Negative wave refraction violates finite signal speed or causality.

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Our simple argument against negative wave refraction.
Assume for a moment that the line BA represents a signal (not phase) wavefront (perpendicular to the incoming bundle of parallel rays shown as arrows) marching forward at speed = c/refractive index. This is like a row of soldiers marching, each soldier has orders to march at a fixed speed defined by the index. Suppose this row of soldiers can refract negatively AND keep marching forward, i.e., BA goes to BC and then to BD in some finite amount of time.

But then soldier B has not moved at all while the soldier A went from A to C to D. Since B is supposed to keep moving at a fixed speed, A has zero time to go to C and then to D, i.e., A must move infinitely fast to C to D. This violates relativity. Hince no real signal or energy front can travel causally forward and negatively refract.

As was done in all previous LHM literature, if the phase speed assumed to approximate the signal speed, negative (phase) index on right would mean light should go backward in LHM, appearing on right before it arrives at the interface from left. That would violate causality.

However, if the "front" BD in NIM on right is seen as a phase front for a single-frequency carrier travelling backwards, there is no problem because only modulations carry information. By itself, each (non-normalizable) single frequency infinite plane wave component is forever and carries no markers so causality arguments do not apply.

See Avi showing how Positive forward signal refraction is fine, but Negative forward signal refraction is not allowed.
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