Does light remember its past to follow the least time path?

I read about the principle of least time. However, I think, for light to follow this principle, light would have to remember its past path before deciding where to go for its future path. Why I think that:

Suppose I shine a torch in the air medium, directed toward the water medium. I shine it at point A. It hits the surface of the water at point B. Here's light deciding where to go next from B while following the principle of least time:

For simplicity, let's consider two choices for the next point to visit (two nearby points to B):

1. Point C- Light thinks: "If I go to C, I'd have traveled the path A-B-C overall. But there was a shorter path to C : (say) A-D-C. So I should have traveled through A-D-C all along if I wanted to go to C. Since, I journeyed through AB instead of AD, the point C is out of option now"

So light rejects C as its future point.

1. Point E- Light thinks- "If I got to E from here, I'd have traveled along A-B-E overall. Since, there's no shorter path from A to E, so I can go to E"

Obviously, light can't think but I had to add that to make my argument clear. So all this requires light remembering that it traveled through AB before deciding where to go next from B. Is this true that light remembers its past path while deciding its future path?

To be clear, light can neither think nor decide, but do the physical laws governing propagation of light take light's past into account?

EDIT- That link does not answer my question, because my question is not limited to refraction. We can make the same argument as in my post by picking any point B in the middle of light's journey. The same argument can also be made for 'principle of least action' in mechanics in general.

My question is "Do all these principles of least things rely on remembering the past path to decide the future path?"

Also, is the argument in my post clear? I tried to explain it best by a scenario where light is thinking where to go next.

• Does this answer your question? Why does light bend? Commented Sep 2, 2020 at 17:14

3 Answers

It seems to me that Fermat's least time is applicable only for the cases of mirror reflection and refraction of a wave front from one medium to another.

It seems to me Fermat's least time doesn't cover the optics of a diffraction grating.

For simplicity let's take the case of a double slit setup. We observe that on the screen an interfererence pattern arises. The interference pattern consists of multiple parallel bands of alternating bright and dark.

In the case of a two slit setup the interference pattern is attributed to the fact that for each position on the screen the wavefront has two paths to reach it. The very reason that the interference pattern arises is that the two paths have different length.

As we know: for positions on the screen where there is a bright band the path length difference is a multiple of the wavelength of the light. (Conversely, path length difference of half a wavelength: destructive interference.)

Generalizing to diffraction gratings: for the interference pattern to arise the wavefront must travel multiple paths, which means not only the path of least time but paths of longer time too.

In that sense there is no such thing as a 'principle of least time'.

Fermat's least time is restricted to the cases of mirror reflection and refraction of a wave front from one medium to another.

Wavefront propagation

Then again, in another sense we can meaningfully talk about the past leaving some imprint.

When describing propagation of a wavefront you have to take into account that light ariving at a particular location on a screen can arrive there from multiple different directions. That spatial aspect of wave propagation does make that the set of obstacles that the wavefront has negotiated can completely determine how the light illuminates the screen.

Anyway, whether the case is wave propagation or motion of a point mass, the phenomenon is fully described with a differential equation. The nature of a differential equation is that it describes motion that proceeds from instant to instant, down to infinitisimally short instants.

You asked in a comment to another answer about Hamilton's stationary action (often referred to as 'principle of least action'). Hamilton's stationary action does not change the fact that the phenomenon is fully described with a differential equation.

Light takes the shortest path between two points, not three.

Consider this: when you see something, light originating at a light source is bouncing off the object and into your eyes. Obviously, the shorter path is directly from the source into your eyes, and if there was some kind of global shortest-path-finding algorithm that was applied to light, you'd never have any light traveling to the object, and you'd never be able to see anything except a very bright light from the original source.

• I've considered that in the post. The reason light rejected the path A-B-C was because A-D-C was the shorter path from A to C. So I'm only considering the shortest path between two points (A and C). Again, the reason light accepted point E as the way to go from B because A-B-E was the shortest path from A to E Commented Sep 2, 2020 at 15:00
• Go through the Wikipedia article for Fermat's principle, especially the second paragraph and see if it helps answer your question?
– pho
Commented Sep 2, 2020 at 15:07
• @RyderRude, and/or the section on Fermat's Principle in this lecture by Feynman
– pho
Commented Sep 2, 2020 at 15:31

The least time path is just for us to understand how light goes. The path light takes is not because it is the shortest path. Just that the shortest time path and the path light takes are same, since they evolve from the same root cause. Why light takes that path can be explained by use of wave optics more clearly than using ray optics.

• Can this also be said about the 'principle of least action' in mechanics? Are all these 'principles of least things' not the cause of physical phenomena but merely consequences (as these principles seem to rely on past)? Commented Sep 3, 2020 at 2:41