Polarization and energy ellipsoids for an introductory visualization of tensors

TL;DR

This paper explores two visualization methods for the polarization tensor—energy ellipsoid and polarization ellipsoid—using geometric representations to aid understanding of tensor properties in an introductory physics context.

Contribution

It introduces an alternative visualization approach for the polarization tensor based on polarization ellipsoids, comparing it with the traditional energy ellipsoid method.

Findings

The polarization ellipsoid offers a new geometric visualization of the polarization tensor.

Comparison shows differences and similarities between the two visualization methods.

The approach enhances understanding of tensor properties in introductory physics.

Abstract

In ``The Feynman Lectures on Physics'' is discussed an introduction to tensors by means of the polarization tensor, including a way of ``visualizing'' this tensor via the energy ellipsoid, which is drawn by the electric fields which produce the same polarization energy density in an anisotropic crystal. Here, we discuss an alternative way of visualizing the polarization tensor, by means of the polarization ellipsoid, which is based on the ideas of Lam\'e's stress ellipsoid and is drawn by the polarization vectors produced by electric fields having the same magnitude. We compare both ellipsoids as a first introductory way of visualizing the polarization tensor.