Microscopic Theory of the Refractive Index

TL;DR

This paper reexamines the microscopic foundations of the refractive index, challenging standard formulas and deriving a more accurate relation based on fundamental electromagnetic response tensors.

Contribution

It provides a theoretical justification for using n^2 ≈ ε_r at optical wavelengths and clarifies the connection between covariant response tensors and optical properties.

Findings

Standard formula n^2 = ε_r μ_r conflicts with microscopic principles.

The relation n^2 ≈ ε_r is justified at optical wavelengths.

A general theorem links the wave equation to the microscopic dielectric tensor.

Abstract

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.