= Refractive index and electromagnetic response
{title2=$n^2=\varepsilon_r\mu_r$}
For a uniform linear medium with <isotropy> and local constitutive response, <Maxwell equations> give $k^2=\omega^2\varepsilon(\omega)\mu(\omega)$. Here $\varepsilon$ is <permittivity> and $\mu$ is <magnetic permeability>. Since $c^{-2}=\varepsilon_0\mu_0$, the <relative permittivity> $\varepsilon_r=\varepsilon/\varepsilon_0$ and the relative <magnetic permeability> $\mu_r=\mu/\mu_0$ obey
$$
n^2=\varepsilon_r\mu_r.
$$
If both responses are real and positive with negligible loss, and the ordinary positive-index branch is intended, then $n=+\sqrt{\varepsilon_r\mu_r}$. If additionally $\mu_r\simeq1$, then $n\simeq\sqrt{\varepsilon_r}$. All these responses must be evaluated at the <electromagnetic wave>'s <frequency>; a static <relative permittivity> cannot simply be substituted at an optical <frequency>. <RP Photonics Encyclopedia by Rüdiger Paschotta/Refractive index>[RP Photonics Encyclopedia by Rüdiger Paschotta; What is a Refractive Index?: squared constitutive relation and frequency-specific material responses]{ref}
With loss, use a complex index $\widetilde n=n'+i\kappa$ and explicitly fix the <phase convention for complex amplitudes>. For $E\propto\exp[i(kz-\omega t)]$ with $k=\widetilde n\omega/c$, its <wave amplitude> has factor $\exp(-\kappa\omega z/c)$. A forward attenuating branch has $\kappa\geq0$, while its <wave phase> is controlled by $n'$. The complex relation $\widetilde n^2=\varepsilon_r\mu_r$ alone leaves a choice of sign: the physical solution also needs <causality> and the appropriate radiation or decay condition. In negative-index media the physically selected real part can be negative. The positive square root is therefore not a universal rule for complex or negative-index media. Direction and <polarization of an electromagnetic wave> also matter in a <plane wave in an anisotropic dielectric>, for which one scalar index need not describe every propagation branch.
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