What is a metamaterial?

(This is part of a lead-in to a brief discussion I'd like to do of two papers that just came out.) The wikipedia entry for metamaterial is actually rather good, but doesn't really give the "big picture".  As you will hopefully see, that wording is a bit ironic.

"Ordinary" materials are built up out of atoms or molecules.  The electronic, optical, and mechanical properties of a solid or liquid come about from the properties of the individual constituents, and how those constituents are spatially arranged and coupled together into the whole.   On the length scale of the constituents (the size of atoms, say, in a piece of silicon), the local properties like electron density and local electric field vary enormously.  However,  on length scales large compared to the individual constituent atoms or molecules, it makes sense to think of the material as having some spatially-averaged "bulk" properties, like an index of refraction (describing how light propagates through the material), or a magnetic permeability (how the magnetic induction \(\mathbf{B}\) inside a material responds to an externally applied magnetic field \(\mathbf{H}\)), or an elastic modulus (how a material deforms in response to an applied stress).

A "metamaterial" takes this idea a step further.  A metamaterial is build up out of some constituent building blocks such as dielectric spheres or metallic rods.  The properties of an individual building block arise as above from their own constituent atoms, of course.  However, the properties of the metamaterial, on length scales long compared to the size of the building blocks, are emergent from the properties of those building blocks and how the building blocks are then arranged and coupled to each other.   The most common metamaterials are probably dielectric mirrors, which are a subset of photonic band gap systems.  You can take thin layers of nominally transparent dielectrics, stack them up in a periodic way, and end up with a mirror that is incredibly reflective at certain particular wavelengths - an emergent optical property that is not at all obvious at first glance from the properties of the constituent layers.

Depending on what property you're trying to engineer in the final metamaterial, you will need to structure the system on different length scales.  If you want to mess with optical properties, generally the right ballpark distance scale is around a quarter of the wavelength (within the building block constituent) of the light.  For microwaves, this can be the cm range; for visible light, its tens to hundreds of nm.  If you want to make an acoustic metamaterial, you need to make building blocks on a scale comparable to a fraction of the wavelength of the sound you want to manipulate.  Mechanical metamaterials, which have large-scale elastic properties far different than those of their individual building blocks, are trickier, and should be thought about as something more akin to a small truss or origami framework.  These differ from optical and acoustic metamaterials because the latter rely crucially on interference phenomena between waves to build up their optical or acoustic properties, while structural systems rely on local properties (e.g., bending at vertices).

Bottom line:  We now know a lot about how to build up larger structures from smaller building blocks, so that the resulting structures can have very different and interesting properties compared to those of the constituents themselves.