Refraction of Light at Meta Material Interface with Negative Index

Typically, materials possess a refractive index $n > 1$. Hence, whenever a light ray travels from air into a naturally occurring substance, according to Snell's law, $\frac{\sin \theta_1}{\sin \theta_2} = \frac{n_2}{n_1}$, the refracted ray bends toward the normal. However, it never emerges on the same side of the normal as the incident ray. From the theory of electromagnetism, the refractive index of a medium is expressed by the relation $n = \left(\frac{c}{v}\right) = \pm \sqrt{\varepsilon_r \mu_r},$ in which $c$ denotes the speed of electromagnetic waves in vacuum, $v$ is its speed in the medium, and $\varepsilon_r, \mu_r$ represent the medium's relative permittivity and relative permeability, respectively.

In conventional substances, both $\varepsilon_r$ and $\mu_r$ are positive, implying a positive value of $n$ for the medium. When both $\varepsilon_r$ and $\mu_r$ are negative, one must choose the negative root of $n$. Such negative refractive index materials can now be synthesized artificially and are called meta-materials. They exhibit significantly different optical behavior, without violating any physical laws. Since $n$ is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.