  We examine the refractive index from the viewpoint of modern first-principles
materials physics. We first argue that the standard formula, $n^2 =
\varepsilon_{\mathrm r} \mu_{\mathrm r}$, is generally in conflict with
fundamental principles on the microscopic level. Instead, it turns out that an
allegedly approximate relation, $n^2 = \varepsilon_{\mathrm r}$, which is
already being used for most practical purposes, can be justified theoretically
at optical wavelengths. More generally, starting from the fundamental,
Lorentz-covariant electromagnetic wave equation in materials as used in plasma
physics, we rederive a well-known, three-dimensional form of the wave equation
in materials and thereby clarify the connection between the covariant
fundamental response tensor and the various cartesian tensors used to describe
optical properties. Finally, we prove a general theorem by which the
fundamental, covariant wave equation can be reformulated concisely in terms of
the microscopic dielectric tensor.
