# Physical meaning of conductor refractive index

While looking at some data on refractiveindex.info I noticed something odd about the listed refractive indices they provided. For dielectrics they are as one would expect, but for a conductor, say silver at 500nm, they list the refractive index as $$0.05$$. This confused me as I thought that the refractive index represented the speed of light over the phase velocity of light through that material, and other than some special cases I did not think it could be much less than 1. To ensure the math was correct, I converted the relative permittivity they provided ($$-9.8+0.31309i$$ in this case) to the refractive index using $$n = \sqrt{\epsilon_r}$$ and got the same result of $$0.05+3.13089i$$. As far as I know the real part of this result is the regular refractive index and the imaginary part is silver's extinction coefficient (which I don't think would be relevant to its phase velocity).

Based on this, does the refractive index of a conductor have some sort of special meaning that differs from the meaning it has with dielectrics or is this just an alternative mathematical representation of it? I did notice additionally that if I convert the refractive index to a reflectance at normal incidence using the Fresnel equations (with the other medium assumed to be a vacuum) I get a function that hits the expected reflectance for silver at 500nm of $$0.98166$$ at $$n=0.05,$$ but also hits the value another time at $$n=216.098$$. Does this mean that $$216.098$$ is the actual physical refractive index properly representing the phase velocity of light through the metal?

• How do you get 0.05 by square rooting a complex number with a real part of 9.8? The refractive index is complex. May 2, 2020 at 8:05
• May 2, 2020 at 8:17
• $\sqrt{-9.8+0.31309i}\approx 0.05+3.13089i$, so the math regarding that part checks out @RobJeffries. May 2, 2020 at 11:02
• @zonksoft Exactly. It's a complex number. May 2, 2020 at 11:34
• @RobJeffries Well I was under the impression that the real part of the complex refractive index was equivalent to the normal one, the imaginary part is just the material's extinction coeffecient as far as I know and isn't related to it's phase velocity. May 2, 2020 at 14:46

## 2 Answers

Usually when people talk about the speed of light being higher than c, they mean the phase velocity. What is the phase velocity? This means that if you draw a clean cosine in the material and look at the location of a specific peak in cosine and measure its velocity (= this is the phase velocity) you will get moving faster than the speed of light.

But it's absurd, apparently nothing can get past the speed of light, isn't it? So I'll divide the answer into 2: Why are there things that can go through the speed of light, and then why specifically the phase speed is one of them.

Part 1 - The prohibition on passing the speed of light basically results from the fact that information cannot pass between 2 points faster than light. But things that don't carry information are no problem. For example, if I stand at night and shine with a laser on one end of the moon, and quickly decide to turn my hand so that I shine with the other side, you can do the calculation and see that the point of light on the moon moves faster than the speed of light. No problem with that - if you think about it, nothing really moves from one end of the moon to the other. If you think of the ray of light as being made up of a lot of little balls that come one after the other (it's not like that, but it will simplify this specific explanation), no ball really moves along the moon - but just at one point one ball reached one point, and when I moved my hand another ball reached the second point. In other words, if there was a human being on one end of the moon who would like to pass a message to a human being on the other end, he could not because he did not really transmit anything. It's a series of balls that came from different places, each to a different point. Hopefully I managed to get the point across.

Part 2 - Phase velocity. Imagine you and I are holding a sheet at its 2 ends, and one of us decides to rock it up and down. A wave will advance according to the first drawing I attached (the black lines represent the "peaks" in the sheet)

.

Now let's say you look at the diagonal line in the sheet (which is almost horizontal in the drawing) and ask about the speed of the peak progress along this line. Even then you will see that the peak velocity (the phase velocity along this axis) is higher than the velocity the wave itself is moving at, but that's fine - because it's not really the same peak, it's different parts of the same peak line that came from the previous one that just cross the diagonal - so nothing Really moving along this line faster than the wave, but only a mathematical concept called the peak.

Now that we've become more acquainted with the concept that there are things that can go through the speed of light and that's fine because nothing really moves down that line faster, I want to talk about plasmas (or things with free charge) and substances where the phase speed is really greater than the speed of light. What is the origin of the refractive index of the material? Microscopically, it can be shown that when the wave arrives, it is momentarily absorbed in the atom (an electric dipole oscillates) and the same dipole then radiates the energy back. But the same dipole would not necessarily radiate a wave with the same phase as it forced it to oscillate in the first place, but with a little delay.

This means that if you take a "clean" cosine in the material it seems effective that it is moving slower than the same cosine in the blank, but the light coming out of the atom is not exactly the same light coming in. So far it's fine and intuitive, but who said that the light emitted by the dipole must be lagging in phase? If for some reason the light that is re-radiated is "ahead" in phase actually, it would seem as if the cosine peaks are effectively moving faster in the material - but again, that's fine!

What is really important is that if I send a pulse, its leading edge will never move faster than the speed of light, because it necessarily carries information. I know this is a confusing explanation, but even undergraduates in second year get involved with it and it takes time for them to understand the concept so don't be ashamed to ask any more questions if the explanation is unsatisfactory! :)

• The laser pointer moon thing is a quite interesting example, I never had thought about it like that. Out of curiosity do you know why things like conductors specifically exhibit the behavior of speeding up the phase like that while dielectrics do not? Is it something to do with their atomic structure? (I know for example conductors tend to absorb much more light as well due to something like that which is why their extinction coefficient is so high) May 2, 2020 at 21:14
• Dielectrics actually do also exhibit weird refractive index properties similar to the kind baffling you, but they happen really near their resonances. I don't have much experience with fields inside metals and conductors but I'm sure that if you search fields in plasmas (or anything with free electrons) you'll find an answer :) May 2, 2020 at 21:32
• @OfekGillon this. I'm quite sure that all resonance phenomena show a negative, then a positive phase shift while passing over the absorption frequency (generalized from the harmonic oscillator. I wanted to research that for an answer to this question but took too much time :) May 3, 2020 at 12:20
• @zonksoft That is interesting and definitely makes sense given the data. Indeed the resonance frequency for silver is around 315nm apparently and that is also where its refractive index goes from >1 to somewhere rather low, so it does match how that graph with the harmonic oscillator behaves. Thanks for the info! May 4, 2020 at 1:25
• It's also the frequency where the transmission is nonzero, at least according to the plot at refractiveindex.info. Btw I would be very interested in a model that describes this, I'm sure it exists. I know of the Plasma frequency, but we need a plot of Re[n] and Im[n] in the end. May 4, 2020 at 10:21

Phase speed of light can be greater than $$c$$, so phase speed of light waves in silver is 20x greater than light speed in vacuum. Not only that, but refractive index can be negative too ! If refractive index is negative this means that light phase speed in that material is also negative. This is usually achieved in meta-materials. Meta-materials bends light in reversed angle :

EDIT

it's not bizarre to have a phase speed of wave greater than $$c$$. Wave phase speed just shows how fast wave extremes are moving, but this is not wave movement speed itself. Group speed is related to phase speed by such relation : $$\sqrt {v_g \cdot \, v_p} = c$$

Only in vacuum, $$v_g=v_p=c$$, but in materials, this is not the case.

Group speed can also exceed $$c$$, but this does not carry information too. Btw, keep in mind that refractive index measures phase speed of wave.

However information in wave is transmitted with wavefront, which speed cannot exceed $$c$$. I. e. pulse wavefront speed is what is most important for information transfer.

Btw, it's interesting to note that metal is transparent to electromagnetic wave frequency which is higher than plasma frequency of metal :

$$\omega _{\mathrm {pf} }={\sqrt {\frac {n_{\mathrm {e} }\,e^{2}}{m_e\,\varepsilon _{0}}}}$$

Where $$n_e$$ is number density of electrons. And of course in transparency region of waves, where $$\omega > \omega _{\mathrm {pf} }$$, we can now expect refractive index to be $$n > 1$$. That's why it is hard to make an ultraviolet or x-rays laser,- because metal mirrors becomes transparent to this spectral range of waves ! Well, one mirror can be partially transparent, because you need to pass-through generated laser beam, but another must be 100% reflective. Anyway, all in all, your question has to do with crossing plasma frequency of metal from transparent to opaque state or in reverse.

• Is that really the case though, like I said I do know there are some odd cases which allow faster than light phase velocities but I thought those were just very minor technicalities. For example in the negative refractive index case I'd imagine that is just the observation of the light's behavior on a macro scale due to it being a complex meta-material, something like silver however is fairly standard and I don't see why it'd cause such bizarre things to happen. May 2, 2020 at 15:42
• Explanation about group speed added, see my edit. May 2, 2020 at 16:33