# Kramers Kronig transforms for refractive index

• christianjb

#### christianjb

I'm puzzled (as usual).

The Kramers Kronig transform is commonly used to relate real and imaginary components of the electric susceptibility- \chi(\omega)

It appears from reading papers that it also works for relating real and imaginary components of the refractive index.

But why? That means that the refractive index can be written as the FT of a response function. What is that response function?

## Answers and Replies

Well, there must be some misunderstanding.

The refractive index is not really a primary concepts, it related wavevector and frequency in a solution of a dispersion relation.

The primary concept is that of the permittivity tensor, and it is a response function.
The permittivity tensor satisfies indeed the Kramers Kronig relations.
This has consequence on the propagating modes, that are solutions of the dispersion relation, and therefore it has a consequence on the refractive index.

But, of course, it make no sense to say that the refractive index satisfies in any way the http://en.wikipedia.org/wiki/Kramers-Kronig_relations" [Broken].

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Well, there must be some misunderstanding.

The refractive index is not really a primary concepts, it related wavevector and frequency in a solution of a dispersion relation.

The primary concept is that of the permittivity tensor, and it is a response function.
The permittivity tensor satisfies indeed the Kramers Kronig relations.
This has consequence on the propagating modes, that are solutions of the dispersion relation, and therefore it has a consequence on the refractive index.

But, of course, it make no sense to say that the refractive index satisfies in any way the http://en.wikipedia.org/wiki/Kramers-Kronig_relations" [Broken].

The real and imaginary portions of the index of refraction are related by the Kramers-Kronig transformations, not jus the real and imaginary parts of $$\chi$$

Replace the real part by $$n - 1$$ and the imaginary part by $$k$$. More publications dealing with this topic work with $$n$$ and $$k$$ than with $$\chi$$.

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Dr Transport,

Since, in the simplest situations (waves in vacuum), the refractive index is the square root of the permitivity, I have difficulties to imagine that it could satisfy the KK relation in general. Moreover, I does not really matter since anyway the KK relation on the permitivity should cover the whole effect of causality in electrodynamics.

Moreover, my definition of the refractive index is associated with any particular solution of the dispersion relation. For example, for longitudinal oscillations in an unmagnetised plasma, the dispersion relation is simply

epslo(w,k)=0 (where epslo is the longitudinal part of the permitivity)​

Solving for w provides the dispersion curve of the longitudinal mode and the so-called refraction index
n = Re(k)/w​
for this mode.

I can imagine however that the KK relation on the permitivity implies some particular properties for the refractive index. But I would consider that as a freedom of language.

As the question assumed, the refractive index is in no way a transfer function and assuming the KK are a property of causal transfer functions, it may be a bit excessive to say that the KK applies to the refractive index.

So it seems more a question of definitions than really a question of physics.

Michel

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