Hello all,
I am working on metramaterial these days,, and I got confused about Negative phase velocity and positive group velocity in metamaterials. I know how to obtain the equations but I can't understand how a wave can have Negative phase velocity (the direction of the wave), but the power propagates in the opposite direction. How would it be possible?
This happens in backward wave structures, which have been used for 80 years.
These are 1D metamaterials, where the wave travels along a bent path next to an electron beam, for instance, and comes back to the electron beam at regular intervals, for instance after just less than a wavelength along the bent path. The electron beam sees a phase change of 300 degrees after every interval, which looks like -60 degrees. So the electron beam sees a wave going backwards slowly (phase velocity) whereas the structure sees a wave going forwards fast (group velocity). The power is going forwards and any not coupled to the electron beam will arrive at a load at the end and make it hot. The power goes in the direction of the group velocity. The pattern that is seen or used goes in the direction of the phase velocity.
See https://en.wikipedia.org/wiki/Backward-wave_oscillator for instance.
In metamaterials the signal is slowed down or diverted in each periodic part of the structure so that in the gaps between the parts the pattern looks like it is going backwards. The pattern is like Moire fringes, which can be made by holding two combs against each other.
This shows fringes going up as one comb is moved down!
https://en.wikipedia.org/wiki/Moir%C3%A9_pattern#/media/File:070309-moire-a5-a5-upward-movement.gif
which is from https://en.wikipedia.org/wiki/Moir%C3%A9_pattern
It takes two periodic things together to do this. In metamaterials the two things are the periodic structure, and the periodic electromagnetic wave. It then generates the phase pattern, which can go forwards, backwards, faster than light, or stay still.
Negative phase velocity means negative refraction index. It is some sort of backward propagating wave
You better read:
V.G. Veselago. ‘‘The electrodynamics of Substances with Simultaneously Negative Values of Img Align= Absmiddle Alt= ϵ Eps/Img and μ.’’ Physics-Uspekhi, vol. 10, pp. 509-514, 1968.
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