What are the physical optical limitations photonic metamaterials can offer? Modern technology has introduced new photonic metamaterials that can extend optical properties beyond what standard materials can offer. For example a negative index of refraction is now possible which can lead to extraordinary abilities such as invisibility (optical cloaking) of massive objects, etc.
So we once believed that our ability to resolve an image was limited to, for example the diffraction limit derived by Abbe which is a function of the wavelength of the light, and index of refraction of the material.
Given these new metamaterials, do these laws still apply? Can we, again for example, just plug-in the negative index of refraction to Abbe's expression for diffraction limit? If so, how does one interpret a negative diffraction limit?
It also leads me to the question of what does a zero index of refraction mean? What physically amazing things might a zero index entail, and is a zero index of refraction even possible to engineer with metamaterials?
 A: Given these new metamaterials, do these laws still apply? Can we, again for example, just plug-in the negative index of refraction to Abbe's expression for diffraction limit? If so, how does one interpret a negative diffraction limit?
The reason that the diffraction limit actually occurs is due to the decay of evanescent EM fields which decay exponentially from the surface of the material, and thus are lost in the far field. However, a material with a refractive index of $-1$ will (as proven by John Pendry) actually cause these evanescent fields to increase in strength (they carry no energy, and therefore conservation of energy is not broken). This is the basis of the so-called super lens (for more information on how this field has developed, take a look at hyper lenses, or meta lenses). It should be pointed out, however, that no-one has yet made a metamaterial with sufficiently exact properties to realise the super lens fully. We have achieved sub-wavelength imagery, but we have not yet made the perfect lens.
I haven't looked into work which aims to have a refractive index of zero, however we certainly can get refractive indices of less than $1$ (see, for instance, $-0.6$, I believe, in a paper by Gunnar Dolling, achieved at $780nm$). In this instance, the wave moves faster than the speed of light. However, causality is not compromised because it is only the phase velocity of the light which is moving faster than the speed of light, and not the group velocity, meaning that no information would be transferred faster (you can only transfer information along a wave by modulating that wave somehow, say by blocking it and unblocking in Morse code). In another paper by the same author, they also note a negative group velocity, but I don't remember if that has a magnitude of more or less than $1$.
