Exploring Refractive Index & Di-Electric Constant Relationship

[neelakash]

The Maxwell's relation in electromagnetic theory states that refractive index is equal to the square root over the di-electric constant.

This relation holds for some simple gases.But more generally,it does not.My question is why this is the case.
Eugene Hecht says that this is because n (refractive index) is a function of frequency.But I do not understand.After all, di-electric constant is also frequency dependent quantity.

What I think is that for di-electric substances,due to the very structure of the substances,the frequency is only measurable at low frequencies.So,that when we try to match the calculated n (observed in visual range of EM spectrum) with the di-electric constant =Limit (frequency--->0) [K(w)] (where K is the di-electric constant and w is the frequency),the two values do not match.

Please confirm if I am correct.

 

[lalbatros]

neelakash,

You are not correct.

You should first define what you call the refractive index.
A general definition of the refractive index is obtained from:

- the wavevector (k) of a wave propagating in the given medium
- the frequency (\omega = 2{\pi}f) of the same wave

the definition is:

\mathbf n = \frac {\mathbf{k}}{\omega}

as you can see, it is a vector, in general.
This vector is the same for all directions in isotropic materials.
Note that it might depend also on the polarisation of the wave.

Now the crucial question is: how do you get the wavevector (k) ?
The answer is simple: solve the Maxwell's equations.

In the Maxwell's equations, you need to include the currents and the charges as well as the external excitations and boundary conditions. The internal currents and charges are related to the wave-fields. When the motions of the electric charges are small enough, and the response of the medium to the excitation can be assumed as a linear response, then the dielectric tensor gives the link between the excititation and the response of the medium. When solving the equations, there is usually no stationary solution (solution is zero!) unless the (linear) equations satisfy a certain equation called the dispersion relation that has (for linear systems) the general forme f(k,w)=0 .

In non-isotropic media, there are usually several solutions to the dispersion relation. These different solutions are sometimes called modes or branches or simply the xxxx-wave (xxxx can be any wave name!). For each of these modes the refractive index can be calculated.

Of course, in isotropic media, all these things are much simpler. So much simpler that the refractive index looks like a primary concept, while actually it is not really a primary concept. the primary concept is the concept of response of the electric charges to an excitation. For small responses (linear assumption, very oftne a valid assumption), the dielectric tensor represents this link between response and excitation. For very simple system, the dielectric tensor is the refractive index. In general, there is a longer way from the dielectric tensor to the refractive index.

Finally, let me indicate that the response of the electric charge depends on the frequency of the exciting wave. For example, heavier charges (ions) cannot respond as fast as electrons. Therefore (in plasmas) very high frequencies with excite electron motions but not ion motions. Many other effects can give a big role to the frequency. I like the case when there are external magnetic fields: in this case the wave can be resonant with the rotational motion of charged particle around magnetic field lines, this leads to so-called cyclotron waves (io- or electron-). In condensed media the atomic and molecular structure plays a very complicated role, not to mention also semiconductors. Therefore, it should be no surprise that the refractive index depends on the frequency, since the refractive index depends on how charged particles respond to the exciting field.

To be very clear:
The dieletric tensor (or constant) does really depend on the frequency, but also on other things, like the direction of the wave.
The refractive index is related to the dielectric tensor and does therefore also depend on the frequency.

 

[Claude Bile]

What I think is that for di-electric substances,due to the very structure of the substances,the frequency is only measurable at low frequencies.So,that when we try to match the calculated n (observed in visual range of EM spectrum) with the di-electric constant =Limit (frequency--->0) [K(w)] (where K is the di-electric constant and w is the frequency),the two values do not match.

Please confirm if I am correct.

You are correct in saying that the dielectric "constant" for most materials is measured at a very low frequency (typically kHz), and thus would be expected to deviate from the value calculated at optical frequencies (100's of THz), due to the frequency dependence that has already been pointed out.

Claude.

 

[neelakash]

I did not understood what lalbatros said.
And what is the refractive index?The formula is w=ck and the formula for refractive index is n=c/v.This absolute index of refraction should not be a vector..However,it depends on frequency.

 

[lalbatros]

neelakash ,

Just to give you a first idea, you can have a quick look at this page:

http://farside.ph.utexas.edu/teaching/plasma/lectures/node44.html

You will find there how the dielectric tensor for a cold plasma can be derived from the equations of motion of charged particles. As you may imagine, this is rather simple as compared to the hot and magnetised plasma dielectric. The illustration by the plasma theory is convenient because it simply reffers to classical physics. But of course, for most common materials quantum physics should be applied.

If you read further from the same site, you will understand what I meant. See this page:

http://farside.ph.utexas.edu/teaching/plasma/lectures/node41.html

where you will find a few possible different wave that may propagate in a cold magnetised plasma. Just by reading the list, you will see that the index of refraction depend on the orientation of the wavevector: for example it can be parallel or perpendicular to the magnetic field.

For the definition of the refractive index, in such a context, you could check this page:

http://farside.ph.utexas.edu/teaching/plasma/lectures/node45.html

Finally, let me note that we did not discuss here the effect of different absorption (damping) mechanism on the wave. This corresponds to the imaginary part of the refractice index. In the web page I mentioned, no absorption mechanism is considered.