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Negative refraction in active and passive media: a discussion of various criteria

There are three widely-used conditions for characterizing negative refraction in isotropic dielectric-magnetic materials. Here we demonstrate that whilst all the different conditions are equivalent for purely passive media, they are distinct if active media are considered. Further, these criteria can also be applied to negative refraction in acoustic materials, where we might replace the dielectric permittivity $ε$ with a bulk modulus $κ$, and the magnetic permeability $μ$ with the mass density $ρ$.