Refraction-Speed difference at different wavelengths

[u0362565]

Hi all,

I've read that when light undergoes refraction into a medium with higher refractive index it changes speed and this is explained by the electrons of the medium absorbing the photon energy, they hold onto it then eventually re-emit the light if the frequency of light doesn't match the resonant frequency of the medium. The electrons hold onto the energy from light of shorter wavelengths for longer and hence shorter wavelength light slows down more than higher longer wavelengths. What's the physical basis explaining why shorter wavelength light is re-emitted at a slower rate compared to longer wavelengths? The texts I've read don't seem to go into this.

Thanks for the help.

 

[mfb]

There is no "resonant frequency of the medium". There can be many frequencies or frequency ranges where a medium absorbs or reflects light.

electrons of the medium absorbing the photon energy, they hold onto it then eventually re-emit the light

That is a problematic model. It seems to give an explanation why light is slower, but it cannot explain anything else.

The wavelength dependence of the refractive index (and therefore the speed of light) depends on the material and the frequency range. There is no general rule. Very high-energetic photons (x-rays, gamma rays) are usually faster than visible light.
To make things more complicated, materials do not have a single speed of light. They have a phase velocity, a group velocity and a signal velocity. The phase velocity can be faster than the speed of light in vacuum, or even negative.

 

[u0362565]

I see that's interesting, I have never heard of phase, signal or group velocity of light. If there's no general rule it does seem higher energy em radiation slows down to a greater extent, so is there no explanation for why this happens?

Can light transmission through a medium be thought of anything like a jablonaki diagram where light can excite electrons to higher energy states? energy loss causes the electron to move back to ground emitting a photon.

 

[mfb]

so is there no explanation for why this happens?

Sure there is, but it is more complicated to predict actual numbers. In metals, plasmons play an important role, for example.

Can light transmission through a medium be thought of anything like a jablonaki diagram where light can excite electrons to higher energy states?

For wavelengths where this is possible, transmission is often very bad because the material absorbs or reflects the light.

 

[Philip Wood]

A model which explains quite a lot is the classical one in which electrons in the material have various natural frequencies. An e-m wave entering the medium will excite electrons to oscillate (by means of its oscillating electric field). The oscillating electrons will re-emit e-m radiation, and the phase of this radiation will lag that of the incident radiation. The wave as it progresses through the medium will be the vector sum of incident and and re-radiated radiation, so will progressively lag in phase behind what it would be doing if traveling in a vacuum. Hence the lower (phase) velocity. The effect becomes more pronounced the closer the frequency is to a natural frequency, because the electron oscillations will have large amplitude, so the (phase-retarded) re-emitted radiation will be stronger. That's the cause of dispersion: the dependence of phase velocity on frequency. [The fact that the phase lag also depends on the incident frequency relative to the natural frequency is of less importance than the re-emitted amplitude being greater near resonance.]

 

[morrobay]

The wave as it progresses through the medium will be the vector sum of incident and and re-radiated radiation, so will progressively lag in phase behind what it would be doing if traveling in a vacuum. Hence the lower (phase) velocity. [The fact that the phase lag also depends on the incident frequency relative to the natural frequency is of less importance than the re-emitted amplitude being greater near resonance.]

Does this resultant wave that is vector sum / superposition of incident and re-emitted radiation with phase lag also have a shorter wavelength with same frequency , hence a slowing of the waves travel in the medium ? If so can you show how superposition of waves with different amplitude and phases produces resultant wave with shorter wavelength ?

 

[mfb]

You can analyze each wavelengths frequency separately, as the medium does not change wavelengths frequencies (at least not at the level we discuss light/matter interactions here).

(Edited for clarity)

 

[Philip Wood]

I don't think I agree with mfb (but am happy to be corrected). As I see it the wave velocity, v, is made less than c, so $\lambda$ is reduced below its value in a vacuum, for a given frequency [$\lambda = \frac{v}{f}$]. The frequency is unaffected because you can't lose or gain cycles. A crude analogy for why the wave speed is reduced is that of an ideal relay race where the runners all run at speed c (suspend disbelief!) but the baton hand-overs all take time.

 

[mfb]

Sorry, my post was quite misleading in that aspect. I should have said frequency.

Yes the wavelength changes, but different wavelengths coming in will still be different wavelengths in the medium, that's what I meant.

 

[morrobay]

I don't think I agree with mfb (but am happy to be corrected). As I see it the wave velocity, v, is made less than c, so $\lambda$ is reduced below its value in a vacuum, for a given frequency [$\lambda = \frac{v}{f}$]. The frequency is unaffected because you can't lose or gain cycles. A crude analogy for why the wave speed is reduced is that of an ideal relay race where the runners all run at speed c (suspend disbelief!) but the baton hand-overs all take time.

With both wavelength and phase velocity functions of frequency : λ = v_p/ƒ , ƒ = v_p /λ and v_p = λƒ , ω/k = ƒλ
Is the reduced phase velocity from a shorter wavelength or shorter wavelength producing a reduced phase velocity ?
And I take from analogy above that while individual waves travel at c it is the summation of those waves ( phase velocity ) that is < c.
The underlying question : Is the shorter wavelength in medium just an artifact resulting from phase velocity < c .
My understanding is that light waves always travel at c

 

[Philip Wood]

I don't think you can distinguish between cause and effect here. The velocity is the distance gone by a point of constant phase in the time for one cycle, divided by the time per cycle. But the distance referred to is the wavelength. For what it's worth, if I'm trying to explain why light doesn't travel as fast in air as in a vacuum I concentrate on the idea that a point of constant phase doesn't go as far in any given time, so I suppose I look more at v_ph rather than specifically at $v_{ph} T$, that is λ. But I don't think this shows that it's the change in v which causes the change in λ.

Quote to ponder: "There is no cause nor effect in nature. Nature simply is." [Ernst Mach]