Fermat's principle spec case

In summary, Fermat's principle states that the path of light is determined by the "path of least time principle" which means that the light ray takes the local minimum path. This principle is also supported by quantum physics which says that light takes all possible paths and only the minimum remains. When examining specific cases, such as cutting the object in different lengths, the principle still holds and the light will always choose the shortest path in time. This contrasts with the explanation based on substance properties, such as the refractive index, which may result in a different path for the light. However, the "least time" principle is believed to be the correct explanation.
  • #1
zrek
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Fermat's principle is the "path of least time principle" or we can say that "he path a light ray takes is a local minimum". Quantum physics also says that the light take (test) all possible paths and only the minimal remains.
Is it working in special cases like this too?
Let's examine the fig. 1-3 below.

fermat_spec.png


Fig1. is the normal case, the light goes from A to B and finds the "least time" path through the object.

Question 1:
What if we cut the object (Fig. 2), resulting a possible better path with less time? Will the light go unchanged, and arrives to the B as the dotted line shows, or will it find the less-time-path, even if this means that it must go through the object in a different angle?

Question 2:
What happens if we cut the object even shorter (Fig. 3), crossing the path of the ray goes normally through the object? Let's assume that in this case the "least time" path is if the light does not enter the object, but goes on the surface. Is this possible, and the path will be the orange one? Or in this case the light will do something else, like the reflecion inside, and will reach the B' ?

My question in general:
Is the "least time between two points" principle is an always working principle (rule 1), or the light is "not so smart" and can decide only locally on the entering points and can't "think further" (rule 2).
If the "rule 2" is the real one, why QP says that the light take all the possible paths?

Thank you!
 
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  • #2
welcome to pf!

hi zrek! welcome to pf! :smile:
zrek said:
What happens if we cut the object even shorter (Fig. 3), crossing the path of the ray goes normally through the object? Let's assume that in this case the "least time" path is if the light does not enter the object, but goes on the surface.

no, it will always be quicker for the light to go through the glass, very near the corner …

check it and see! :wink:
 
  • #3
Ok, tiny-tim, I see that on he corner it will enter into the glass, but on a small part of the path it will follow the surface, right? Of course only if this is the case of the principle is working "in general", which was my main question.
May I understand your answer that you are sure that the principle working "in general", so the light is "smart" and "knows" that the glass will end soon and will choose a different, "unusual" angle (fig 2)?

This behavior surely different to the explanations of the "general physics", that is about the substance property and the refractive index, this is why I think that this topic is about the Quantum Physics. This behavior explained by the QF and the calculation result of the all possible paths, am I right?

Unfortunately I have not found answers for my special cases on the Internet, I have found only explanations and examples of the cases where the "least time" principle resulted the same as the "refractive index" calculations. It would be nice if someone shows me a link where I can find an explanation on the net that shows an example where these two are different.

Please strenghten my tought that in case of the specially cut object (fig2 and fig3) the light will choose a different way and is able "to see before its path" and find always the least time path.

Thank you!
 
  • #4
zrek said:
Ok, tiny-tim, I see that on he corner it will enter into the glass, but on a small part of the path it will follow the surface, right?

nooo …

then it would be going through the air round two sides of a triangle, wouldn't it? :wink:

i think once you've convinced yourself that fermat works for this example, you'll be happy applying it generally :smile:
 
  • #5
I'm sorry but I don't get what do you mean on "round two sides of a triangle" :-(
I see that it is strange to follow the surface and then suddenly enter the object, but I don't know why, it seems to me that this is the least time path. The other possible explanation is that in this case the light will be reflected, or something else, but it can't reach the "B" (there is no possible path at all).

I'm not yet convinced myself, I still need some help.

...and what about the fig 2? Is the orange path possible? Is it possible that the light goes through in a different angle than in the fig 1?
 
  • #6
zrek said:
I'm sorry but I don't get what do you mean on "round two sides of a triangle" :-(

i mean if the light starts at A, goes through the air straight to B, then through the air along the edge of the glass from B to C, then to D E F etc,

then the light could go quicker if it went ACDEF etc, leaving out the detour via B :smile:
 
  • #7
I really appreciate your help, tiny-tim, but I'm afraid of I don't understand you clearly :confused:
I'm not native english, maybe I don't get the information behind the sarcasm and I'm not always understand if someone is kidding me. I think I understand only the straight answers...

So once again I'd like to try to explain what is my problem and where I need some help:

There are 2 kinds of interpretation of the path of the light:
1) Based on the substance property, the refractive index: if the light enters to a material with different refractive index, this definies clearly how it will behave, which angle will be the direction of the light.
2) Based on the QF and the Fermat's principle: from A to B the path will be always the shortest in time.

I hardly believe that this two explanation results the same in every case. I do believe however the second one, the "least time" is the correct one. Now I'd like to find somewhere an example which shows the difference between this two approach of the problem.

Let's see this figure:
fermat_spec2.png


Between A and B there are several objects with different refraction indexes. As you can see that this is some kind of a maze.
Let's shot a light ray from the point 'A' to a random direction and examine where it will go...
Let's try to calculate the path by the simple way: when we reach an object, calculate the new angle by the refraction index, and step forward. The orange line shows this path.

Now let's calculate the all of the possible paths and let's select the "least time" one. The dotted line represents this solution.

I think that if the task is to find the least time path throught this labyrinth, we can't succeed easily, only by following the refraction index rule. We need much more work to find the way with the shortest time.

Am I right or wrong?
 
  • #8
The paths you drew through that coloured maze were not calculated, were they? You just made them up, I think.
But, if you consider a lens, there are many different paths through, from object to (focussed) image. They are all the same (shortest possible) length so there may well be multiple possible paths through that maze - as long as they all take the same time. Nothing needs to be 'violated'.
 
  • #9
sophiecentaur said:
The paths you drew through that coloured maze were not calculated, were they? You just made them up, I think.
Of course the image is only a demo, not based on calculation.
"Nothing needs to be 'violated'"
I too think so, but I feel that I misunderstood something.
If it is working as I explained, I'm sure that you can't find an optimal way through a maze just by analyzing the next step. This will not work.

But now I think that maybe I know what is this all about. The principle is not about finding the least time path between the A and B, but finding the least time path which fullfills the rule of the step-by step way given by the refractive index.
I mean that the following selection is explained by Fermat:
From the point "A" we shoot the light to every direction (360 degrees) and calculate the (step-by-stem, refractive index based) path for every possible case that finally leads to "B". Finally we will select only the least time path from these: this will be that path the light will go on. By this, we can determine that which angle the "B" will receive the light from.

Am I right, or just getting more confused ? :uhh:
 
  • #10
You are describing the common 'ray tracing' method that is commonly used in optics designs. And yes you can determine which angle(s plural)) the light will arrive from.
 
  • #11
I slept on it, and still I don't feel that I got answers to my questions. Not easy to tell what is my problem, but in general I don't get what the Fermat principle is trying to explain.
I found lots of pages about explaining Snell's law by it, and seems that simply the "least time" principle is enough to calculate the path, no other criterion is necessary.
http://cnx.org/content/m12895/latest/
Sometimes it seems that they explain some behavior about the the wave-particle duality by it, like here:
http://en.wikipedia.org/wiki/File:Snells_law_wavefronts.gif
Also clear that it is working even with lasers:
http://simple.wikipedia.org/wiki/Snell's_law

Seems also that the principle should work in difficult cases too, where there are many entering points or inhomogene substance, in this case it is calculated by the calculus of variations:
http://en.wikipedia.org/wiki/Calculus_of_variations#Fermat.27s_principle

Not easy to explain what I still don't understand, maybe tomorrow I'll try to show you in a longer post.
 
  • #12
zrek said:
I found lots of pages about explaining Snell's law by it, and seems that simply the "least time" principle is enough to calculate the path, no other criterion is necessary.
http://cnx.org/content/m12895/latest/

Note that for the first example given in that link, the path of least time is the one that goes directly from A to B and does not reflect at all. Yet both rays (AB and ACB) are legitimate paths for the light to take, even though AB clearly takes less time that ACB. Feynman explains this in the Feynman lectures (vol I) - I recommend you read it as he is much clearer than I could ever be. The upshot is that the paths that the light will take are those for which small changes in the path produces "second order" (very small - smaller than "first order") changes in the propagation time; in other words, there are many nearby paths with almost the same propagation time.

Also, whenever you are solving these kinds of ray tracing problems Snell's law must hold. If you are using some technique to calculate paths that violates Snell's law then you know you are doing something wrong ...

jason
 
  • #13
jasonRF said:
Note that for the first example given in that link, the path of least time is the one that goes directly from A to B and does not reflect at all.
Yes, in other examples they used to draw a barrier between A and B to prevent the direct path.
jasonRF said:
The upshot is that the paths that the light will take are those for which small changes in the path produces "second order" (very small - smaller than "first order") changes in the propagation time; in other words, there are many nearby paths with almost the same propagation time.
If I understand it well, this is the point when they used to say that the "interference" kills the other, non-optimal paths, right?
jasonRF said:
Also, whenever you are solving these kinds of ray tracing problems Snell's law must hold. If you are using some technique to calculate paths that violates Snell's law then you know you are doing something wrong ...

I'd be happy to agree with you, but I think I missed something. There must be an additional criteria that prevents the non-Snell-like behavior. If I use only the "least time" principle for the calculation, sometimes other path would be optimal compared to the Snell's law says.
For example in the post #1 the fig2. orange path why is not good?
I also miss the "critical angle" explained by the Fermat's principle.

(But these are not my main problems, I'll explain it later on)
I'd like to see this principle clearly, this is why I asking your opinion, thank you!
 
  • #14
I think you have the wrong idea about what Fermat is actually saying. If you start off with a 'ray' in a certain direction, Fermat will tell you where it will end up. Fermat doesn't imply that there's only one path between A and B. Where there is partial reflection, Fermat works just as well. Can you think of any reason why it should not apply to Total Internal Reflection?
 
  • #15
sophiecentaur said:
I think you have the wrong idea about what Fermat is actually saying.
I'm pretty sure that I miss something about the Fermat's principle, but I don't know what. :frown: :smile:
sophiecentaur said:
Fermat doesn't imply that there's only one path between A and B.
It is clear for me that there may be several proper paths between A and B but they must be finally all optimal, with the same minimal time. (am I right?)
sophiecentaur said:
If you start off with a 'ray' in a certain direction, Fermat will tell you where it will end up.
I assume that it is working even if we are thinking of the light as wave, and Fermat finally determines if it is arrived to B, how it is appear as "ray", particle.
http://en.wikipedia.org/wiki/File:Snells_law_wavefronts.gif

I think I understand this part.
--------
I still don't get why the orange path between A and B is not allowed (post #1 fig.2)
As I see the orange path is time-optimal, the normal path (Snell's law) takes more time in that case.
Am I right?
 
  • #16
Of course there can be many paths. Just think of how a paraboloid works. The focus us where all path lengths are equal.

Have you proved that the orange path is shortest time? But the shortest time (that's minimum phase delay) from the point of incidence is given by Smells law. Fermat will operate over each infinitessimal step and give Snell. There can always be shorter ways through that involve extra paths that the light can't know about - but you can't see round corners; you have to use a mirror. ;-)
 
  • #17
I just thought. Is it possible that you are assuming that Fermat works 'globally'? In fact, it works in small steps and it just has the effect of looking as thought it worked it all out on a large scale. In the example of your figure, as always, you need to satisfy phase coherence as well as Fermat. There is no way that the direction could suddenly take a sharp left turn, any more than it could go into the glass at any other than the Snell predicted direction or be reflected asymetrically.
I think you are assuming that Fermat is the prime idea here but imo, it is only a principle which helps in making predictions. Remember, it is pretty ancient!
 
  • #18
zrek said:
Yes, in other examples they used to draw a barrier between A and B to prevent the direct path.

ah …

there is a path that goes via the endpoint, C, of the barrier …

light is diffracted at C, and some of it does get to B ! :wink:

(remember, fermat's principle doesn't say that all light from A gets to B)
 
  • #19
sophiecentaur said:
Is it possible that you are assuming that Fermat works 'globally'?
Yes, in several examples I saw that they suggested that (the modern version of) Fermat is (1) general and (2) working in every case, and not only in case of 2 homogene substances. If the path is going through difficult material combination, they use the calculus of variations ( http://en.wikipedia.org/wiki/Calculus_of_variations#Fermat.27s_principle ) to determine the path.
sophiecentaur said:
There is no way that the direction could suddenly take a sharp left turn, any more than it could go into the glass at any other than the Snell predicted direction or be reflected asymetrically.
I agree, but my problem is that they explain the Snell with Fermat but I have not found that where the Fermat says that this is impossible. I miss an an additional restriction that prevents this behavior. Is it come from the phase coherence that you mentioned? But how?
sophiecentaur said:
I think you are assuming that Fermat is the prime idea here but imo, it is only a principle which helps in making predictions. Remember, it is pretty ancient!
I don't agree with you in this point, since as I read, the Fermat is the deeper rule and the base of the others (this principle drew Feynmann's attention to use it in QP too) Snell's law even more ancient.
"There can always be shorter ways through that involve extra paths that the light can't know about"
This is what I assume: the Fermat is all about that the light will always "knows" the shortest time path between any A and B (if there was a light traveled between A and B, then that path was surely the least time available of the all the possible)

I know that my knowledge is not perfect about this, and maybe soon I'll find what is missing.

The wiki says:
"...modern statement of the principle is that rays of light traverse the path of stationary optical length with respect to variations of the path.In other words, a ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse"

I think I know what the "variations of the path" means, but what they mean on "stationary optical length" and "arbitrarily nearby" exactly? It would be nice of you if you try to explain it to me (or show me a link for it), thank you.

(I'll open a new thread here, in the next post, but please try to answer this too, thank you! :smile:)
 
  • #20
Here is an example, shows that the Fermat is better than Snell.
https://www.physicsforums.com/attachments/58104
fermat_spec3.jpg

(The fig. 21 shows that I assumed that the orange path is not available, since in the point B0 the path is not "least time optimal". But I changed my mind, this would be not relevant, only the final result counts: in the point B we measured the light ray, so it must be optimal to that point and irrelevant if during the path there are "seemingly not optimal" points)

Let's see the fig. 22.
To make it simple, the A is far in the infinity.
If we are not counting with the strangely cut part, the path#1 is the correct one, fits to Snell's law.
But if you take a look at the path#2, you can see that it is the least time path (and still fits to Snell!).
Am I right that the ray of light will go in the path#2?
 
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  • #21
tiny-tim said:
(remember, fermat's principle doesn't say that all light from A gets to B)
OK, sure, Fermat says that if the light arrived to B then the path was the "less time" optimal. :smile:
 
  • #22
zrek said:
Here is an example, shows that the Fermat is better than Snell.

You seem to be treating this as some sort of a competition. :confused:

Tiny Tim has introduced the idea of diffraction and makes the point that Fermat doesn't insist that all the energy goes 'his way'. I think you should stop worrying about this and be confident enough to accept that a number of alternative treatments may or may not apparently yield exactly the same result when you pursue them to the bitter end.

If you really want to confuse yourself then sit and think why a ray of light just carries on in the same direction and why it doesn't just split up into all directions at all points on its path. Huygens principle will come to your aid and takes you into the realms of diffraction (which is the basis of all wave propagation, even when it doesn't appear to be).
 
  • #23
sophiecentaur said:
You seem to be treating this as some sort of a competition. :confused:
Please don't take this seriously. I think that it is normal, that some theorem are better than others, this is what physicists working on: to create new, better ones (maybe based on other ones). I only tried to show an example.

sophiecentaur said:
Tiny Tim has introduced the idea of diffraction and makes the point that Fermat doesn't insist that all the energy goes 'his way'.
But this is not that case. Fermat says clearly that if the ray goes from A to B then the least time path will be used, others will be not used (the other alternatives will be blanked by the interference or something like that).
This is normal and logical: you are in the point "B" and the ray will arrive into your eyes from a specific angle. Only in those cases, when there are more possible "least time paths" you will see the source paralelly from different angles. But this is not that case, the path#1 is clearly takes more time, so the path#2 will be the right one. Snell can't explain this, but Fermat can.
Am I right or wrong?
 
  • #24
Imo, you are 'wrong' to follow Fermat slavishly. To me, that's just not a helpful attitude. For a start, you cannot ignore the fact that diffraction occurs. Fermat is rule that sums up the main result of diffraction and that is all. Fermat is the result of the phase addition of the Huygens secondary wavelets (later, described in terms of the diffraction integrals) and that is where Snell comes from, too.
The only time that Fermat totally applies to a Snell's Law situation (or a simple reflection) is when the aperture is infinitely wide. You are just dismissing this in your phrase "blanked by the interference or something like that", which is not rigorous enough if you really want to get this straight. You cannot be so selective.
Every optical system is diffraction limited, in the end. Are you going to say that you can't analyse a lense using Fermat, just because there are some alternative paths through the system that seem not to behave that way?
You are falling into that terrible "Categorisation Trap", where madness and confusion lie.
 
  • #25
zrek said:
It is clear for me that there may be several proper paths between A and B but they must be finally all optimal, with the same minimal time. (am I right?)
Almost right. True, in a general problem, light will take several possible paths between A and B, and they must all be optimal in the sense that they 'extremise' the total time, but they don't necessarily have the same minimal time. This is related to the 'variation of path' concept that you were reading about on wiki. When the path is changed slightly, then to lowest order, the total time taken must not change. This is what I meant by 'extremise' the total time. (sorry extremise is not an actual verb, but I couldn't think of a good real verb to use).

Start with a simple example, if you have a candle in a room, and a mirror on the wall, then the light from the candle can reach you two ways: it goes in the straight line towards your eyes. And also it bounces off the mirror, then into your eyes (as long as the mirror has been positioned correctly so that the incident angle can be equal to the reflected angle). So here, the light has two different paths. Both of them 'extremise' the total time, but the straight line path takes less total time. The light takes both paths, so you see that what is important is 'extremising' the total time, not choosing the path of least time. (I think someone already said it, this is something Feynman was saying in one of his lectures). The phrase 'choose the path of least time' is much more intuitive, but what physicists are actually doing is making the functional derivative equal to zero. These two things are almost the same, but not quite.

Also, one more important thing, is that all of this relies on an approximation. And we can only use the idea of 'rays of light' as an approximation. Generally, the approximation only holds when the typical length of objects is much greater than the wavelength of the light.
 
  • #26
sophiecentaur said:
...that's just not a helpful attitude ... You cannot be so selective ... You are falling into that terrible "Categorisation Trap", where madness and confusion lie.

Please don't take it as my attitude, but only as my lack of knowledge. I don't want to make selection or fall into trap by my hopeless stupidity, it is only because I'm uninformed. I accept that I wrong in my conclusions, but I'd like to understand why. Please give me some time and take my examples as some kind of questions (I'm sorry if my wording is not appropriate, I never want to be impolite -- this is because of my lack of english) I appreciate your help, please try to stay with me, thank you :smile:

sophiecentaur said:
For a start, you cannot ignore the fact that diffraction occurs. Fermat is rule that sums up the main result of diffraction and that is all. Fermat is the result of the phase addition of the Huygens secondary wavelets (later, described in terms of the diffraction integrals) and that is where Snell comes from, too.

I don't know, I just found these explanations about the Fermat' principle in the docs, like this picture:
http://en.wikipedia.org/wiki/File:Snells_law_wavefronts.gif
...also I found somewhere that Fermat is about to find the "trajectory" of the ray.
And I found that calculus of variations and this for example:
http://en.wikipedia.org/wiki/Calculus_of_variations#Fermat.27s_principle
... etc... .. and I don't found the limits of the principle. This is why I assumed that it is working in case of larger scales, and not only about the diffraction. (now I'll try to answer BruceW and then I'll ask again, please try to stay with me, thank you)
 
  • #27
BruceW said:
...The phrase 'choose the path of least time' is much more intuitive, but what physicists are actually doing is making the functional derivative equal to zero. These two things are almost the same, but not quite...

Ok, thank you BruceW, I think I'm getting closer.
I still have questions (in the next posts), please try to answer, thank you!
 
  • #28
I understand now (still not completely clear for me however), that in the fig.21 the orange path is not availabe, and in the fig. 22 the ray will choose both the path#1 and #2 (in the point B we will see the point A in both directions)
Now I'll try to not assume something, only asking.

My question is:
Is there a shape available that by changing the lower surface of it, will change the entrance point of light ray? (And the light will still go from A to B) (Any kind of complexity is allowed, even mirrors or meta-materials, the only restriction is that the first, entrance surface must be untouched and flat)
See fig. 23.
fermat_spec4_b.png
 
  • #29
good question. The best way to think about it is by thinking about the functional derivative, since this is the thing that must equal zero for any path [itex]y_{(x)}[/itex] that the light is allowed to take. The functional derivative is defined as this:
[tex]\frac{T[y_{(x)}+ \epsilon \phi_{(x)}] - T[y_{(x)}]}{\epsilon} [/tex]
In the limit as epsilon goes to zero, and T is the total time (a functional), and phi is an arbitrary function of x. So this is the lowest order change to the total time.

So how does this answer your question? Well, in the original case where the lower surface is flat, we know the single answer that gives an extreme path y(x). And if we change the lower surface? We still have the possibility of the entrance point of the light ray being in the same place. After, all, up until the lower surface, it definitely gives an extreme value for the total time up until then. So as long as we can find a continuation of this path that still is extreme, then the original entrance point of the light ray is still allowed.

But now since we have changed the lower surface, there is also the possibility of other extreme paths, since the geometry of the problem is not as simple anymore. In the original case, we knew there was a unique answer because the geometry was simple, but now it is not so simple, we can't say there is a single path that the light will take.

Also, we might change the lower surface such that the original path cannot be continued on an extreme path. (For example, we might even put an opaque object on the lower surface where the original path would have been, so the original path is not even allowed, let alone an extreme path). 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.

One last thing, this post was a bit long because your question "Is there a shape available that by changing the lower surface of it, will change the entrance point of light ray?" Doesn't really have a direct answer, because 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.
 
  • #30
zrek said:
Please don't take it as my attitude, but only as my lack of knowledge. I don't want to make selection or fall into trap by my hopeless stupidity, it is only because I'm uninformed. I accept that I wrong in my conclusions, but I'd like to understand why. Please give me some time and take my examples as some kind of questions (I'm sorry if my wording is not appropriate, I never want to be impolite -- this is because of my lack of english) I appreciate your help, please try to stay with me, thank you :smile:



I don't know, I just found these explanations about the Fermat' principle in the docs, like this picture:
http://en.wikipedia.org/wiki/File:Snells_law_wavefronts.gif
...also I found somewhere that Fermat is about to find the "trajectory" of the ray.
And I found that calculus of variations and this for example:
http://en.wikipedia.org/wiki/Calculus_of_variations#Fermat.27s_principle
... etc... .. and I don't found the limits of the principle. This is why I assumed that it is working in case of larger scales, and not only about the diffraction. (now I'll try to answer BruceW and then I'll ask again, please try to stay with me, thank you)

Sorry. i didn't mean "attitude" in the sense of lary teenager attitude. I meant it in the sense of 'viewpoint' or 'approach'.

That figure (Snells Law) of the waves changing wavelength and direction as they cross a boundary is good and it can be put down entirely to the effect of diffraction / Huygen's construction as the wave speed changes - It also seems to account well enough, in a geometric way, for Fermat's principle of shortest time. But it only works where there is a big enough width of aperture for diffraction to produce a well defined ray. Remember, a 'ray' is only a simplification of the fuller description of propagation which diffraction will give you.

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 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.
 
  • #31
Wow, thank you, now I'll try to analyze your answer :smile:

BruceW said:
... 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)

BruceW said:
... 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?
 
  • #32
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).
 
  • #33
sophiecentaur said:
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.

sophiecentaur said:
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.
 
  • #34
zrek said:
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').

zrek said:
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.
 
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  • #35
sophiecentaur said:
zrek said:
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?
 
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<h2>What is Fermat's principle spec case?</h2><p>Fermat's principle spec case is a principle in optics that states that light will travel between two points along the path that takes the least time.</p><h2>Who discovered Fermat's principle spec case?</h2><p>Fermat's principle spec case was first proposed by French mathematician Pierre de Fermat in the 17th century.</p><h2>What is the significance of Fermat's principle spec case?</h2><p>Fermat's principle spec case is important in understanding the behavior of light and its interactions with different materials. It also serves as the basis for many optical theories and applications.</p><h2>How is Fermat's principle spec case applied in real life?</h2><p>Fermat's principle spec case is applied in various fields such as optics, engineering, and physics. It is used in designing optical systems, predicting the behavior of light in different mediums, and developing technologies such as fiber optics.</p><h2>Are there any limitations to Fermat's principle spec case?</h2><p>While Fermat's principle spec case is a useful tool in understanding light, it has its limitations. It assumes that light travels in a straight line and does not account for phenomena such as diffraction and interference. It also does not take into account the effects of changing refractive indices along the light's path.</p>

What is Fermat's principle spec case?

Fermat's principle spec case is a principle in optics that states that light will travel between two points along the path that takes the least time.

Who discovered Fermat's principle spec case?

Fermat's principle spec case was first proposed by French mathematician Pierre de Fermat in the 17th century.

What is the significance of Fermat's principle spec case?

Fermat's principle spec case is important in understanding the behavior of light and its interactions with different materials. It also serves as the basis for many optical theories and applications.

How is Fermat's principle spec case applied in real life?

Fermat's principle spec case is applied in various fields such as optics, engineering, and physics. It is used in designing optical systems, predicting the behavior of light in different mediums, and developing technologies such as fiber optics.

Are there any limitations to Fermat's principle spec case?

While Fermat's principle spec case is a useful tool in understanding light, it has its limitations. It assumes that light travels in a straight line and does not account for phenomena such as diffraction and interference. It also does not take into account the effects of changing refractive indices along the light's path.

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