J. Eur. Opt. Society-Rapid Publ. 2025, 21, 21 Ó The Author( s), published by EDP Sciences, 2025 https:// doi. org / 10.1051 / jeos / 2025019 Available online at: https:// jeos. edpsciences. org
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE
Electromagnetic optics theory of light at the isotropic interface: illustrating the behavior of the electric field
Salvador Bosch * and Oriol Arteaga Universitat de Barcelona, Dep. Física Aplicada, Martí i Franqués 1, 08028 Barcelona, Spain
Received 3 February 2025 / Accepted 8 April 2025
Abstract. 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 Appendix.
Keywords: Electromagnetic optics, Fresnel formulas, Inhomogeneous waves, Evanescent waves.
1 Introduction and aim
A fundamental topic in optics( and physics) is the application of electromagnetic theory( Maxwell’ s equations) to the light reflection and refraction at a plane boundary between transparent materials. Understanding how light rays bend at an interface and determining the ratio of the amplitudes of reflected and refracted rays to the incident one are essential concepts in this field. The classical analysis, based on the principle of continuity of tangential electric and magnetic fields at the interface, leads to the well-known Fresnel equations. These equations represent a remarkable achievement in physics, made even more impressive by the fact that they were firstformulatedin1823 [ 1 ]. At the level of a physics degree, providing a clear and comprehensive explanation of the phenomenon is an important objective. This constitutes the central purpose of the present paper.
By convention, the Fresnel coefficients at an interface are defined as the ratios of the electric field amplitudes of the reflected and refracted waves to that of the incident wave. The demonstration of the formulas is done by assuming the coexistence of three homogeneous plane waves( incident, reflected and transmitted ones, see Fig. 1) for the case of external reflection between transparent materials, going from a refractive index of the incident medium n i to a refractive index of the emergent medium n t, with n = n t / n i > 1 [ 2 – 7 ]. By imposing the continuity of the
* Corresponding author: sbosch @ ub. edu tangential components of the electric and magnetic fields through geometric projection on the interface, a system of four equations is obtained. Following the formulation in [ 3 ], the Fresnel formulas can be written as:
r? E? r
E? i
¼ n i cos h i � n t cos h t n i cos h i þ n t cos h t
; r jj Ejj r ¼ n i cos h t � n t cos h i
;
E jj n i i cos h t þ n t cos h i
t? E? t
E? i
¼
2n i cos h i n i cos h i þ n t cos h t
; t jj Ejj t 2n i cos h i ¼
:
E jj n i i cos h t þ n t cos h i ð1Þ
ð2Þ
ð3Þ
ð4Þ
It should be noted that there are no major differences in the procedures developed in references [ 2 – 7 ] for obtaining the formulas and we will not compare them. The only significant( but obvious) detail is to note that the sign of r || in textbooks depends on the convention chosen for the positive direction of the reflected electric field [ 8 ]. It is evident that the geometrical procedure outlined above for the derivation of the Fresnel formulas is not applicable in two specific situations:( i) internal reflection beyond the critical angle, and( ii) when the second medium is absorbing. These cases
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