Dear Ezeddin Mohajerani , thanks for sharing this interesting question. Of course, while the laser source power remains constant, the intensity of the fringes will be reduced as the dye concentration is encreased. The arm with the dye solution would absorb more and therefore less light will arribe to the detector from this arm, causing a fainter interference.

Normally addition of solute to a solution would cause a rise in refractive index, depending of the solvent and solute refractive indexes.

These variations could be relatively small, but important compared to the reference arm where you only has air.

The higher the refractive index of the dye solution, the larger the paths differences.

Hope this helps. Good luck with your research work and best wishes.

As Manuel Gómez says, increasing the concentration of dye in a solution will usually increase the refractive index.

However, when the absorption changes rapidly with optical frequency, as will be the case with a narrow absorption band, there can be a reduction in refractive index near to the absorption peak.

https://www.rp-photonics.com/kramers_kronig_relations.html

https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations

https://en.wikipedia.org/wiki/Refractive_index#Complex_refractive_index

Article Designing Refractive Index Fluids using the Kramers-Kronig Relations

https:// pubs.rsc.org/en/content/articlelanding/2020/fd/d0fd00027b

Article Relationship between the Kramers-Kronig relations and negati...

Alan Robinson has provided references, but has not emphasised it in the text.

In the spectral region of an absorption (imaginary component of the refractive index) the real part of the refractive index also undergoes significant change, which can be related to the absorption band by the Kramer-Kronig relationship.

Thus, the optical pathlenght of that arm of the interferometer will change but in a strongly wavelength/frquency dependent manner.

First I should thank Manuel, Alan and Hugh for their valuable contributions. Still something is not clear to me.

I better not to emphasize on the concentration. Assume we have a solution in red color which absorbs green light. if we put this solution in one arm of an interferometer, we can see the interference fringes at lower intensity comparing with a clear solution. Now theoretically Kramer-Kronig relation says that the real part of refractive index follows an anomalous dispersion. In this case we might observe very high or very low refractive index. If we use a tunable laser around the absorption peak of the solution, can we really observe such changes in the refractive index in the interference fringes.

Really I haven't seen any report on this type of observation.

I have never done it, so it could be worth a try. Alternatively, it should be easy to simulate.

Yes, it is not very easy to try that without a tunable laser around the absorption peak of the medium. And simulation also uses Kramers-Kronig relation which is not difficult to guess the result. But really I am not convinced with that result because in reality the light which is not absorbed by the medium just is not absorbed and they are among those lucky lights which just bounced the charges and involved in a reduction in speed of those transmitted lights.

Ezeddin Mohajerani

I don't have a peer-reviewed reference, but here is a demonstration using white light and a prism filled with aqueous solution of 4% KMnO4

https://www.millersville.edu/physics/experiments/042/

I will let you decide how convincing this.