# Representation Independent Boundary Conditions for a Piecewise-Homogeneous Linear Magneto-dielectric Medium

## TL;DR

This paper derives electromagnetic boundary conditions and Fresnel relations in a representation-independent manner for piecewise-homogeneous linear magneto-dielectric media, ensuring broad applicability across different formulations of Maxwell's equations.

## Contribution

It provides a novel, representation-independent derivation of boundary conditions and Fresnel relations, extending their validity beyond Minkowski's formulation.

## Key findings

- Derived boundary conditions using energy conservation and Stokes's theorem.
- Established the general applicability of Fresnel relations across different formulations.
- Ensured the derivations are valid for non-Minkowski continuum electrodynamics.

## Abstract

At a boundary between two transparent, linear, isotropic, homogeneous materials, derivations of the electromagnetic boundary conditions and the Fresnel relations typically proceed from the Minkowski {E,B,D,H} representation of the macroscopic Maxwell equations. However, equations of motion for macroscopic fields in a transparent linear medium can be written using Ampere {E,B}, Chu {E,H}, Lorentz, Minkowski, Peierls, Einstein-Laub, and other formulations of continuum electrodynamics. We present a representation-independent derivation of electromagnetic boundary conditions and Fresnel relations for the propagation of monochromatic radiation through a piecewise-homogeneous, transparent, linear, magneto-dielectric medium. The electromagnetic boundary conditions and the Fresnel relations are derived from energy conservation coupled with the application of Stokes's theorem to the wave equation. Our representation-independent formalism guarantees the general applicability of the Fresnel relations. Specifically, the new derivation is necessary so that a valid derivation of the Fresnel equations exists for alternative, non-Minkowski formulations of the macroscopic Maxwell field equations.