Negative refraction

[1] Negative refraction occurs at interfaces between materials at which one has an ordinary positive phase velocity (i.e., a positive refractive index), and the other has the more exotic negative phase velocity (a negative refractive index).

Negative phase velocity (NPV) is a property of light propagation in a medium.

While this restriction is not practically significant, the criterion has been generalized into a covariant form.

[3] Veselago NPV media are also called "left-handed (meta)materials", as the components of plane waves passing through (electric field, magnetic field, and wave vector) follow the left-hand rule instead of the right-hand rule.

The terms "left-handed" and "right-handed" are generally avoided as they are also used to refer to chiral media.

One can choose to avoid directly considering the Poynting vector and wave vector of a propagating light field, and instead directly consider the response of the materials.

Assuming the material is achiral, one can consider what values of permittivity (ε) and permeability (μ) result in negative phase velocity (NPV).

In these materials, the criterion for negative phase velocity is derived by Depine and Lakhtakia to be where

is determined using where by convention the positive square root is chosen for

However, in NPV materials, the negative square root is chosen to mimic the fact that the wave vector and phase velocity are also reversed.

The refractive index is a derived quantity that describes how the wavevector is related to the optical frequency and propagation direction of the light; thus, the sign of

A negative refractive index due to chirality was predicted by Pendry and Tretyakov et al.,[8][9] and first observed simultaneously and independently by Plum et al. and Zhang et al. in 2009.

[10][11] The consequence of negative refraction is light rays are refracted on the same side of the normal on entering the material, as indicated in the diagram, and by a general form of Snell's law.