When a ray of light crosses a boundary from air to water, glass or other transparent material, it bends, and the degree of bending is determined by the index of refraction.

Air has an index of 1. Water’s index of refraction is about 1.3. That is why rippling water waves distort the view of a pond bottom, for instance. It is refraction that makes a straw in a glass of water look as if it is bending toward the surface, and fish swimming in a pond look closer to the surface than they really are.

Diamonds have a refractive index of 2.4, giving them their sparkling beauty.

For visible light, transparent materials like glass, water and diamonds all have an index of 1 or higher, meaning that when the light enters, its path bends inward, closer to the perpendicular. Because the index is uniform throughout a material, the bending occurs only as the light crosses a boundary.

But with metamaterials, scientists can now also create indexes of refraction from 0 to 1. In the Duke cloaking device, the index actually varies smoothly from 0, at the inside surface of the cylinder, to 1, at the outside surface. That causes the path of light to curve not just at the boundaries, but also as it passes through the metamaterial.

Metamaterials first took center stage in a scientific spat a few years ago over a startling claim that the index of refraction could be not just less than 1, but also negative, less than 0. Light entering such a material would take a sharp turn, almost as if it had bounced off an invisible mirror as it crossed the boundary.

The refractive index depends on the response of a material to electric and magnetic fields. Typically within a material, electrons flow in a way to minimize the effects of an external electric field, producing an internal electrical field in the opposite direction. But that is not universally true. For some metals like silver, an oscillating electric field induces a field in the same, not opposite, direction.

Victor G. Veselago, a Russian physicist, realized in the 1960s that if it were possible to find a material that responded in a contrarian way not just to electric fields and but also magnetic fields, a result would be a negative index of refraction.