# The magnetic permeability in Fresnel's equation

## TL;DR

This paper evaluates the accuracy of Fresnel's equation using magnetic permeabilities derived for periodic media, showing it works well for 2D metamaterials mimicking magnetism but less so for 1D structures or high frequencies.

## Contribution

It provides a systematic comparison between Fresnel's equation predictions and FDTD simulations for different media, clarifying the conditions under which Fresnel's equation is valid.

## Key findings

- Fresnel's equation accurately predicts reflection for 2D metamaterials in certain frequency ranges.
- The correspondence between Fresnel's equation and simulations is poor for 1D layered structures.
- Local permittivity and permeability do not guarantee the validity of Fresnel's equation.

## Abstract

Magnetic permeabilities derived for infinite, periodic media are used in the Fresnel equation to calculate the reflection from corresponding semi-infinite media. By comparison to finite-difference-time-domain (FDTD) simulations, we find that the Fresnel equation gives accurate results for 2D metamaterials which mimic natural magnetism, in a frequency range where the magnetic moment density dominates the $\mathcal O(k^2)$ part of the total Landau--Lifshitz permittivity. For a 1D layered structure, or for large frequencies, the correspondence is poor. We also demonstrate that even if a medium is described accurately by a local permittivity and permeability, the Fresnel equation is not necessarily valid.