Editing Fermat's principle (section)

{{short description|Principle of least time}}
{{for-multi|other theorems named after Pierre de Fermat|Fermat's theorem}}
{{anchor|Fig1}}[[File:Fermat Snellius.svg|thumb|'''Fig.{{nnbsp}}1''':{{big|&nbsp;}}Fermat's principle in the case of refraction of light at a flat surface between (say) air and water. Given an object-point ''A'' in the air, and an observation point ''B'' in the water, the refraction point ''P'' is that which minimizes the time taken by the light to travel the path ''APB''. If we seek the required value of ''x'', we find that the angles ''α'' and ''β'' satisfy [[Snell's&nbsp;law]].]]

'''Fermat's principle''', also known as the '''principle of least time''', is the link between [[geometrical optics|ray&nbsp;optics]] and [[physical optics|wave&nbsp;optics]]. Fermat's principle states that the path taken by a [[Ray (optics)|ray]] between two given points is the path that can be traveled in the least time.

First proposed by the French mathematician [[Pierre de Fermat]] in 1662, as a means of explaining the [[Snell's law|ordinary law of refraction]] of light (Fig.{{nnbsp}}1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature. Not until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves.<ref>Cf.{{tsp}} [[#Young1809|Young, 1809]], p.{{nnbsp}}342; [[#SecMem|Fresnel, 1827, tr.&nbsp;Hobson]], {{nowrap|pp.{{tsp}}294–6,{{tsp}}310–11}}; [[#deWitte|De Witte, 1959]], p.{{nnbsp}}293n.</ref> If points ''A'' and ''B'' are given, a [[wavefront]] expanding from ''A'' sweeps all possible ray paths radiating from ''A'', whether they pass through ''B'' or not. If the wavefront reaches point ''B'', it sweeps not only the ''ray'' path(s) from ''A'' to ''B'', but also an infinitude of nearby paths with the same endpoints. Fermat's principle describes any ray that happens to reach point&nbsp;''B''; there is no implication that the ray "knew" the quickest path or "intended" to take that path.

{{anchor|Fig2}}[[File:Fermat-Huygens principle.svg|thumb|320px|'''Fig.{{nnbsp}}2''':{{big|&nbsp;}}Two points {{mvar|P}} and {{mvar|P&prime;}} on a path from ''A'' to ''B''. For the purposes of Fermat's principle, the propagation time from {{mvar|P}} to {{mvar|P&prime;}} is taken as for a point-source at {{mvar|P}}, not (e.g.) for an arbitrary wavefront ''W'' passing through {{mvar|P}}. The surface ''&Sigma;''{{hsp}} (with unit normal '''n̂''' at {{mvar|P&prime;}}) is the locus of points that a disturbance at {{mvar|P}} can reach in the same time that it takes to reach {{mvar|P&prime;}}; in other words, ''&Sigma;'' is the secondary wavefront with radius {{mvar|PP&prime;}}. (The medium is ''not'' assumed to be homogeneous or [[isotropy|isotropic]].)]]

In its original "strong" form,<ref>Cf. [[#BW|Born & Wolf, 2002]], p.{{nnbsp}}876.</ref> Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "[[stationary point|stationary]]" with respect to variations of the path &ndash; so that a deviation in the path causes, at most, a ''second-order'' change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in ''very'' close times. It [[#A ray as a signal path (line of sight)|can be shown]] that this technical definition corresponds to more intuitive notions of a ray, such as a [[Line-of-sight propagation|line of sight]] or the path of a [[narrow beam]].

For the purpose of comparing traversal times, the time from one point to the next nominated point is taken as if the first point were a ''point-source''.<ref>[[#deWitte|De Witte (1959)]] invokes the point-source condition at the outset (p.{{nnbsp}}294, col.{{nnbsp}}1).</ref> Without this condition, the traversal time would be ambiguous; for example, if the propagation time from {{mvar|P}} to {{mvar|P&prime;}} were reckoned from an arbitrary wavefront ''W'' containing {{mvar|P}}{{hsp}} (Fig.{{nnbsp}}2), that time could be made arbitrarily small by suitably angling the wavefront.

Treating a point on the path as a source is the minimum requirement of [[Huygens' principle]], and is part of the [[#Derivation|explanation]] of Fermat's principle. But it [[#Equivalence to Huygens' construction|can also be shown]] that the geometric ''construction'' by which [[Christiaan Huygens|Huygens]] tried to apply his own principle (as distinct from the principle itself) is simply an invocation of Fermat's principle.<ref>[[#deWitte|De Witte (1959)]] gives a proof based on [[calculus of variations]]. The present article offers a [[#Equivalence to Huygens' construction|simpler explanation]].</ref> Hence all the conclusions that Huygens drew from that construction &ndash; including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of "[[Iceland spar|Iceland crystal]]" (calcite) &ndash; are also consequences of Fermat's principle.