Applicability of Kramer Kronig relation in the context of normal dispersion

[skmaidulhaque]

In the context of complex refractive index, is it possible to apply the Kramer-Kronig relation in the region of normal dispersion ? What I want to do is the following.

I can measure real part of complex refractive index of a material, n in the limited range of 200-1000 nm wavelength where refractive index shows normal dispersion behaviour namely Cauchy behaviour. Now, by applying Kramer-Kronig relation, is it possible to calculate the imaginary part of complex refractive in the same region.
If yes, please indicate how?

 

[Born2bwire]

Not without making some kind of value judgement about the behavior of the real part for the rest of the frequency spectrum. However, this is not an uncommon approximation to make. If you take a look at various basic dielectric dispersive models (Debye model, Cole-Cole model, ect.) you will find similar approximations. That is, in a Debye model, we assume that the dielectric has a single resonance. The resulting real and imaginary parts are chosen to still satisfy the Kramers-Kronig. Despite such a rudimentary approximation, the resonant behavior is largely confined around the frequency of interest and thus can serve as a limited bandwidth model of actual materials.

That being said, the information beyond your measurement spectrum can still affect the unknown imaginary part of that same spectrum because the Kramers-Kronig relation relates the entire spectrum of the real part to each point in the imaginary part (and vice-versa). So how you choose to model your permittivity outside the measurement can affect your estimated Hilbert transform.

Since you only have a bandwidth that demonstrates normal dispersion, I can see how this could be tricky because you now have to make some kind of guess as to how the refractive index relaxes back down (obviously one must have at least one point where the derivative reverts the sign so that the index will approach the vacuum permittivity again as we expect of materials when we go to infinite frequency).

So probably the simplest thing to do is to allow your real part to smoothly relax back to the vacuum (or general bulk dielectric) permittivity. I can't say whether or not you can do this intelligently on the data that you have. Maybe another thing to do is see if you can't fit one of the single resonance models (like the Debye or Cole-Cole) into your data. The normal dispersion could match up with one side of a resonant permittivity where the resonant frequency is at some frequency above your bandwidth.

EDIT: Personally, I wonder if it wouldn't be easier and more reliable to find a way to estimate the imaginary part simply from measurement. I guess the real part is easy to measure simply via refractory measurements but the lossy part might be crudely estimated from the loss in intensity over the path length in the material. One might rig up a photodetector like a photodiode and use a highly collimated light source like a laser to measure the loss in intensity over the path through the material versus air. One might then obtain a simple approximation to the imaginary part from this.

 

[skmaidulhaque]

Thanks for your reply.
I could not follow this part of your reply. "(obviously one must have at least one point where the derivative reverts the sign so that the index will approach the vacuum permittivity again as we expect of materials when we go to infinite frequency)"

Regarding the measurement part, what you said is correct only. But, regarding the direct measurement of lossy part, the technique you have described suffers due to not encountering reflection or scattering loss from the front surface.

What I want to do ...is to get a direct functional relationship (may be under certain approximation) between real and imaginary part of the complex refractive index to be used in normal dispersion region of dielectric.

Any more suggestion is welcome.

 

[Born2bwire]

There are several ways that I can think of that you can account for reflection. You can use another meter to measure the specular reflection from the top surface. You can then use that information coupled with the refraction information to come up with the reflection coefficient. Then you can calculate the transmission coefficient for the entire slab and take that into account. Of course you won't be able to track the phase shift in the reflection and transmission coefficients. But if your slab thickness is much greater than the wavelength (which I would assume would be the case as it seems like you are working with visible light) then the phase information would not be useful.

There are two limiting cases for the permittivity and permeability, that at DC and infinite frequency. These two cases have to be finite and in the case for infinite frequency it is generally taken that the permittivity and permeability approach vacuum. That is, we see that for higher and higher energy radiation that the radiation interacts less with the objects (think of how X-rays pass through dielectric tissue). If you have a normal dispersion, then your permittivity is increasing with frequency. Thus, there must be a point in the higher frequencies where the dispersion become anomalous so that the permittivity comes back down. In essence you may be able to simply model your limited bandwidth measurement as being one side of a single resonance. Which is why I think you may be able to fit one of the resonant dielectric models. If you can make a good fit with the real part of one of the models, then you can instantly get a rough estimation of the imaginary part.

 

[pmacias]

The Kramer-Kronig relation is a mathematical relationship that connects the real and imaginary parts of a complex function. It is commonly used in the study of optical properties of materials, including the complex refractive index. In the context of normal dispersion, where the refractive index decreases with increasing wavelength, the Kramer-Kronig relation can be applied to calculate the imaginary part of the complex refractive index.

In your specific case, where you have measured the real part of the complex refractive index in the range of 200-1000 nm and want to calculate the imaginary part using the Kramer-Kronig relation, it is possible to do so. However, there are some considerations to keep in mind.

Firstly, the Kramer-Kronig relation requires a complete set of data points for the real part of the complex function, which in this case is the refractive index. This means that your measured values should cover the entire range of wavelengths in the region of interest, from 200-1000 nm. If there are any gaps in your data or if the measurements are not precise enough, it may affect the accuracy of the calculated imaginary part.

Secondly, the Kramer-Kronig relation is an integral equation that involves the integration of the real part of the complex function over all wavelengths. This can be a complex mathematical process and may require specialized software or programming to perform the calculation accurately.

In summary, it is possible to apply the Kramer-Kronig relation in the context of normal dispersion to calculate the imaginary part of the complex refractive index. However, it is important to ensure that your data is complete and accurate, and to use appropriate methods for performing the integration.