Optics is an important branch of physics and a required foundational course for students majoring in physics, optical engineering, and optoelectronics, among others. Snell's law and Fresnel's formulas are crucial topics in optical teaching, but recent research has revealed anomalous reflection and refraction phenomena at non-uniform interfaces, rendering traditional Snell's law obsolete. While researchers have derived the generalized Snell's law using Fermat's principle to calculate reflection and refraction angles at non-uniform interfaces, the generalized Fresnel formula for calculating the reflection coefficients has remained unexplored. In this paper, we derive the generalized Snell's law and Fresnel's formulas using Maxwell's equations and boundary conditions, providing a comprehensive framework for describing the relationships between reflection and refraction angles at both uniform and non-uniform interfaces, as well as calculating the efficiency of reflection and refraction. The theoretical formulas derived align with simulation results, confirming the correctness of the generalized Snell's law and Fresnel's formulas. This work presents a valuable attempt at innovating the teaching content of optics in universities.
Chen et al. (Sun,) studied this question.