Negative Refraction - Negative Refractive Index

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 are the real valued parts of ε and µ, respectively. However, negative refraction (negative refractive index) and negative phase velocity can be distinct from each other, even in passive materials, but also in active materials.

Typically, the refractive index n is determined using, where by convention the positive square root is chosen for n. However, in NPV materials, we reverse that convention and pick the negative sign to mimic the fact that the wavevector (and hence phase velocity) 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 n must be chosen to match the physical situation.