How does Fermat's principle explain the shape of a lens in Feynman's derivation?

[neworder1]

I have a problem in understanding Feynman's derivation of a lens' shape using Fermat's principle. Feynman writes that we have to choose such a surface of the lens that all optical ways from the source S to the focal point F will have the same length (so all will be taken by the traveling light). However, how do we know that the light chooses its path so as to minimize the distance from S to F and not to another point G, or any other point? When Feynman derives Snell's law while source and target points are fized, it's OK, bu here tHe light doesn't "know" that there is a focal point, it could go anywhere (i.e. destination point is not fixed in any way). So I don't see how we can use Fermat principle here.

 

[mgb_phys]

The same way water 'knows' to flow down hill or electricty 'knows' to flow through the lowest resistance.
Fermat's principle is either the greatest idea in physics - in a way it pre-invented both relativity and quantum mechanics, or it's just a convenient short cut to get the right answers.
It depends on your philosphical veiw of the nature of physics really.

 

[astrokreng]

Fermat's principle states that light will take the path that minimizes the travel time between two points. In the case of a lens, the source point S and the focal point F are fixed points, so the light will naturally choose the path that minimizes the distance between these two points. This is because the light travels at a constant speed, so minimizing the travel time also minimizes the distance traveled.

In Feynman's derivation, he is showing that for a lens to focus light at a single point (the focal point), the surface of the lens must be shaped in a specific way. This shape is chosen in such a way that all possible paths from the source to the focal point have the same length, ensuring that the light will be focused at the focal point. This is a consequence of Fermat's principle, as the shortest path between two points is a straight line.

It is important to note that Fermat's principle does not require the light to "know" anything about the destination point. Rather, it is a fundamental principle that describes the behavior of light in a given medium. In the case of a lens, the shape of the lens is determined by the requirement that all possible paths from the source to the focal point have the same length, thus ensuring that the light will be focused at the focal point.