Does Fermat's Principle Apply in All Special Cases of Light Path Alteration?

[zrek]

Wow, thank you, now I'll try to analyze your answer

... And we can make other changes to the lower surface, so that a different extreme path is possible. Light will take this path, and so in this sense, we can change the entrance point of the light ray by changing the lower surface.

Am I understand it right, that still the new path with the new entrance point have to fit to Snell's law? (I'll try to draw a concrete example with a new image that shows the old and the new lower surface/path, but I'm not sure that I can do it easily, if you have an idea, please help me on this too, thank you)

... your question implies that there is only one path. Light will take any and all paths which extremise the total time. So there is not just one possible path that we are changing. We solve for the functional derivative to be equal to zero, and all the solutions give the paths that light will take.

Now I'll try to analyze your words and draw some (maybe wrong) conclusions (my imagination added). Please correct me...
1. In case of the simple rectangle (cuboid) the light path from A to B is clear, can be calculated easily by Snell's law and is the result of Fermat's principle, but it does not really matter, because that path is not much more real than the other possible paths...
2. ... because we can't be sure wether the light (package) is behaving as a wave or particle ...
3. ... but if there is a detector at the point B which determines the photon itself and its arriving angle, then we can calculate back and can determine the entering point into the surface ...
4. ... which is also not relevant, only a imaginary property, since the light itself interacted only in the point B, so nothing is changed at the entering point.
5. If we change the lower surface, the arriving angle may change and also the entering point -- that is not really matters, since the entering point is not a physical property. The light calculates and chooses, "knows" this imaginary path before it actually goes on it.
6. If we are thinking about the light as a particle, the light propagates not step by step, and decides its new angle when it hits the actual surface, but its path determined before its movement.

Am I too dreamy?

 

[sophiecentaur]

Be very careful about talking about 'light energy on the move' as photons. They are nothing like little bullets and can only really be considered when actually interacting with an emitter or absorber. Waves is the appropriate may of dealing with EM radiation except at either end - even when you are down to 'one at a time' photon densities. Duality is not what it seems and hasn't been for a long time in Physics. The tempting pictures of photons that many people cherish in their minds have not been accepted by the Science establishment for many decades and are very misleading.

My question is:
Is there a shape available that by changing the lower surface of it, will change the entrance point of light ray?

This suggestion would seem to violate causality except, perhaps, in the case of a resonant cavity (optical or RF).

 

[zrek]

I still feel you are looking for a distinction between things where it need not exist. There is no disagreement between the results of the approaches, in principle. Ifaics, Fermat merely gives a way of working out where a ray (an approximation of the description of EM propagation) will go in a large enough system for other diffraction effects not to be significant.
Have you explored how the propagation of light can be treated in terms of diffraction (not just two slits and a pinhole)?

I like the two slits experiments, they are really interesting. I also examined the delayed choice experiments (quantum erasers) too.

I think that I expected too much from the Fermat's principle, but I still not given up with it, I think that there is much more in it than I understand.

I really feel that if you want to get into optics then you need to be prepared for a wave approach for the best model. Rays are totally fine for most practical applications but every optical system departs from 'rays' eventually and demands diffraction.

Sure, but I stuck with Fermat, and I thought that I make it clear for me once and for all.

 

[BruceW]

Am I understand it right, that still the new path with the new entrance point have to fit to Snell's law? (I'll try to draw a concrete example with a new image that shows the old and the new lower surface/path, but I'm not sure that I can do it easily, if you have an idea, please help me on this too, thank you)

Yes. Good question. As long as we are assuming that the 'ray approximation' works, then at a sharp boundary between two materials, Snell's law is always going to work. You can reason it like this: we can always 'zoom in' on a bit of the path where the light ray crosses the boundary, so in this small section of space, there is simply a flat boundary between two materials. Now, if we change the path in this section by a small amount, then the first-order change to total time must be zero. So Snell's law must be obeyed at every boundary, even though there may be other stuff happening in other parts of the experiment. (again, this all assumes the 'ray approximation').

Now I'll try to analyze your words and draw some (maybe wrong) conclusions (my imagination added). Please correct me...
1. In case of the simple rectangle (cuboid) the light path from A to B is clear, can be calculated easily by Snell's law and is the result of Fermat's principle, but it does not really matter, because that path is not much more real than the other possible paths...
2. ... because we can't be sure wether the light (package) is behaving as a wave or particle ...
3. ... but if there is a detector at the point B which determines the photon itself and its arriving angle, then we can calculate back and can determine the entering point into the surface ...
4. ... which is also not relevant, only a imaginary property, since the light itself interacted only in the point B, so nothing is changed at the entering point.
5. If we change the lower surface, the arriving angle may change and also the entering point -- that is not really matters, since the entering point is not a physical property. The light calculates and chooses, "knows" this imaginary path before it actually goes on it.
6. If we are thinking about the light as a particle, the light propagates not step by step, and decides its new angle when it hits the actual surface, but its path determined before its movement.

Am I too dreamy?

I don't know if you are dreamy, but I am not sure what you mean in several places. Also, you seem to dive right into quantum concepts, which you need to be a bit careful with, because the 'ray approximation' and the Feynman 'sum over paths' are quite analogous, but they are not exactly the same thing. The Feynman 'sum over paths' is an exact quantum calculation, and the 'ray approximation' is an approximate classical calculation.
1. Uh... I'm guessing you are talking about the quantum idea of how the photon's path is a superposition of all possible paths? In this case, there is only one path which leads to constructive interference, which is the 'classical' path corresponding to the 'ray approximation'.
5. Ah, not sure what you mean here. But the entrance point of the ray of light is a physical thing in this case. We are talking about the 'ray approximation', in other words, the path which corresponds to constructive interference. If we were talking about the actual path of a photon, then yes, this is not a physical thing, because the state vector is made up of a superposition of paths. But we are not talking about the path of a photon, we are talking about the path corresponding to a constructive interference.

edit: well, the entrance point is not really a physical thing.. I really meant that it can make sense using only classical physics. In other words, it is not something that we have to resort to quantum mechanics to explain.

 

[BruceW]

My question is:
Is there a shape available that by changing the lower surface of it, will change the entrance point of light ray?

This suggestion would seem to violate causality except, perhaps, in the case of a resonant cavity (optical or RF).

No, I think it is OK. For example, if we had a bunch of mirrors below the lower surface, and only the final mirror (right next to the person's eye), is turned to the wrong angle, then if we just turn that final mirror the correct way, then light will quickly enter that person's eye, only having to go that very short distance between the final mirror and the person's eye. The light that goes into the person's eye has already completed most of its journey by the time we turn that final mirror to the correct angle.

edit: also, is it bad form to use double quotes?

 

[tiny-tim]

… is it bad form to use double quotes?

there's no quota on quotiness

(but it is bad form for a quote to quote itself)

 

[zrek]

As long as we are assuming that the 'ray approximation' works, then at a sharp boundary between two materials, Snell's law is always going to work.

Good, if the Snell's law is working, then we can use it to create a concrete example for the question above. It seems to be not so difficult, almost already solved on the fig 22. I created a new example, see the fig. 24.

If the glass is not cut, the path#1 is the calculated. But if we cut it as the figure shows, the light will still go from A to B, the surface is untouched, still the entrance point will be changed, see the path#2. The point A not necessary to be in the infinity, the cut angle must be the appropriate.
Is this example correct?

... the 'ray approximation' and the Feynman 'sum over paths' are quite analogous, but they are not exactly the same thing. The Feynman 'sum over paths' is an exact quantum calculation, and the 'ray approximation' is an approximate classical calculation.

Thanks for mentioning this, I think I need to look for a reading about this. Maybe this is explained in the book "Feynman lectures on physics", I'll borrow it somehow.

 

[zrek]

tiny-tim, sophiecentaur and BruceW, thanks the help all of you, you gave me the direction where I have to increase my knowledge, now I know that I expected too much from Fermat. I still think that some of my questions are interesting, but this is not matter because the real problem and the solution is lying elsewhere, I need more examination of the topic. Thanks again, have a nice day!

 

[sophiecentaur]

It's been a good conversation. Good luck with taking this further.