Does reflection depend on the ratio of refractive index ratios or on impedance ratios? These are not the same thing:

,

while

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My basic optics textbook (Guenther, 1990) gives two formulas for reflection at normal incidence. They look identical, except that one is in terms of refractive index ratios and the other is in terms of impedance ratios. In non-magnetic media (or in non-dielectric media), the expressions give the same total reflection. But which is correct in general?

Well, working through the boundary value problem shows that impedance version is the correct one. What does this mean? It means that* the refractive index is not intrinsically linked to the reflection and refraction coefficients*.

It turns out that if you can choose the magnetic permeability of your material then these two properties are completely decoupled. Who knew?