# Controllable faster-than-light phase velocity

This is not another question about faster-than-light travel or superluminal communication. I totally appreciate the speed limit capped by physical laws (or theories.)

Just curious, since there is no limit for phase velocity, is there any way to generate a wave with controllable phase velocity higher than speed of light?

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@skywaddler-- see my question-- What does Lijun Wang's experiment about supraluminal speed of light in a medium mean? – Gordon Feb 18 '11 at 7:35

Yes. The phase velocity $v_\phi$ of light in a medium is equal to $c_0/n$, where $c_0$ is the speed of light in vacuum, and n is the material's refractive index. The index is usually greater than 1, so normally this doesn't happen; but occasionally it is smaller for certain frequencies of light (Wikipedia gives the examples of X-rays and frequencies near absorption resonances), giving a phase velocity greater than $c_0$.

It is possible to change the refractive index of a material: see for example the Pockels effect and the Kerr effect. I don't know if this means that it would actually be possible to build a material with a "phase velocity knob" though, as suggested in the comment below.

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OP wanted to know if you could control the phase velocity. I.E. build a "phase velocity dial" to turn it up and down at will, I think. – Mark Eichenlaub Dec 22 '10 at 9:22
@Mark Eichenlaub, I understood the question to be whether you could design a material with any particular phase velocity that you wanted. But see the edit to my answer. – ptomato Dec 22 '10 at 11:06
Interesting, thanks. – Mark Eichenlaub Dec 22 '10 at 11:28
Doesn't the tunnel effect provide an in principle tunable mechanism? – Raskolnikov Dec 22 '10 at 13:30

The phase velocity of light in a fully ionized plasma is strictly superluminal; see equation 1156 of Fitzpatrick. Like all media in which the speed of light is not $c$, plasmas are dispersive, but the phase velocity is still superluminal over a large bandwidth (from the plasma frequency on up).

The phase velocity can be tuned by tuning the density of the plasma. My old group used this denstiy-tuned index of refraction to make transient micro-optics out of plasma.

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The highest-profile schemes for manipulating the speed of light to make "slow light" use quantum phenomena like electromagnetically induced transparency to manipulate the index of refraction. The "slow light" effects come not from having a large index, but rather a steep slope in the index as a function of frequency. They produce this by pumping the medium with additional light so that the combination of the two light fields puts the atoms in the medium into a particular quantum superposition. The exact final speed can be tuned by choosing the properties of the pump field appropriately.

"Fast light" experiments can work in a similar manner-- for example, this paper on superluminal pulse propagation (arxiv version) uses a cesium vapor pumped into a particular state as the medium through which the pulse propagates faster than the normal speed of light. The resulting speed depends on the parameters of the laser used to excite the vapor, and thus can be tuned by varying those parameters.

These experiments generally deal with the group velocity rather than the phase velocity, as that's the thing that you can measure in an experiment. I'm not sure if that invalidates these as responses to your question, but that's what comes to mind on the topic of varying the speed of light.

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If I understand the abstract of that paper correctly, it appears to claim that the signal is emitted from the receiving side before it is actually sent! – Robert Harvey Mar 31 '11 at 18:50