Negative refraction is the name for an electromagnetic phenomenon where light rays are refracted at an interface in the reverse sense to that normally expected. Such an effect can be obtained using a metamaterial which has been designed to achieve a negative value for both permittivity ε and permeability μ, as in such cases the material can be assigned a negative refractive index. Such materials are sometimes called "double negative" materials. Negative refraction occurs at interfaces between materials at which one has an ordinary positive phase velocity, and the other has the more exotic [|negative phase velocity].
Negative phase velocity is a property oflight propagation in a medium. There are different definitions of NPV, the most common being Veselago's original proposal of opposition of wavevector and Poynting vector, i.e. E×H; other choices are opposition of wavevector to group velocity, or to energy to velocity. The use of "phase velocity" in the naming convention, as opposed to the perhaps more appropriate "wave vector", follows since phase velocity has the same sign as the wavevector. A typical criterion used to determine Veselago NPV is that the dot product of the Poynting vector and wavevector is negative ; however this definition is not covariant. Whilst this restriction is rarely of practical significance, the criterion has nevertheless been generalized into a covariant form. For plane waves propagating in a Veselago NPV medium, the electric field, magnetic field and wave vector follow a left-hand rule, rather than the usual right-hand rule. This gives rise to the name "left-handed materials". However, the terms left-handed and right-handed can also arise in the study of chiral media, so this terminology is best avoided.
One can choose to avoid directly considering the Poynting vector and wavevector or a propagating light field, and consider instead the response of the materials directly. Assuming an achiral material, one can consider what values of permittivity ε and permeability µ result in negative phase velocity. Since both ε and µ are in general complex, their imaginary parts do not have to be negative for a passive material to display negative refraction. In these materials, the criterion for negative phase velocity was derived by Depine and Lakhtakia to be where are the real valued parts of ε and µ, respectively. For active materials, the criterion is different. The occurrence of negative phase velocity does not necessarily imply negative refraction. Typically, the refractive index is determined using where by convention the positive square root is chosen for. However, in NPV materials, we reverse that convention and pick the negative sign to mimic the fact that the wavevector are likewise reversed. Strictly speaking, the refractive index is a derived quantity telling us how the wavevector is related to the optical frequency and propagation direction of the light, thus the sign of must be chosen to match the physical situation. In case of chiral materials, the refractive index also depends on the chirality parameter, resulting in distinct values for left and rightcircularly polarized waves, given by It can be seen that a negative index will occur for one polarization if >. In this case, it is not necessary that either or both and be negative to achieve a negative index of refraction. A negative refractive index due to chirality was predicted by Pendry and Tretyakov et al., and first observed simultaneously and independently by Plum et al. and Zhang et al. in 2009.
Refraction
The principal symptom of negative refraction is just that – light rays are refracted on the same side of the normal on entering the material, as indicated in the diagram, and by a suitably general form of Snell's law.
