Fermat's principle

In optics, Fermat's principle or the principle of least time is the idea that "the path taken between two points by a ray of light is the path that can be traversed in the least time." This principle is sometimes taken as the definition of a ray of light. [Arthur Schuster, "An Introduction to the Theory of Optics", London: Edward Arnold, 1904 [http://books.google.com/books?vid=OCLC03146755&id=X0AcBd-bcCwC&pg=PA41&lpg=PA41&dq=fermat%27s-principle online] .]

Fermat's Principle can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection. It can be deduced from Huygens' principle, and can be used to derive Snell's law of refraction and the law of reflection.

Modern version

The historical form due to Fermat is incomplete. The modern, full version of Fermat's Principle states that the optical path length must be "stationary", 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 a planar mirror. Maxima occur in gravitational lensing. A point of inflection describes the path light takes when it is reflected off an elliptical mirrored surface. The incompleteness of the principle can be explained in the following experiment: Place a glass brick between a light source and an observer. The optically shortest path would be a segmented line from the source around the brick to the observer's eye, just along the surface of the brick. In fact, the light source is seen through the brick - the light beam is straight. The statement "stationary" instead of "minimal" means, mathematically, that the principle doesn't apply to the extrema where the dependence is not contiguous.

History

Hero of Alexandria (Heron) (c. 60) described a principle of reflection, which stated that a ray of light that goes from point A to point B, suffering any number of reflections on flat mirrors, in the same medium, has a smaller path length than any nearby path.

Ibn al-Haytham (Alhacen), in his "Book of Optics" (1021), expanded the principle to both reflection and refraction, and expressed an early version of the principle of least time. [Pavlos Mihas (2005). [http://www.ihpst2005.leeds.ac.uk/papers/Mihas.pdf Use of History in Developing ideas of refraction, lenses and rainbow] , Demokritus University, Thrace, Greece.]

The generalized principle of least time in its modern form was stated by Pierre de Fermat in a letter dated January 1, 1662, to Cureau de la Chambre. 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. Amongst 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."

Indeed Fermat's statement does not hold standing alone, as it directly attributes the property of intention and choice to a beam of light. However, Fermat's principle is in fact correct if one considers it to be a result rather than the original cause.

Derivation

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

Fermat's principle can be derived from the main principle of Quantum Electrodynamics stating that any particle (e.g. a photon or electron) propagates over all possible paths and the interference (sum) of all possible wavefunctions (at the point of observer or detector) gives the correct probability of detection of this particle (at this point). Thus all paths except extremal (shortest, longest or stationary) cancel each other out.

In the classic mechanics of waves, Fermat's principle follows from the extremum principle of mechanics (see variational principle).

References