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How do we deduce Snell's law using Fermat's principal in case of metamaterials?

Metamaterials have negative refractive index. This makes the refracted ray of light bend on same side of normal as incident ray.

But according to Fermat principle, the light could have taken a shorter path. Without the light 'bending backward', it would have a path which took lesser time, like one of the possible path I have shown below in red.

enter image description here

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What is the "travel time" of a beam of light in a medium with negative refractive index? Do you save time by traveling a greater distance? Time = distance/speed, and speed = c/n. When $n<0$, the part of the trajectory in the meta-material will contribute "negative time" to the over all travel time of the beam.

Remember, Fermat's "shortest" means least time, not least distance. And it's just a mathematical construct.

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  • $\begingroup$ so is Fermats' principle a general result from some other more basic principle? which? And, can we say time and speed as being negative? @Floris $\endgroup$ – alto santa May 4 '17 at 15:52
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    $\begingroup$ Yes - Fermat's principle is derived from calculus of variation. See for example this paper, section 16.4 . For the purpose of that calculation, we do have to say time appears to be going backwards in the meta-material. But that's a mathematical oddity, like "negative effective mass" for holes in semiconductors. $\endgroup$ – Floris May 4 '17 at 15:55
  • $\begingroup$ this is possible if Fermats' principle is purely mathemtical. So 'algebraic' sum of time is least. and speed is taken as 'c/n' with sign. Am I correct? $\endgroup$ – alto santa May 4 '17 at 16:00
  • $\begingroup$ That's my understanding, yes. $\endgroup$ – Floris May 4 '17 at 16:17

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