K^2-1))

ɛ = ɛr+ i ɛi

ɛr = n2-K2 (**)

ɛi = 2nk

σopt= αnc/4П

Dear May Ahmed, before commenting on the above calculation steps (will have to make up my mind for it), have a look at the following link (same website as previous). It should be at least somewhat related to your material. Quite a few comments on how measurements were acquired are given, along with the reference in the literature.

https://refractiveindex.info/?shelf=other&book=Al-ZnO&page=Treharne

And I have a few additional questions, since you have a thin film:

- is it supported or support-free (self supported)

- if supported, what is the support? How do you take its optical properties into account?

Can you share how the films are produced and how homogeneous they are, what is their structral characteristics (might range from amorphous to single crystalline...). How did you characterize film thickness and homogeneity (of both, material and thickness)? What is actually the thickness?

What is the measurement setup and geometry?

And with respect to the calculations you did (you call them "known calculations"): what is your reference for these? Have you followed and understood the derivations leading to these formulae?

Several datasets on ZnO available on refractiveindex.info, e.g.:

https://refractiveindex.info/?shelf=main&book=ZnO&page=Stelling

Do look them up and discover that they do not match each other arbitrarily well... still, it gives you something to compare to. One of the references being pretty recent, you may discover more material by going through the references of this article.

Dear May Ahmed,

referring to the first of your eqations,

α= 1/d ln 1/T

it is intended to deliver the attenuation coefficient from the measured transmission. Is that correct? In a subsequent step, you want to use this to determine the imaginary part of the refractive index taking the third equation:

K = αλ/4П (where λ is the wavelength in vacuum/air and П is "Pi")

Now, while the latter equation appears to be correct to me (it is what I get), I see one possible pitfall and one conceptional error (with the first equation).

The pitfall would be arising if α and λ were not expressed in the same units, e.g. because λ and d were not expressed in the same units. However, this is just a matter of being careful with units in equations.

The conceptional error in equation 1 is to take the symbol "T" in the equation to be the same as the measured transmission in your experiment. It is not!

The equation is equivalent to Lambert's law which describes the attenuation of the intensity of light along its travel through an absorbing medium according to

I(z) = I(0).exp(-αz) [where I(0) = I(z=0)]

In your (and any) experiment, the light is actually impinging from the air and partially reflected at the surface (after all, you can measure a reflectivity spectrum, can't you). If the film is free-standing (no support) then there is also reflectivity at the second surface. Actually, there will be multiple reflections etc. and all this affects the quantitative result of both your transmission and reflection measurements. Depending on film thickness, you might even have to consider interference effects.

If the film is supported (e.g. on glass) then there is yet an additional interface you need to consider.

As long as you don't correct for these you cannot expect any of your results to be reliable. And in fact, there is no simple recipe for generating the complex refractive index from reflectivity and transmission experiments alone. (This is, actually, why people do ellipsometry).

This is not to say you can't do anything meaningful with transmission and reflectivity measurements, but you need to ponder precisely what you can and what you cannot conclude. Your equations above are not most helpful. I will comment on the fourth equation in a follow-up post.

Dear May Ahmed,

the fourth of your above equations stems from the law of reflection in normal incidence geometry from a single interface between vacuum (or air, neglecting the small difference) and some material described by the complex index of refraction N = n + ik. This law derives from the more general Fresnel's equations. Under the mentioned conditions, it reads:

R = |(N-1)/(N+1)|2

R= [(n-1)2+k2] / [(n+1)2+k2]

which you can resolve for n2, thus resulting, in a first instance, in a quadratic equation for n. As you know, such an equation has two solutions. Which one do you choose? Why the one given in your post above? [Note that it contains also a sign error: the last parenthesis in the sqrt expression ought to contain "+" instead of "-".] Now, what is your measurement geometry? Is it close enough to normal incidence so that chosing this simplified approach makes sense to begin with?

More severely, we again run in the problem of an incompatibility between the equation and what it describes on the one hand, and what happens in your experiment, on the other. As I indicated previously, you will not measure the reflection from a single interface, especially in the transparency regime of your specimens. Therefore the equation simply cannot be applied.

Overall: these equations can be useful, but you need to understand their underlying assumptions, what physical situations they actually describe and the limitations and approximations that go along with using them in your situation (and where you better refrain from doing so).

It can then be a better idea to expand on the experimental side, e.g. starting with undoped ZnO films and varying the dopant concentration. If the nanostructure and microstructure does not vary much with dopant concentration, you might concentrate on alternative ways to evaluate your results, entailing less errors.

As for your last equation, concerning the optical conductivity, I don't know where you have taken this expression from. Pleas compare with

• wikipedia: https://en.wikipedia.org/wiki/Optical_conductivity
• e.g. lecture notes: http://physics.unl.edu/tsymbal/teaching/SSP-927/Section%2013_Optical_Properties_of_Solids.pdf
• Wooten's book: http://www.phys.ufl.edu/~tanner/Wooten-OpticalPropertiesOfSolids-2up.pdf

(The link to the lecture notes is actually copied from the wikipedia page. From a comparison you will note that the relation between ɛ and σopt seems to depend on the unit system [SI=MKSA vs. cgs]! Awkward but such is life...)

The book by Wooten is old (and a bit old fashioned from today's perspective) but I still find it a text very much worthwhile studying.

Dear Kai Fauth, thank you very much for your comments, which do not know how much you helping me. I used the link for refractive index data calculation as you recommended, i found that my calculation is close to what i found on it. I will take your tips into account and now i will study what you sent me. If you do not mind, I may do some other queries later.

I am really grateful for your interest and patience even though it is the vacation days of the feast.

I wish you happy new year.

I recalculated again, extended the range and the result was as follows

Dear May Ahmed, what is the database for the extension? Did you have the data from the beginning? You know, that the sudden jump right into the previous position is most unlikely...

I hoped to have shown you that several of your steps taken in evaluating your data into determining ε from reflectivity and transmission data were incorrect. What did you change?

What did you learn from the literature study?

In order to make this a more valid discussion you should consider showing plots with all datasets:

• T and R
• n and k
• εr and εi

Here some a useful resources. Check with Chapter 7 in:

https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwjU18b5qcHfAhXMLlAKHdDwBH8QFjAAegQICRAC&url=https%3A%2F%2Fwww.researchgate.net%2Fprofile%2FJerome_Polesel%2Fpost%2FWhat_is_the_correct_band_gap_determined_by_a_Tauc_plot%2Fattachment%2F59d620fc79197b807797f6da%2FAS%253A293973668188172%25401447100192588%2Fdownload%2FThe%2BPhysics%2Bof%2BThin%2BFilm%2BOptical%2BSpectra_%2BAn%2BIntroduction.pdf&usg=AOvVaw31Ht7GF4cs3jMfQU_EXQzy

and here fom PE:

https://www.perkinelmer.com/lab-solutions/resources/docs/APP_Thin-films.pdf

plus a glimpse in yet another Stenzel book:

http://www.ciando.com/img/books/extract/3642540635_lp.pdf

To quote from Chap.7.4.6 of Stenzel's book:

" As shown in the previous sections, it is possible to provide explicit expressions for the spectra of a thin film sample if geometry and optical constants are known. It is however impossible to obtain explicit expressions for the optical constants as a function of the measured data."

Quite amazing what online resources already exist:

https://www.google.de/search?client=opera&q=thin+film+optical+online+calculator&sourceid=opera&ie=UTF-8&oe=UTF-8

One could generate artifical data and see whether one is able to recover the original refractive index data from evaluation.

As with all scientific software: (i) use wisely (ii) do not forget that there is no SW without bugs.

Here's a link to a previous, related discussion:

https://www.researchgate.net/post/Is_there_any_free_software_that_can_calculate_optical_constants_from_thin_films_reflect_and_or_transmetance_spectra_data

And yet another good read:

Article Methods for the determination of the optical constants of th...

Dear Mai,

The real dielectric constant according to your curve shows a very shallow dip between 3 and 4 eV of photon energy. What can be the cause of this dip?

The material in the optical range of frequencies have only electronic polarization which is more or less independent of frequency until the electron resonance frequency.

So, you have to show also the preliminary dielectric constant. Both curves may give more complete picture aiding in interpreting the results.

Till you give more data, one can only say that you can consider this very shallow dip as negligible and the real dielectric constant is more or less independent on the the photon energy in the measuring range.

Best wishes

Dear Abdelhalim abdelnaby Zekry,

of course, I cannot speak (write) on the behalf of May Ahmed. However, we did have some "private" discussion and from this I can tell you that at the stage shown in the plots of this thread the plotted frequency dependence of εr essentially results from erroneous evaluations. And while what appears as "the dip" does relate to real features of the spectra (albeit only in the first version of the plot) the underlying calculations were not correct.

Again, quoting Stenzel's book (reference given in a previous post):

"It is however impossible to obtain explicit expressions for the optical constants as a function of the measured data."

You have a pointing vector for the wave. Changing (epsilon relative) means that the impedance has changed.