Electromagnetic optics theory of light at the isotropic interface: illustrating the behavior of the electric field

The electromagnetic behavior of light at the interface between isotropic materials is governed by the Fresnel formulas. These formulas primarily describe the electric field and are straightforward to interpret when dealing with transparent materials and incidence angles below the critical angle. However, when the incidence angle exceeds the critical angle or when the transmitted wave propagates through an absorbing medium, the mathematical description becomes more complex, and the physical behavior appears less intuitive. The aim of this work is to clarify these challenging scenarios at the undergraduate and graduate level by providing a practical mathematical formulation complemented by insightful graphical illustrations. We believe this work may also be a valuable resource for researchers and professionals. Therefore, for completeness, the mathematical treatment of inhomogeneous plane waves – often necessary for the second medium – is provided in <ns0:xref ref-type="app" rid="APP1">Appendix</ns0:xref>.