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I see this claim being originally made by Vesselago (the discoverer of the principles of metamaterials) and indeed in contemporary papers. It means that such a metamaterial would be pulled towards its illuminating light source. Why does this not violate conservation of momentum?

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This does not violate conservation of momentum, because the momentum behaves in a counter-intuitive manner in a negative refractive index medium.

In most media, the refractive index is positive, and the Poynting vector $\vec{S}$ and the wave vector $\vec{k}$ point in the same direction. In a medium with a negative refractive index, the Poynting vector points in the opposite direction from the wave vector.

The wave vector of the light is what you control when you set up the light source. If you point the source towards the positive $x$ axis, for example, then $\vec{k} = |\vec{k}|\hat x$. On the other hand, the Poynting vector is what controls the light wave's momentum, and therefore the radiation pressure.

The result is (from the APS Meeting paper you cite):

a perfect mirror illuminated with a plane wave would experience a negative radiation pressure (pull) when immersed in a left-handed medium [Emphasis added]

I've emphasized that the mirror must be immersed in a left-handed medium, rather that simply made from a left-handed material, because the light wave must actually be in the negative refractive index medium for this effect to occur.

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