Detailed derivation of the generalized Snell–Descartes laws from Fermat’s principle

Beginning with Fermat’s principle, we provide a detailed derivation of the generalized laws of refraction and reflection for a geometry realizing a metasurface. We first solve the Euler–Lagrange equations for a light ray propagating across the metasurface. The ray-path equation is found analytically, and the results are supported by numerical calculations. We get generalized laws of refraction and reflection that have three main features: (i) They are relevant in gradient-index optics and in geometrical optics; (ii) A collection of rays emerges from the metasurface as a result of multiple reflections inside the metasurface; and (iii) The laws, although derived from Fermat’s principle, differ from previously published results.