Upon checking the optical properties of different dielectrics, I found the interesting case of $Al_2O_3$. It seems to be reported with a refractive index below 1 in the infrared range of $10 - 12~\mu m$, for example by Kischkat, Boidin and Querry. Also the absolute value of the complex index of refraction $\bar n = \sqrt{n^2 + \kappa^2}$ is below 1 at certain wavelengths.

enter image description here

(red: $n$, blue: $\kappa$)

Basic physics tells me that the phase velocity of a EM wave in a dielectric is related to the real part of the refractive index by $$v_{ph} = \frac{c_0}{n}$$

Now obviously, the phase velocity can be above $c_0$, but mostly this happens only in 'strange' cases: in plasmas, active media, metamaterials or magnetic substances. $Al_2O_3$ seems to be neither of those.

Another clue is that this happens around the region where $n$ and $\kappa$ cross - that is also the point where $\varepsilon = 0$ in a Drude metal, and hence the wavelength of the bulk plasmon. Could this therefore just be an effect of the $Al_2O_3$ becoming metallic, as mentioned by @boyfarrell?

What's going on here?

  • $\begingroup$ Lots of free charge charges i.e. material behaving as a metal will cause this $\endgroup$
    – boyfarrell
    Oct 23 '18 at 11:01
  • 1
    $\begingroup$ @boyfarrell Ah, right, I absolutely didn't think of this. So Al2O3 just behaves in this range more like a metal. Feel free to add it as an answer. $\endgroup$
    – ahemmetter
    Oct 23 '18 at 11:04
  • $\begingroup$ Yeah could correspond to a spectral region near some sort of resonance. Not really an expert on metals. Would be interested in a good answer too. $\endgroup$
    – boyfarrell
    Oct 23 '18 at 11:06

What you are describing is anomalous dispersion. This happens when a material becomes strongly absorbing, typically near an absorption line, and the refractive index becomes complex.

In these circumstances the phase and group velocities are different and indeed the group velocity can be greater than $c$. This isn't a problem since the group velocity is no longer the speed at which information travels.

Materials don't need to be strange to show this. It occurs at absorption lines in all materials.

  • $\begingroup$ It seems the maximum of $\kappa$ does not coincide with the region where $n < 1$, no I'm not sure in how far absorption is applicable here. Clearly $n$ decreases with increasing wavelength, but the question is rather about the value of it, not its derivative. $\endgroup$
    – ahemmetter
    Oct 23 '18 at 11:12
  • $\begingroup$ "the group velocity is no longer the speed at which information travels" - how is this possible though? Could you please explain this in more detail? Or give a good reference, if you can, that would work as well $\endgroup$
    – Yuriy S
    Oct 23 '18 at 19:05
  • $\begingroup$ @YuriyS The answer to this question deals with that $\endgroup$
    – ahemmetter
    Oct 23 '18 at 20:51

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