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This question already has an answer here:

Generally the refractive index of any lens is greater than air. My question is what will happen if the refractive index of lens is less than unity?

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marked as duplicate by Kyle Kanos, user191954, Yashas, Peter Diehr, Jon Custer Oct 29 '18 at 20:35

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  • $\begingroup$ @Steeven $c$ is only the phase velocity, which can indeed be greater than in the vacuum. $n$ is less than unity for example in metamaterials, plasmas, metals and dielectrics at certain wavelengths. See my question here for a real life example of $n < 1$. $\endgroup$ – ahemmetter Oct 26 '18 at 7:06
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Steeven is correct in comments when he says that there are no materials with $n<1$, because this would imply light propagating faster than $c$ in these materials.

However, you can imagine a situation where a "lens" made of a low-index material is embedded in a high-index medium, for example a bubble of water ($n\approx 1.33$) in a body of oil ($n\approx 1.5$).

In this case you should consider what is the "optical" length of the path different rays travel through the lens, depending on their distance from the optical axis. Since a thicker path through the lens results in a shorter optical length, you'll find that convex shapes form converging lenses and concave shapes produce diverging lenses, the opposite of the situation for high-index lenses in air.

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    $\begingroup$ Actually, it is possible for the index of refraction $n$ to be less than 1. $n<1$ for many materials at x-ray wavelengths in the 10's of keV range. And, no, that does not imply faster than light propagation of information according to the equation $n=c/v$ because the $v$ here refers to the phase velocity, which can exceed the speed of light. $\endgroup$ – Samuel Weir Oct 26 '18 at 6:55
  • $\begingroup$ Of course the phase velocity $c$ can be higher than in a vacuum - this happens in plasmas (ionized gases, metals and even most dielectrics near the absorption maximum). Adjusting the arrangement of areas of different $\varepsilon$ and $\mu$ even allows the refractive index to be negative (which corresponds to a backward wave propagation) in the case of metamaterials or even simple transmission lines. $\endgroup$ – ahemmetter Oct 26 '18 at 8:02

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