# Why is it necessary for all the light rays to reach the focus of a spherical mirror at the same time?

I've heard people say this stems out of fermat's principle of least time, if so, how?

Note:I don't have great knowledge of ray optics, please explain as you would to a person just been introduced to ray optics

• 1) The actual shape that reflects achieves this is a parabola/parabaloid. One can locally approximate a parabola with a circle that best fits the minimum. 2) This result only applies to rays parallel with the axis of the parabola. For rays close to the axis this doesn't matter as much since as above it can be modelled as spherical so the axis isn't well defined. – jacob1729 May 19 at 14:00
• Possible duplicate: physics.stackexchange.com/questions/323701/… – jacob1729 May 19 at 18:44
• There is certainly a close relationship between this question and the proposed duplicate, but I think that this one is much more direct and clear in asking how Fermat's principle related to the foundation of rays optics. – dmckee May 19 at 19:16
• I read your answer to the possible duplicate, which was very satisfactory, one thing I want to ask is that does fermat's principle say that if all rays coming from a source meet at the same point, they would've all done it in the same time(i.e the minimum time path of one ray is equal to that of any other ray)? – Amritansh Joshi May 21 at 8:36

Fermat's principle holds that the physical path of light is the one out of all conceivable paths that (locally) minimizes the travel time.1

Now, in the case of a ray optics system that focuses, we assert that the light actually travels along multiple paths.

The only way that both of these things ("multiple paths" and "actual path takes the least time") can be true is if all the actual paths require the same amount of time.2

1 Don't worry about the "locally" addendum at first. It is not necessary in simple examples, but becomes useful later on.

2 Which means that grammatically I should have stated Fermat's principle as "the physical paths of light are the the ones out of all conceivable paths that (locally) minimize the travel time". Details, details.

• So you mean if many light rays converge to a single point, they would've done it at the same time (provided they're coming from the same source)? So that means the "least time" paths of all light rays consume the same time? – Amritansh Joshi May 20 at 9:50

Just think of this way that only thing fermat's principle do is to prove that incident and reflected angles are same. It is also well known that for parabola mirror light rats converges. For spherical mirror it does not it from caustic curve you can see it in cups etc. You can google individual key words. Bottom line is that fermat's principle is only used to prove incident angle and reflected angles are same. Side note-fermat's principle can be proven by path integral of quantum mechanics. Hopefully I have given enough keywords for Googleing.