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Fermat's principle, also known as the principle of least time, can be used to derive Snell's law, so does Huygens' principle. The former used a single ray of light from point A in medium 1 to point B in medium 2 while the latter used wavefront to derive. I can see that wiki said that Huygens' construction and Fermat's principle are geometrically equivalent and wavefronts are always perpendicular to the rays of light. In Fig. 1.7.5 of Huygens', it assumes that time traveled in BB'/v1 = AA'/v2 or directly we can assume that BB'=λ1 and AA'=λ2, so that the refracted wavefronts are in sync (same phase, no delay between BB' and AA'). But with all that, have we proved that Huygens' wavefront takes the least time?

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  • $\begingroup$ Actually, a lot of assumptions go into such things. People of the time definitely could not do direct experiments to show definitively which way it is. For example, Newton thought, via his corpuscular theory, that instead of slowing down in denser media, it is speeding up. There is simply no way at the time for us to know if the ratio of longitudinal velocity v.s. transverse velocities are increasing or decreasing. i.e. all you can do is prove within an internally consistent framework that the premises imply such an interpretation is correct. $\endgroup$ Aug 18, 2023 at 3:15
  • $\begingroup$ What is available, I think is the following two: 1) Demonstration that Fermat's stationary time, if granted, implies Snell's law, 2) Demonsration that Huygens' wavefront hypothesis, if granted, implies Snell's law. From that it is inferred that the two must be equivalent. Other than that: that which is accessible to measurement is expressed by Snell's law. That which is described by some supposition is not accessible to direct measurement. Demonstrations always work from supposition to physical implication. That is why, I think, people take the inference as sufficient. $\endgroup$
    – Cleonis
    Aug 18, 2023 at 10:04

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As pointed out in the comment by contributor naturrallyinconsistent: until the experiments by Fizeau and Foucault it wasn't certain in which medium the speed of ligth is fastest.

While it is clearly more plausible that in air light propagates faster than in glass, the possibility of the reverse of that could not be excluded.


Ratio of speeds

The principle only allows you to infer that there is a ratio of speeds. If the light is faster in glass then instead of Fermat's least time you get Fermat's most time.

I recommend that you verify that for yourself; it's a worthwhile exercise. Assume the ratio of index of refraction is a ratio where in glass light propagates faster, and then find the path of most time. That will reproduce Snell's law.

So in retrospect we see that the concept of Fermat time is a concept of Fermat stationary time; Stationary time covers both cases.

Stationary time means:
Introduce variation of the path and then take the derivative of the start-to-end time with respect to variation of the path. The actual path of the light coincides with the point in variation space such that that derivative is zero.


Stationary time

There is another angle that shows that Fermat time must be stated in terms of stationary time.

The following three diagrams are screenshots from an interactive diagram that is on my website. (A link to my website is available on my stackexchange profile page.)

enter image description here
Diagram 1.
Ellipse, reflection from one focus to the other focus

In Diagram 1. the ellipse is drawn such that the two points at (-1, 0) and (1, 0) are the two focussus of that ellipse. For all paths from one focus to the other the time-from-start-to-end is the same.

The next two diagrams are for the case that the reflecting surface is more concave than the ellips of diagram 1.

enter image description here
Diagram 2.
The reflecting surface is more concave than the ellips of Diagram 1

enter image description here
Diagram 3.
The time from start-to-end is shorter than in Diagram 2.


Diagrams 2. and 3. together illustrate that when the reflecting surface is more concave than an ellipse then the path that the light takes is the path of most time.

We see that the case of an ellipse, with light propagating from focus to focus, is a critical case. When the reflecting surface is closer to being flat the path that the light takes is the path of least time. However, when the reflecting surface is more concave than the ellipse then the path that the light takes is the path of most time. The ellipse is the critical case where the inversion from least time to most time occurs.

In order to cover all cases: assume that it is intrinsically about stationary time.

I copy the earlier paragraph:
Introduce variation of the path and then take the derivative of the start-to-end time with respect to variation of the path. The actual path of the light coincides with the point in variation space such that that derivative is zero.



Fresnel lens refraction

Finally, I want to point out a case that I encountered in the book 'Optics', by Frank L. Pedrotti, Leno S. Pedrotti, Leno M. Pedrotti.

enter image description here
Diagram 4
Schematic representation of refraction by a Fresnel lens

The most widely known application of Fresnel lenses is in the beam-forming optics of lighthouses. However, there are also imaging Fresnel lenses. That is: a lens that is designed to take light from one focal plane to another.

The diagram illustrates that from the start point to the end point three paths are available. Clearly each path will have a different time from-start-to-end. So there is a path of least time, a path of most time, and a path of intermediate time. The light takes all three paths. That shows that in order to account for the paths that the light takes an additional hypothesis is required.

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