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VortexLattice

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I'm trying to solve a problem in which I have a uniaxial (uniform in two directions, different in the other) crystal with the optical axis normal to the plane of reflection. I call the permittivity parallel to the optical axis [itex]ε_par[/itex] and the permittivity in the other two directions (in the plane perpendicular to the optical axis [itex]ε_perp[/itex]. The magnetic permeability is isotropic and uniform.

Now, I want to investigate what happens when the index of refraction is negative in ONE direction, not both. This means that we need to say the magnetic permeability is negative (say -1 here) and the permittivity in the direction we want the negative refractive index is also negative. Then we take the negative square root: [itex] n = \sqrt(εμ)[/itex] and everything is dandy.

But that leaves the index in the other direction -- This one has positive ε and negative μ, so its refractive index is imaginary, which usually means it is absorbed by the material. And a paper I've been reading does say this, as long as the it is polarized in this direction. But what if it's propagating in between these two axes? It seems like the component of the electric field parallel to the imaginary refractive index should just die, but the other component should happily continue.

I'm really confused. Can anyone help??

Thanks!