Fermat's principle
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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.[1]
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.
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[edit] 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 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 off of an elliptical mirrored surface.
[edit] History
Fermat's principle is the generalization, to include refraction, of the Principle of Hero of Alexandria. Around 60 AD, Hero (also known as Heron) proved 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.
The general 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. 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 principle is in fact correct if one considers it to be a result rather than the original cause.
[edit] 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 main principle of Quantum Electrodynamics stating that any particle (for example, a photon or an electron, etc.) 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 classic mechanics of waves Fermat principle follows from the extremum principle of mechanics (see variational principle).
