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If so, what happens in that case? Can we also think about this with the quantum picture of light in mind? Is convex lens an example? Note: Not confined to euclidean space. Also, let's not talk about mirrors here. P.S.: I have an intuition for Fermat's principle (from Huygens' principle).

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The best example is a lens focusing light. Every path through the lens takes the same time (but does not create different images). The center of the lens is thicker and the speed of light in glass is lower. This compensates for the longer path thought the edges of the lens where it is thinner. As a result, it takes light exactly the same time to pass through any part of the lens to get from an object to its image in the lens focus.

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  • $\begingroup$ I think I agree. So, let's move on to the photon picture of light: What does a photon do? Is there a amplitude(probability) associated with the paths? $\endgroup$ – Rusty Slinger Jan 12 '18 at 19:54
  • $\begingroup$ From the source to the lens, it is the probability of emission, which is usually random and is rather irrelevant to your question. From the lens to the focused image, yes, the quantum probability defines the path. Some photons will not be in focus, but most will be. Also, the focal point will be blurred by diffraction, which essentially represents the quantum probability distribution for photons. $\endgroup$ – safesphere Jan 12 '18 at 20:03
  • $\begingroup$ "the focal point will be blurred by diffraction" : Can you elaborate? $\endgroup$ – Rusty Slinger Jan 13 '18 at 13:31
  • $\begingroup$ @RustySlinger In the quantum view, you have a wave function that defines the probability distribution of photons. The position of each photon is random, but statistically they follow the wave distribution. Diffraction is a wave effect, it applies to the wave function as well. So, when you look at diffraction in the quantum way, you get photons randomly hitting the screen around the focal point in a pattern that follows the probability distribution of the wave function, which in turn exhibit the wave properties, including interference, diffraction, etc. Classical or quantum, a wave is a wave. $\endgroup$ – safesphere Jan 13 '18 at 19:39
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Yes. For a physical proof, consider mirages. Light from the sun "reflects" off the temperature gradient of the air near the ground, making it look like light is reflecting off the water. But light from the sun also goes straight into your eye. Both the straight-line path and the bent mirage path are extremal.

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Yes, it can happen, and does happen in astronomy. What happens in this case is you see multiple images, one for each path. In the classic Einstein cross you're seeing four images of the same thing where the light followed four different locally optimal paths to get to us.

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