Fermat's principle

Fermat's principle leads to Snell's law; when the sines of the angles in the different media are in the same proportion as the propagation velocities, the time to get from P to Q is minimized.

In optics, Fermat's principle or the principle of least time is the principle 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.^[1] However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary, not minimal, time.

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 of the French mathematician Pierre de 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 when a wave passes from medium into another refraction and in the reflection of light from 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.

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.^[2]

The generalized principle of least time in its modern form was stated by 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.^{[citation needed]}

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).

Notes

1. ^ Arthur Schuster, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 online.
2. ^ Pavlos Mihas (2005). Use of History in Developing ideas of refraction, lenses and rainbow, Demokritus University, Thrace, Greece.