Introduction to Nonimaging Optics, 2nd Edition

14

Lagrangian and Hamiltonian Geometrical Optics

14.1 Fermat’s Principle

All geometrical optics can be derived from Fermat’s Principle, which states that, given a light ray between two points and its travel time between them, any adjacent path close to it should have the same travel time. Being a principle, it is not demonstrated, but accepted as being true and used to derive the entire mathematical framework of geometrical optics. It is, however, possible to infer why it describes the behavior of light by analyzing reflection and refraction.

The law of reflection has long been known. Back in the Hellenistic Age, the Hero of Alexandria stated that light travels along the shortest path in a homogeneous medium.¹ His reasoning is illustrated ...