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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 (electric) permittivity ε and (magnetic) permeability μ, as in such cases the material can be assigned a negative refractive index. Such materials are sometimes called "double negative" materials.
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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
Negative phase velocity (NPV) is a property of light propagation in a medium. There are different definitions of NPV, the most common being Veselago's original proposal of opposition of wavevector and (Abraham) Poynting vector, i.e. E×H; other common choices are opposition of wavevector to group velocity, or to energy 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 (i.e. that
Negative refractive index
We 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: that is, we consider what values of permittivity ε and permeability µ result in negative phase velocity (NPV). Since both ε and µ are in general complex, their imaginary parts do not have to be negative for a passive (i.e. lossy) material to display negative refraction. The most general Veselago criterion applying to ε and µ is that of Depine and Lakhtakia, although other less general forms exist. The Depine-Lakhtakia criterion for negative phase velocity is
where
Typically, the refractive index n is determined using
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.