Simulating the Action Principle in Optics

Abstract

Light has a fascinating property: it always travels the path that takes the least time between any two points. This is the motivating property behind optical phenomena such as reflection and refraction. The unreasonable economic efficiency of light is captured by a single proposition: the principle of least action (PLA) in optics. Unlike reflection and refraction, which emerge from optimizing a one-dimensional function, the PLA emerges from optimizing an infinite-dimensional functional. The PLA can be difficult for students to comprehend, as the formulation of the Lagrangian is often left unexplained. To this end, this paper presents various simulations to demonstrate the action principle, including a numerical solution to a generalization of the brachistochrone problem to an arbitrary refractive profile. The interactive simulations discussed in the paper are available at Ref. 1.