Computational model of Fermat’s Principle via Path Integration to illustrate the Law of Reflection, Snell’s Law and dispersion in a prism

Fermat’s Principle, or the Principle of Least Time, is used to computationally model common optical phenomena, including reflection, refraction and dispersion through a prism. Fermat’s Principle states that light will take the path between two points corresponding to the shortest time-of-flight. The time-of-flight of this path can also be associated with the optical path length (OPL). Equivalency is demonstrated between Fermat’s Principle and minimization of the action of a photon via a path integral approach. A computational approach to minimizing the action is taken when applying Fermat’s Principle to simulations of reflection and refraction at a plane. In both cases, simulations using the least-action approach to Fermat’s Principle are in exact agreement with the Law of Reflection and Snell’s Law. Fermat’s Principle and maintaining a consistent OPL for visible wavelengths passing through a prism is then modeled, thereby demonstrating its relation to dispersion of visible light in a material with refractive index that varies with wavelength. The computational approach employed in this paper would be suitable for use as an activity or demonstration for college students studying physics at both the lower- and upper-division levels.