Fermat Principle and Snell’s Law of Refraction

We review the classical Fermat principle, a fundamental result in the calculus of variations. By determining the Euler–Lagrange equations for the optical path length functional in homogeneous, isotropic media, we review how to obtain the classical ray equations in terms of the refractive index of a material. Staying in the variational framework, we discuss how the Weierstrass–Erdmann corner condition for parametric variational integrals leads to the classical Snell’s Law of refraction. We then discuss many variants of Snell’s Law, looking at both refraction and reflection. Finally, we focus on liquid crystal optics by taking the optical path length functional to be the one with integrand being the effective refractive index obtained in the previous chapter. Connections to recent results in the literature are provided along the way, and we end with a discussion about Snell’s Law with absorption.