The speed of light in a material is defined as $c = \frac{1}{\sqrt{\epsilon \mu}}$. There are metamaterials with negative permittivity $\epsilon < 0$ and permeability $\mu < 0$ at the same time. This leads to a negative refractive index of these materials.

But do (meta-) materials exist with only negative $\epsilon < 0$ and positive $\mu > 0$ or vice versa? This would lead to a complex speed of light inside such materials.

What would be the consequences of a complex speed of light? Could particles reach unlimited speed inside these materials? Would there still be Cherenkov radiation?

## Best Answer

Complex quantities always denote loss. So if the velocity is imaginary, it is impossible for a wave to travel from one point to another. If you look at the Drude model, for some certain frequency the signal will pass so it behaves like a dielectric at that time, but for frequencies lower than the Plasma frequency it will behave like a metal where no transmission is possible and at that time permittivity is less than zero, so at that time the velocity of the wave is imaginary.

So, in my opinion, imaginary velocity means no transmission.