Constitutive relations for electromagnetic field in a form of $6\times 6$ matrices derived from the geometric algebra

TL;DR

This paper derives a universal matrix form of constitutive relations in electromagnetism from geometric algebra, facilitating analysis of complex media properties in various electromagnetic applications.

Contribution

It transforms relativistic constitutive relations into 6x6 matrices, bridging geometric algebra and vector calculus for diverse media types.

Findings

Derived 6x6 matrix form of constitutive relations

Unified approach for anisotropic and bianisotropic media

Facilitates analysis of electromagnetic wave propagation

Abstract

To have a closed system, the Maxwell equations should be supplemented by constitutive relations which connect the primary electromagnetic fields $(\bE,\bB)$ with the secondary ones $(\bD,\bH)$ induced in a medium. Recently [Opt. Commun. \textbf{354}, 259 (2015)] the allowed shapes of the constitutive relations that follow from the relativistic Maxwell equations formulated in terms of geometric algebra were constructed by author. In this paper the obtained general relativistic relations between $(\bD,\bH)$ and $(\bE,\bB)$ fields are transformed to four $6\times 6$ matrices that are universal in constructing various combinations of constitutive relations in terms of more popular Gibbs-Heaviside vectorial calculus frequently used to investigate the electromagnetic wave propagation in anisotropic, birefringent, bianisotropic, chiral etc media.