Veselago medium

Example

A Veselago medium is a homogeneous, isotropic, lossless medium whose relative permittivity $\varepsilon$ and relative permeability $\mu$ are both negative at the frequency considered. It is named after Victor Veselago, who studied the theory of such media in the 1960s.

Check that it carries waves and that it refracts negatively. Take a plane wave $E = E_0 e^{i(k \cdot x - \omega t)}$, $H = H_0 e^{i(k \cdot x - \omega t)}$ with real vectors $k$, $E_0$, $H_0$ and $k \cdot E_0 = 0$. Maxwell's equations give $k \times E_0 = \omega \mu \mu_0 H_0$ and $k \times H_0 = -\omega \varepsilon \varepsilon_0 E_0$. They have nonzero solutions when $\abs{k}^2 = \varepsilon \mu \, \omega^2 / c^2$, and $\varepsilon \mu > 0$, so plane waves propagate. Since $\mu < 0$, the vector $H_0$ points opposite to $k \times E_0$, so $(E_0, H_0, k)$ is a left-handed triple. The time-averaged Poynting vector is $$\tfrac12 E_0 \times H_0 = \frac{E_0 \times (k \times E_0)}{2 \omega \mu \mu_0} = \frac{\abs{E_0}^2}{2 \omega \mu \mu_0} \, k,$$ which points opposite to $k$. So energy flows against the phase velocity. Since energy must flow away from a source, the refractive index is $n = -\sqrt{\varepsilon \mu}$. At an interface with vacuum, Snell's law $\sin \theta_1 = n \sin \theta_2$ then puts the refracted ray on the same side of the normal as the incident ray. This is negative refraction.