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The actual ray path between two points is the one for which the optical path length is stationary with respect to variations of the path. Can you explain me what is extremum path?

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  • $\begingroup$ Extremum means maximum or minimum. In the Fermat principle, it is usually (but not absolutely always) the path that takes the minimum time. Because the derivative (speed of change) in the extremum is zero, the path is stationary (no change due to small variations). $\endgroup$ – safesphere Sep 22 '17 at 5:36
  • $\begingroup$ Take a look in my answer therein : Why one should follow Snell's law for shortest time? May be help you. $\endgroup$ – Frobenius Sep 22 '17 at 6:04
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  1. An global (local) extremum is a global (local) maximum or minimum, respectively.

  2. Note that the principle of least action is more accurately a 'principle of stationary action', since we are looking for stationary/critical paths rather than minimum paths.

  3. Similarly, Fermat's principle aka. the principle of least time is more accurately a 'principle of stationary time'.

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As an example suppose you want to find the path that light takes in going from £A£ to $B$ as a result of a reflection.

There are an infinite number of possible paths and below I have shown three.

enter image description here

You will note that the tame taken for the light to travel from $A$ to $B$ depends on the path taken but one of the paths $AYZ$ takes a minimum (extremum) path.
Fermat says that that is the path taken by the light and it also happens to be the path such that the angle of incidence is equal to the angle of reflection.

Feynman devotes a whole chapter "Optics: The Principle of Least Time" on this topic.

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