# How is phase velocity related to deflection angle?

• A
binis
Refractive index is a function of velocity in the medium. How is this related to deviation angle inside the medium? I am not asking for the known formula, but for the mechanism behind it.

Homework Helper
Did you google anything at all? E.g. Snell's law?

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vanhees71
binis
Did you google anything at all? E.g. Snell's law?
Of course I did. Should I had asked it in Quandum physics section?

Homework Helper
Of course I did
So you found the relationship that your post is asking for !
Or is there an interpretation for 'deviation angle' I am not aware of ?

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vanhees71
2022 Award
Of course I did.
So what's wrong with the explanation on Wikipedia, for example?

hutchphd and vanhees71
binis
So you found the relationship that your post is asking for !
Snell's law is a rule, not explanation

binis
So what's wrong with the explanation on Wikipedia, for example?
Do you mean the Fermat's principle?

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vanhees71 and hutchphd
Homework Helper
Snell's law is a rule, not explanation
I see. And the derivation (from Fermat principle) is not understood, not acceptable, something else ?
In which case there are three other derivations, just below (that's why I wondered somewhat ironically in post #2).

I vaguely remember optics lectures (1971 or 72) that followed the 'boundary conditions' path. The old lecture notes should be in the attic somewhere, but nowadays I find the internet quite adequate ...

Perhaps you can be a bit more specific: PF is good at answering focused questions, but the format isn't ideal for rehashing textbook material. (However, we do have 'Insights' !)

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binis and vanhees71
binis
What don't you understand about its explanation?
Many issues arise. 1. Since wave is spreading according to Huygens law in wavefronts, how it reaches the interface between the materials at an angle? The figure shows an arbitrarily cuted section of wavefronts to create a "light ray". 2. Interface in microscale is not flat but consists of distant molecules or atoms (see Bragg's law). 3. How is the interference resultant wave neither reinforced or canceled? 4. How is the resultant wave not Compton shifted? 5. Oscillation is an assumption inconsistent with electronic cloud. And what about opaque materials? By this explanation must be also transparent. 6. There must be a little loss of energy because the wave is totally absorbed by the glass (or quartz or liquid) at last, if medium's width is quite long (Lambert's law).

Gold Member
2022 Award
Ibix, BvU and binis
2022 Award
. 1. Since wave is spreading according to Huygens law in wavefronts, how it reaches the interface between the materials at an angle? The figure shows an arbitrarily cuted section of wavefronts to create a "light ray".
Because it's a plane wave, probably with a Gaussian intensity profile. Apply Huygens' principle to a plane wave in free space and you'll get a plane wave out.
2. Interface in microscale is not flat but consists of distant molecules or atoms (see Bragg's law).
Bragg's law isn't relevant here - the wavelength of light is far too long. Scratches and imperfections on the surface on the 0.1##\mu##m scale will cause diffraction.
3. How is the interference resultant wave neither reinforced or canceled?
Don't understand what you are asking.
4. How is the resultant wave not Compton shifted?
I don't think there are any nearly free electrons in glass. And even if there are have you worked out the Compton shift and compared it to the wavelength of light? Would you be able to see it?
5. Oscillation is an assumption inconsistent with electronic cloud. And what about opaque materials? By this explanation must be also transparent.
No idea what you mean here.
6. There must be a little loss of energy because the wave is totally absorbed by the glass (or quartz or liquid) at last, if medium's width is quite long (Lambert's law).
Yes. There's also usually some reflection at the surface. So what?

BvU and vanhees71
binis
Perhaps you can be a bit more specific:
I must clarify my OP question: How is Snell's law formula mathematically deduced from the change in phase velocity?
It has an explanation for the slowing down. It has an explanation for the divergence. But it doesn't explain how is Snell's law deduced from the slowing down.
What don't you understand about its explanation?
"may have wave packets that pass an observer at a slower rate" This is an arbitrary assumption. Is this observed elsewhere, in a TV antenna for example?

Yes. There's also usually some reflection at the surface. So what?
Energy decrease means frequency decrease.

2022 Award
It has an explanation for the slowing down. It has an explanation for the divergence. But it doesn't explain how is Snell's law deduced from the slowing down.
It explicitly does so in the second paragraph of the section headed Explanation for bending of light as it enters and exits a medium. What didn't you understand about it?
Energy decrease means frequency decrease.
No. In wave optics energy decrease means amplitude decrease, and amplitude is not related to frequency. If you are thinking of the ##E=h\nu## relation from quantum mechanics, remember that it is the energy of a single photon. The beam is made up of many photons, so can lose energy without changing frequency as individual photons are absorbed.

BvU
binis
No idea what you mean here.
Oscillation of the free electrons (i.g. inside a TV antenna) is known. Oscillation of the orbital electrons inside a material is unknown to me (not aware of QM).

Gold Member
Oscillation of the free electrons (i.g. inside a TV antenna) is known. Oscillation of the orbital electrons inside a material is unknown to me (not aware of QM).
It’s hard to discuss matters involving QM if you don’t know some details. Read around about it. There’s plenty of good material out there.

binis
binis
What didn't you understand about it?
"the resulting "combined" wave may have wave packets that pass an observer at a slower rate."
This is an assumption. Is this observed elsewhere, in a TV antenna for example?

Homework Helper
"the resulting "combined" wave may have wave packets that pass an observer at a slower rate."
This is an assumption. Is this observed elsewhere, in a TV antenna for example?
You are misquoting, by accident I must presume
Wiki said:
The resulting "combined" wave has wave packets that pass an observer at a slower rate.
(emphasis mine)

This is not an assumption. And yes, wave guides, klystrons, whatever: they have their own speed of propagation. Of course.

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2022 Award
You are misquoting, by accident I must presume
Search for "wave packet" in the page. You are quoting from the first instance, binis from the second.
This is an assumption.
No it's not, it's a statement of fact about combining two arbitrary waves. The combination may propagate slower, but the obvious counter example is combining two waves with the same frequency and propagation speed, which produces nothing more than a phase offset. But you aren't combining arbitrary waves, you are combining a light wave and the radiation from electrons driven by the wave. The result of that particular combination is a wave that travels slower.

BvU
binis
Oscillation of the orbital electrons inside a material is unknown to me (not aware of QM).
It is known as the fluorescent effect, having a different result.

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