Post on 2024/7/31

By Richard Quan

In Arrival, the breakthrough in scientific communication between humans and seven-limbed barrels was brought about by Fermat's Principle .

Here is the definition of Fermat's principle that we can find on Wikipedia: Fermat's principle, also known as the principle of least time, is the link between Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics . Fermat's principle states that the path taken by a Fermat's principle states that the path taken by a ray Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time.

However, here are some more creative thoughts

Whether it is fluctuation theory or the theory of photons, light does not actually pass along a certain path from the source to the illuminated area. So what is the process by which point A illuminates point B?

According to (the) Huygens principle Huygens' principle we can think of it as an electromagnetic wave emitted from point A to all of space, while the electromagnetic field at point B is influenced by the electromagnetic field everywhere in space. Or the light field elsewhere in space emitted by point A becomes no different from point A's secondary wave sources and point B is affected by these secondary wave sources.

If light is considered to be composed of a large number of photons, then he would not have a particular path. Photons are not particles as we normally see them, they are quanta, and according to Feynman's path integral theory photons go from A to B by summing up all possible paths across space, even for a single photon, much like the fluctuating point of view (though this is inevitable when it comes to that). The quantum point of view will not be repeated later. It is mainly explained from the fluctuation point of view.

Since there is no particular path for light to travel from A to B, what is meant by light inside Fermat's principle?

Indeed, even without the Fermat's principle , we would still perceive light as coming from a line. Suppose that A light source B is illuminated by light source A. If we use a large baffle to separate A and B, B will not receive light from A. If we make a small hole in the baffle, B will not receive light from A. If we make a hole in the baffle, B will not receive light from A. At this point we make a small hole in the baffle. In general, after opening the hole, there will be two cases, one small hole in the M point, B did not respond; the other is a lesser case, the small hole in the N point, B is illuminated. In this case we consider that the light from A to B does not pass through point M but through point N. Assuming that there are many such baffles placed parallel to each other in space, the light is the set of holes made by these baffles that illuminate point B. Conversely, if there is no baffle, if there is a small opaque piece blocking only point N, then point B is not illuminated, whereas if only point M is blocked, point B is not affected. That means that the light field at point N cannot affect B,the light field on the light ray can affect B.

Additionally, according to Fermat's principle, the location of the small hole must make the light path from A to B the shortest run (longest or stable, without delving into the details of this for now). Or the path that can affect the light field to B is the one that makes the light path shortest. In this way we have transformed the problem of the light path into a problem of small holes.

So how does Fermat work? Or why is the location of the small hole that makes point B illuminated just on the path that makes the light travel the shortest.

Even if the hole is small, it has a size, and if it is just an infinitesimal hole, even if this infinitesimal hole is on the abstract light, point B will not light up. all the electromagnetic waves to which point B is exposed all come from the small hole. Now let's take the small hole N that is on the light ray and the small hole M that is not on the light ray, and see how the electromagnetic waves from these two places affect the light field at point B.

In connection with Fig. (1), we look at the position of the baffle in relation to the light path.

When a field emitted from point A reaches point B, the phase of its vibration is related to the light travel, or to the time it takes for the light to fly over.

Compared to the N point, which has the shortest optical range, the relatively unspecific M point has the characteristic that the optical range at each of these places has a continuous process of increasing or decreasing with respect to the small hole change. In contrast, the variation of the light range at each of the N points is very small. And that's the key to the light field at point N being able to affect point B. That is, his slope is the key to the problem.

You can see that the slope of the neighborhood of 0, the change of its light range is very slow, then the phase of the light vibration transmitted to the B place is also basically different, these light vibration between each other to strengthen, so that the B point of the overall vibration of the electromagnetic wave. And there is a slope near point M, he has a different light range everywhere, even if the small hole is very small, but because of the wavelength of light is smaller, so this light range on the light wave vibration has a large phase shift. Thus the phase of light vibration from point M is a continuous process of change.

Successive changes in phase make it so that for point B some make him vibrate upward and some make him vibrate downward, and eventually the total effect disappears.

There are more appropriate mathematical tools (vector arrows) to analyze more closely about the effects caused by phase changes, but that's probably what the above means.

By this point the facts are clear, making the light in the neighborhood of the position where the slope of the light path is 0 have the greatest effect on the receiving point, and therefore it is considered the position through which the light passes. The reason that the slope of 0 has the greatest effect is that all the light in its vicinity has about the same phase to the receiving point, so that the vibrations are not weakened. This is how Fermat's principle works. And the light near places with a certain light-travel-path slope is always weakening each other, making the field at these last places have no effect on the receiving point. (Actually, it is not completely absent, there are places where a little bit of strengthened vibration can be left by integration, so that under certain conditions we open the hole not at point N but at some special location that still illuminates point B. In fluctuation optics point B can be seen as this special location of the secondary diffraction spot (which does not conform to Fermat's principle).