linguistic:paraphrasing, stylistic:tone
Negative refractive index

Negative-index metamaterials (NIM) are characterized by a negative index of refraction. Other terms for NIMs include "left-handed media", "media with a negative refractive index", and "backward-wave media". NIMs where the negative index of refraction arises from simultaneously negative permittivity and negative permeability are also known as double negative metamaterials or double negative materials (DNG).

Assuming a material well-approximated by a real permittivity and permeability, the relationship between permittivity ε r {\displaystyle \varepsilon _{r}}, permeability μ r {\displaystyle \mu _{r}} and refractive index n is given by n = ± ε r μ r {\textstyle n=\pm {\sqrt {\varepsilon _{\mathrm {r} }\mu _{\mathrm {r} }}}}. All known non-metamaterial transparent materials (glass, water,...) possess positive ε r {\displaystyle \varepsilon _{r}} and μ r {\displaystyle \mu _{r}}. By convention the positive square root is used for n. However, some engineered metamaterials have ε r {\displaystyle \varepsilon _{r}} and μ r < 0 {\displaystyle \mu _{r}<0}. Because the product ε r μ r {\displaystyle \varepsilon _{r}\mu _{r}} is positive, n is real. Under such circumstances, it is necessary to take the negative square root for n. When both ε r {\displaystyle \varepsilon _{r}} and μ r {\displaystyle \mu _{r}} are positive (negative), waves travel in the forward (backward) direction. Electromagnetic waves cannot propagate in materials with ε r {\displaystyle \varepsilon _{r}} and μ r {\displaystyle \mu _{r}} of opposite sign as the refractive index becomes imaginary. Such materials are opaque for electromagnetic radiation and examples include plasmonic materials such as metals (gold, silver,...).

The foregoing considerations are simplistic for actual materials, which must have complex-valued ε r {\displaystyle \varepsilon _{r}} and μ r {\displaystyle \mu _{r}}. The real parts of both ε r {\displaystyle \varepsilon _{r}} and μ r {\displaystyle \mu _{r}} do not have to be negative for a passive material to display negative refraction. Indeed, a negative refractive index for circularly polarized waves can also arise from chirality. Metamaterials with negative n have numerous interesting properties:

Negative index of refraction derives mathematically from the vector triplet E, H and k.

For plane waves propagating in electromagnetic metamaterials, the electric field, magnetic field and wave vector follow a left-hand rule, the reverse of the behavior of conventional optical materials.

To date, only metamaterials exhibit a negative index of refraction.