Can the speed of light become complex inside a metamaterial? 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? 
 A: 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.
A: Technically, all of these materials will have (effective) complex dielectric and/or magnetic properties.  Thus, you'll be dealing with complex wave speeds.
Complex speed means lossy transmission.
The amplitude of the wave will decay with a decay constant that is inversely proportional to the imaginary part of the square root of the speed.
Note that if this term is small, or the material is thin enough, then you can have transmission.
A: Since there's only one square root, one would end up with an imaginary velocity rather than the more general case of a complex velocity.
In physics, one occasionally deals with complex energies. I don't know what imaginary velocities will mean and there's not a lot of references in the literature. One paper is "Velocities" in Quantum Mechanics
