Index of refraction vs frequency

[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 Feynman 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]/[(ωο²-ω²) ]
(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/{ωο²-ω²}]

/* 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/{ωο²-ω²}])

The index of refraction is given by:

n=sqrt(ε*μ)/sqrt(ε0*μ0) = ...
n= 1+{Nq²/2mε0}*{1/(ω0²-ω²}

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

 

[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.

 

[e.chaniotakis]

About the Feynman 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!

 

[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.

 

[sardar]

for your question! The relation between the index of refraction and the frequency of an electromagnetic wave is a fundamental concept in optics and material science. Your explanation is a good start in understanding this relationship, but I would like to provide a more detailed and accurate explanation for a high school student.

The index of refraction, denoted by the symbol n, is a measure of how much a material slows down the speed of light passing through it. The higher the index of refraction, the slower the speed of light in that material. This is because the speed of light in a vacuum is the maximum possible speed, and when it enters a material, it has to interact with the atoms and molecules in that material, which slows it down.

Now, let's take a closer look at how the index of refraction is related to the frequency of light. As you mentioned, the material can be thought of as a series of tiny springs (atoms) connected to a central point (nucleus) by tiny strings (electrons). When an electromagnetic wave, such as visible light, enters the material, it forces these electrons to oscillate at the frequency of the wave. This is similar to pushing a swing at its natural frequency to make it oscillate with maximum amplitude.

The frequency at which the electrons in the material oscillate is known as the resonant frequency, denoted by ω0. When the frequency of the electromagnetic wave is close to the resonant frequency of the material, the amplitude of the electron oscillations increases, and the probability of interaction between the light and the electrons also increases. This means that the light takes longer to pass through the material, effectively slowing down its speed and increasing the index of refraction.

To put it simply, when light enters a material, it causes the electrons to vibrate at their natural frequency. If the frequency of light is close to the natural frequency of the material, the electrons vibrate with maximum amplitude, causing more interactions with the light and slowing it down. This is why the index of refraction increases as the frequency of light approaches the resonant frequency of the material.

I hope this explanation helps in understanding the relationship between the index of refraction and frequency. It is important to note that this is a simplified explanation and that the actual calculation involves more complex mathematical equations. However, the concept remains the same, and understanding it is crucial in understanding the behavior of light in different materials.