Fermat's principle

Modern version

The historical form of Fermat's principle (The path between two points taken by a beam of light is the one which is traversed in the least time.) is incomplete. The modern, full version, of Fermat's Principle states that, the optical path length must be extremal, which means that it can be either minimal, maximal or a point of inflection (a saddle point). Minima occur most often, for instance the angle of refraction a wave takes when passing into a different medium or the path light has when reflected off of a planar mirror. Maxima occur in gravitational lensing. A point of inflection describes the path light takes when it is reflected from an elliptical mirror.

History

This principle was first stated in a letter dated January 1st, 1662, to Cureau de la Chambre by Pierre de Fermat. It was immediately met with objections made in May 1662 by Claude Clerselier, an expert in optics and leading spokesman for the Cartesians at that time. Among his objections, Claude states:

... Fermat's principle can not be the cause, for otherwise we would be attributing knowledge to nature: and here, by nature, we understand only that order and lawfulness in the world, such as it is, which acts without foreknowledge, without choice, but by a necessary determination.

Claude is correct. Fermat's statement cannot stand alone because it directly attributes the property of intention and choice to a beam of light. The principle is only strictly correct if one considers it to be a result rather than an original cause.

Derivation

Classically, Fermat's principle can be considered as a mathematical consequence of Huygens' principle. Of all secondary waves (along all possible paths) the waves with the extrema (stationary) paths contribute most due to constructive interference.