# Index of refraction vs frequency

1. Feb 17, 2013

### e.chaniotakis

Hello!
I have a question concerning the relation of the index of refraction of a material with E/M wave frequency.
If we take the classical Feynmann approach, we can envision the material as a series of atoms, with electrons bound with springs to the nuclei.
Let the natural frequencies of the oscillators be called ω0 and the electric field of the wave be E=Eo*e-iωt.
The E/M wave forces the electron to oscillate with frequency ω.
If we solve the differential equation, we will find:
y(t) = (q/m)*[Eo*e-iωt]/[(ωο22) ]
(we consider the damping infinitessimal).

The polarization of the material can be given from :

P=χeE
P=qy(t)
→ P = E*xe = Ε*[Nq2/m]*[1/{ωο22}]

/* N=nr of electrons per unit volume with natural frequency ω0 */

Τhe permitivity ε is given by:
ε=ε0*(1+χe) =
ε=ε0*(1+[Nq2/m]*[1/{ωο22}])

The index of refraction is given by:

n=sqrt(ε*μ)/sqrt(ε0*μ0) = ...
n= 1+{Nq2/2mε0}*{1/(ω022}

For ω<<ω0,

n=1+(Nq2/2mε0)*{ [1/ω02]+[1/ω04]*ω2] }
n=1+A+Bω2

This is the actual derivation of the dispersion formula.
Now, if I wanted to "popularize" the effect to a high school student, how should I phrase it?

My idea was something like the following:
When a photon enters the material, it forces the electrons to oscillate. If ω<ω0, as ω increases , the amplitude increases until it reaches resonance.
As the amplitude increases, the probability of a photon to interact with an electron increases since there are more "meeting points" for a photon and an electron .
Thus, effectively there will be more Rayleigh scattering processes per unit length, and the photon will "take more time" to pass through the material.
Thus its speed effectively decreases or the index of refraction increases.
Is this "popularization" correct?

Thank you

2. Feb 17, 2013

### DrDu

First I don't think that this description is due to Feynman although he may have used it.
Second your intent of popularisation is more complicated than the original.
The original equations consider the interaction of a classical electromagnetic field with some classical oscillators while you start to interpret it in terms of probabilities of photons to interact with electrons. I think there are nice demonstrations of forced oscillations like e.g.

Last edited by a moderator: Sep 25, 2014
3. Feb 17, 2013

### e.chaniotakis

About the Feynmann part, you are most right, I refer to its being a classical textbook.
About the probability part, to my experience, students understand things conceptually better in terms of bouncing balls (the way I treat photons in my description :) ).
I actually want to make an analogy between this effect and the effect of a resistor's resistance increasing with temperature and how can one visualise it microscopically.
Thank you for the video!

4. Feb 17, 2013

### DrDu

I prefer to think of this in terms of two coupled oscillators, e.g. pendulums. There are two normal modes, one where mainly the oscillator with lower frequency is moving and one where mainly the one with higher frequency is moving.
If you want to interpret this quantum mechanically then you can consider the process where a photon converts into a vibrational excitation and this back into a photon and so on and so on.
The point is that the probability for interconversion is quite independent of frequency but the time a vibration can live is given by the time energy uncertainty. If the energy of the vibration does not coincide with the energy of the photon, then the lifetime of the vibrational excitation is proportional to the inverse of this energy difference. Hence far from resonance, a photon will be most of the time be a photon and a vibration will stay most of the time a vibration. However, this is not a statistical process like the scattering of electrons in a resistor.