{"history":[],"article":{"id":"99b41408-d688-4cda-9673-c9ccf6508b8f","title":"Fermat's principle","slug":"fermat-s-principle","content":":::summary\n:::\n\n# \n**Fermat's principle**, also known as the **principle of least time**, is the link between [ray optics](https://bliptext.com/articles/geometrical-optics) and [wave optics](https://bliptext.com/articles/physical-optics).\nFermat's principle states that the path taken by a [ray](https://bliptext.com/articles/ray-optics) between two given points is the path that can be traveled in the least time.\nFirst proposed by the French mathematician [Pierre de Fermat](https://bliptext.com/articles/pierre-de-fermat) in 1662, as a means of explaining the [ordinary law of refraction](https://bliptext.com/articles/snell-s-law) of light (Fig.1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature.\nNot until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves.\nIf points A and B are given, a [wavefront](https://bliptext.com/articles/wavefront) expanding from A sweeps all possible ray paths radiating from A, whether they pass through B or not.\nIf 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.\nFermat's principle describes any ray that happens to reach point B; there is no implication that the ray \"knew\" the quickest path or \"intended\" to take that path.\nIn its original \"strong\" form, 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.\nIn order to be true in all cases, this statement must be weakened by replacing the \"least\" time with a time that is \"[stationary](https://bliptext.com/articles/stationary-point)\" with respect to variations of the path – so that a deviation in the path causes, at most, a second-order change in the traversal time.\nTo put it loosely, a ray path is surrounded by close paths that can be traversed in very close times.\nIt [can be shown](https://bliptext.com/articles/) that this technical definition corresponds to more intuitive notions of a ray, such as a [line of sight](https://bliptext.com/articles/line-of-sight-propagation) or the path of a [narrow beam](https://bliptext.com/articles/narrow-beam).\nFor 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.\nWithout this condition, the traversal time would be ambiguous; for example, if the propagation time from P to P′ were reckoned from an arbitrary wavefront W containing P (Fig.2), that time could be made arbitrarily small by suitably angling the wavefront.\nTreating a point on the path as a source is the minimum requirement of [Huygens' principle](https://bliptext.com/articles/huygens-principle), and is part of the [explanation](https://bliptext.com/articles/) of Fermat's principle.\nBut it [can also be shown](https://bliptext.com/articles/) that the geometric construction by which [Huygens](https://bliptext.com/articles/christiaan-huygens) tried to apply his own principle (as distinct from the principle itself) is simply an invocation of Fermat's principle.\nHence all the conclusions that Huygens drew from that construction – including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of \"[Iceland crystal](https://bliptext.com/articles/iceland-spar)\" (calcite) – are also consequences of Fermat's principle.\n# [Fig.1: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 law. | upload.wikimedia.org/wikipedia/commons/d/d9/Fermat///Snellius.svg]\n# [Fig.2:Two points P and P′ on a path from A to B. For the purposes of Fermat's principle, the propagation time from P to P′ is taken as for a point-source at P, not (e.g.) for an arbitrary wavefront W passing through P. The surface Σ (with unit normal n̂ at P′) is the locus of points that a disturbance at P can reach in the same time that it takes to reach P′; in other words, Σ is the secondary wavefront with radius PP′. (The medium is not assumed to be homogeneous or isotropic.) | upload.wikimedia.org/wikipedia/commons/6/6b/Fermat-Huygens///principle.svg]\n\n# Derivation\n\n## Sufficient conditions\nLet us suppose that:\nThen the various propagation paths from A to B will help each other, or interfere constructively, if their traversal times agree within the said tolerance.\nFor a small tolerance (in the limiting case), the permissible range of variations of the path is maximized if the path is such that its traversal time is stationary with respect to the variations, so that a variation of the path causes at most a second-order change in the traversal time.\nThe most obvious example of a stationarity in traversal time is a (local or global) minimum – that is, a path of least time, as in the \"strong\" form of Fermat's principle.\nBut that condition is not essential to the argument.\nHaving established that a path of stationary traversal time is reinforced by a maximally wide corridor of neighboring paths, we still need to explain how this reinforcement corresponds to intuitive notions of a ray.\nBut, for brevity in the explanations, let us first define a ray path as a path of stationary traversal time.\n\n## A ray as a signal path (line of sight)\nIf the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly alter the disturbance reaching B from A – unlike a similar-sized obstruction outside any such corridor, blocking paths that do not reinforce each other.\nThe former obstruction will significantly disrupt the signal reaching B from A, while the latter will not; thus the ray path marks a signal path.\nIf the signal is visible light, the former obstruction will significantly affect the appearance of an object at A as seen by an observer at B, while the latter will not; so the ray path marks a line of sight.\nIn optical experiments, a line of sight is routinely assumed to be a ray path.\n\n## A ray as an energy path (beam)\nIf the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly affect the energy reaching B from A – unlike a similar-sized obstruction outside any such corridor.\nThus the ray path marks an energy path – as does a beam.\nSuppose that a wavefront expanding from point A passes point P, which lies on a ray path from point A to point B.\nBy definition, all points on the wavefront have the same propagation time from A.\nNow let the wavefront be blocked except for a window, centered on P, and small enough to lie within the corridor of paths that reinforce the ray path from A to B.\nThen all points on the unobstructed portion of the wavefront will have, nearly enough, equal propagation times to B, but not to points in other directions, so that B will be in the direction of peak intensity of the beam admitted through the window.\nSo the ray path marks the beam.\nAnd in optical experiments, a beam is routinely considered as a collection of rays or (if it is narrow) as an approximation to a ray (Fig.3).\n# [Fig.3:An experiment demonstrating refraction (and partial reflection) of rays – approximated by, or contained in, narrow beams | upload.wikimedia.org/wikipedia/commons/3/32/Concave///lens.jpg]\n\n## Analogies\nAccording to the \"strong\" form of Fermat's principle, the problem of finding the path of a light ray from point A in a medium of faster propagation, to point B in a medium of slower propagation ([Fig.1](https://bliptext.com/articles/)), is analogous to the problem faced by a [lifeguard](https://bliptext.com/articles/lifeguard) in deciding where to enter the water in order to reach a drowning swimmer as soon as possible, given that the [lifeguard](https://bliptext.com/articles/lifeguard) can run faster than (s)he can swim.\nBut that analogy falls short of explaining the behavior of the light, because the lifeguard can think about the problem (even if only for an instant) whereas the light presumably cannot.\nThe discovery that ants are capable of similar calculations does not bridge the gap between the animate and the inanimate.\nIn contrast, the above assumptions (1) to (3) hold for any wavelike disturbance and explain Fermat's principle in purely [mechanistic](https://bliptext.com/articles/mechanical-philosophy) terms, without any imputation of knowledge or purpose.\nThe principle applies to waves in general, including (e.g.) sound waves in fluids and elastic waves in solids.\nIn a modified form, it even works for [matter waves](https://bliptext.com/articles/matter-wave): in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), the [classical path](https://bliptext.com/articles/classical-limit) of a particle is obtainable by applying Fermat's principle to the associated wave – except that, because the frequency may vary with the path, the stationarity is in the [phase shift](https://bliptext.com/articles/phase-shift) (or number of cycles) and not necessarily in the time.\nFermat's principle is most familiar, however, in the case of visible [light](https://bliptext.com/articles/light): it is the link between [geometrical optics](https://bliptext.com/articles/geometrical-optics), which describes certain optical phenomena in terms of rays, and the wave theory of [light](https://bliptext.com/articles/light), which explains the same phenomena on the hypothesis that [light](https://bliptext.com/articles/light) consists of waves.\n\n# Equivalence to Huygens' construction\nIn this article we distinguish between Huygens' principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens' construction, which is described below.\nLet the surface W be a wavefront at time t, and let the surface W′ be the same wavefront at the later time\nt + Δt\n(Fig.4).\nLet P be a general point on W. Then, according to Huygens' construction,\n

W′ is the [envelope](https://bliptext.com/articles/envelope-mathematics) (common tangent surface), on the forward side of W, of all the secondary wavefronts each of which would expand in time\nΔt\nfrom a point on W, and
if the secondary wavefront expanding from point P in time\nΔt\ntouches the surface W′ at point P′, then P and P′ lie on a ray.

\nThe construction may be repeated in order to find successive positions of the primary wavefront, and successive points on the ray.\nThe ray direction given by this construction is the radial direction of the secondary wavefront, and may differ from the normal of the secondary wavefront (cf. [Fig.2](https://bliptext.com/articles/)), and therefore from the normal of the primary wavefront at the point of tangency.\nHence the ray velocity, in magnitude and direction, is the radial velocity of an infinitesimal secondary wavefront, and is generally a function of location and direction.\nNow let Q be a point on W close to P, and let Q′ be a point on W′ close to P′.\nThen, by the construction,\n

the time taken for a secondary wavefront from P to reach Q′ has at most a second-order dependence on the displacement P′Q′, and
the time taken for a secondary wavefront to reach P′ from Q has at most a second-order dependence on the displacement PQ.

\nBy (i), the ray path is a path of stationary traversal time from P to W′; and by (ii), it is a path of stationary traversal time from a point on W to P′.\nSo Huygens' construction implicitly defines a ray path as a path of stationary traversal time between successive positions of a wavefront, the time being reckoned from a point-source on the earlier wavefront.\nThis conclusion remains valid if the secondary wavefronts are reflected or refracted by surfaces of discontinuity in the properties of the medium, provided that the comparison is restricted to the affected paths and the affected portions of the wavefronts.\nFermat's principle, however, is conventionally expressed in point-to-point terms, not wavefront-to-wavefront terms.\nAccordingly, let us modify the example by supposing that the wavefront which becomes surface W at time t, and which becomes surface W′ at the later time\nt + Δt\n, is emitted from point A at time\n0\n. Let P be a point on W (as before), and B a point on W′.\nAnd let A, W, W′, and B be given, so that the problem is to find P.\nIf P satisfies Huygens' construction, so that the secondary wavefront from P is tangential to W′ at B, then PB is a path of stationary traversal time from W to B. Adding the fixed time from A to W, we find that APB is the path of stationary traversal time from A to B (possibly with a restricted domain of comparison, as noted above), in accordance with Fermat's principle.\nThe argument works just as well in the converse direction, provided that W′ has a well-defined tangent plane at B. Thus Huygens' construction and Fermat's principle are geometrically equivalent.\nThrough this equivalence, Fermat's principle sustains Huygens' construction and thence all the conclusions that Huygens was able to draw from that construction.\nIn short, \"The laws of geometrical optics may be derived from Fermat's principle\".\nWith the exception of the Fermat–Huygens principle itself, these laws are special cases in the sense that they depend on further assumptions about the media.\nTwo of them are mentioned under the next heading.\n# [Fig.4:Two iterations of Huygens' construction. In the first iteration, the later wavefront W′ is derived from the earlier wavefront W by taking the envelope of all the secondary wavefronts (gray arcs) expanding in a given time from all the points (e.g., P) on W. The arrows show the ray directions. | upload.wikimedia.org/wikipedia/commons/8/8b/Huygens-construction-two-iterations.svg]\n\n# Special cases\n\n## Isotropic media: rays normal to wavefronts\nIn an isotropic medium, because the propagation speed is independent of direction, the secondary wavefronts that expand from points on a primary wavefront in a given infinitesimal time are spherical, so that their radii are normal to their common tangent surface at the points of tangency.\nBut their radii mark the ray directions, and their common tangent surface is a general wavefront.\nThus the rays are normal (orthogonal) to the wavefronts.\nBecause much of the teaching of optics concentrates on isotropic media, treating anisotropic media as an optional topic, the assumption that the rays are normal to the wavefronts can become so pervasive that even Fermat's principle is explained under that assumption, although in fact Fermat's principle is more general.\n\n## Homogeneous media: rectilinear propagation\nIn a homogeneous medium (also called a uniform medium), all the secondary wavefronts that expand from a given primary wavefront W in a given time\nΔt\nare [congruent](https://bliptext.com/articles/congruence-geometry) and similarly oriented, so that their envelope W′ may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center (source) moves over W. If P is its center while P′ is its point of tangency with W′, then P′ moves parallel to P, so that the plane tangential to W′ at P′ is parallel to the plane tangential to W at P. Let another ([congruent](https://bliptext.com/articles/congruence-geometry) and similarly orientated) secondary wavefront be centered on P′, moving with P, and let it meet its envelope W″ at point P″.\nThen, by the same reasoning, the plane tangential to W″ at P″ is parallel to the other two planes.\nHence, due to the congruence and similar orientations, the ray directions PP′ and P′P″ are the same (but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical).\nThis construction can be repeated any number of times, giving a straight ray of any length.\nThus a homogeneous medium admits rectilinear rays.\n\n# Modern version\n\n## Formulation in terms of refractive index\nLet a path\nΓ\nextend from point A to point B. Let s be the arc length measured along the path from A, and let t be the time taken to traverse that arc length at the ray speed (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path).\nThen the traversal time of the entire path\nΓ\nis\n(where A and B simply denote the endpoints and are not to be construed as values of t or s).\nThe condition for\nΓ\nto be a ray path is that the first-order change in T due to a change in\nΓ\nis zero; that is,\nNow let us define the optical length of a given path ([optical path length](https://bliptext.com/articles/optical-path-length), OPL) as the distance traversed by a ray in a homogeneous isotropic reference medium (e.g., a vacuum) in the same time that it takes to traverse the given path at the local ray velocity.\nThen, if c denotes the propagation speed in the reference medium (e.g., the speed of light in vacuum), the optical length of a path traversed in time dt is\ndS = c dt\n, and the optical length of a path traversed in time T is\nS = cT\n. So, multiplying equation **(1)** through by c, we obtain\nwhere is the ray index – that is, the [refractive index](https://bliptext.com/articles/refractive-index) calculated on the ray velocity instead of the usual [phase velocity](https://bliptext.com/articles/phase-velocity) (wave-normal velocity).\nFor an infinitesimal path, we have indicating that the optical length is the physical length multiplied by the ray index: the OPL is a notional geometric quantity, from which time has been factored out.\nIn terms of OPL, the condition for\nΓ\nto be a ray path (Fermat's principle) becomes\nThis has the form of [Maupertuis's principle](https://bliptext.com/articles/maupertuis-s-principle) in [classical mechanics](https://bliptext.com/articles/classical-mechanics) (for a single particle), with the ray index in optics taking the role of momentum or velocity in mechanics.\nIn an isotropic medium, for which the ray velocity is also the phase velocity, we may substitute the usual refractive index n for\nnr\n.\n\n## Relation to Hamilton's principle\nIf\nx\n,\ny\n,\nz\nare Cartesian coordinates and an overdot denotes differentiation with respect to s, Fermat's principle **(2)** may be written\nIn the case of an isotropic medium, we may replace\nnr\nwith the normal refractive index\nn(x, y, z)\n, which is simply a [scalar field](https://bliptext.com/articles/scalar-field).\nIf we then define the optical [Lagrangian](https://bliptext.com/articles/lagrangian-mechanics) as\nFermat's principle becomes\nIf the direction of propagation is always such that we can use z instead of s as the parameter of the path (and the overdot to denote differentiation w.r.t. z instead of s), the optical Lagrangian can instead be written\nso that Fermat's principle becomes\nThis has the form of [Hamilton's principle](https://bliptext.com/articles/hamilton-s-principle) in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics.\nThe optical Lagrangian is the function which, when integrated w.r.t. the parameter of the path, yields the OPL; it is the foundation of [Lagrangian and Hamiltonian optics](https://bliptext.com/articles/hamiltonian-optics).\n\n# History\nIf a ray follows a straight line, it obviously takes the path of least length.\n[Hero of Alexandria](https://bliptext.com/articles/hero-of-alexandria), in his [Catoptrics](https://bliptext.com/articles/catoptrics) (1st century CE), showed that the ordinary [law of reflection](https://bliptext.com/articles/specular-reflection) off a plane surface follows from the premise that the total length of the ray path is a minimum.\n[Ibn al-Haytham](https://bliptext.com/articles/ibn-al-haytham), an 11th-century polymath later extended this principle to refraction, hence giving an early version of the Fermat's principle.\n\n## Fermat vs. the Cartesians\nIn 1657, Pierre de Fermat received from [Marin Cureau de la Chambre](https://bliptext.com/articles/marin-cureau-de-la-chambre) a copy of newly published treatise, in which La Chambre noted Hero's principle and complained that it did not work for refraction.\nFermat replied that refraction might be brought into the same framework by supposing that light took the path of least resistance, and that different media offered different resistances.\nHis eventual solution, described in a letter to La Chambre dated 1 January 1662, construed \"resistance\" as inversely proportional to speed, so that light took the path of least time.\nThat premise yielded the ordinary law of refraction, provided that light traveled more slowly in the optically denser medium.\nFermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a [variational principle](https://bliptext.com/articles/variational-principle) or [action principle](https://bliptext.com/articles/action-physics), setting the precedent for the [principle of least action](https://bliptext.com/articles/principle-of-least-action) in classical mechanics and the corresponding principles in other fields (see History of [variational principle](https://bliptext.com/articles/variational-principle)s in physics).\nIt was the more notable because it used the method of [adequality](https://bliptext.com/articles/adequality), which may be understood in retrospect as finding the point where the slope of an infinitesimally short [chord](https://bliptext.com/articles/chord-geometry) is zero, without the intermediate step of finding a general expression for the slope (the [derivative](https://bliptext.com/articles/derivative)).\nIt was also immediately controversial.\nThe ordinary law of refraction was at that time attributed to [René Descartes](https://bliptext.com/articles/ren-descartes) (d.1650), who had tried to explain it by supposing that light was a force that propagated instantaneously, or that light was analogous to a tennis ball that traveled faster in the denser medium, either premise being inconsistent with Fermat's. Descartes' most prominent defender, [Claude Clerselier](https://bliptext.com/articles/claude-clerselier), criticized Fermat for apparently ascribing knowledge and intent to nature, and for failing to explain why nature should prefer to economize on time rather than distance.\nClerselier wrote in part:\n1. The principle that you take as the basis of your demonstration, namely that nature always acts in the shortest and simplest ways, is merely a moral principle and not a physical one; it is not, and cannot be, the cause of any effect in nature .... For otherwise we would attribute knowledge to nature; but here, by \"nature\", we understand only this order and this law established in the world as it is, which acts without foresight, without choice, and by a necessary determination.\n2. This same principle would make nature irresolute ... For I ask you ... when a ray of light must pass from a point in a rare medium to a point in a dense one, is there not reason for nature to hesitate if, by your principle, it must choose the straight line as soon as the bent one, since if the latter proves shorter in time, the former is shorter and simpler in length?\nWho will decide and who will pronounce?\nFermat, being unaware of the mechanistic foundations of his own principle, was not well placed to defend it, except as a purely geometric and [kinematic](https://bliptext.com/articles/kinematics) proposition.\nThe [wave theory of light](https://bliptext.com/articles/wave-theory-of-light), first proposed by [Robert Hooke](https://bliptext.com/articles/robert-hooke) in the year of Fermat's death, and rapidly improved by [Ignace-Gaston Pardies](https://bliptext.com/articles/ignace-gaston-pardies) and (especially) [Christiaan Huygens](https://bliptext.com/articles/christiaan-huygens), contained the necessary foundations; but the recognition of this fact was surprisingly slow.\n# [Pierre de Fermat (1607–1665) | upload.wikimedia.org/wikipedia/commons/b/bd/Pierre///de///Fermat///%28F.///Poilly%29.jpg]\n\n## Huygens's oversight\nIn 1678, Huygens proposed that every point reached by a luminous disturbance becomes a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time.\nHuygens repeatedly referred to the envelope of his secondary wavefronts as the termination of the movement, meaning that the later wavefront was the outer boundary that the disturbance could reach in a given time, which was therefore the minimum time in which each point on the later wavefront could be reached.\nBut he did not argue that the direction of minimum time was that from the secondary source to the point of tangency; instead, he deduced the ray direction from the extent of the common tangent surface corresponding to a given extent of the initial wavefront.\nHis only endorsement of Fermat's principle was limited in scope: having derived the law of ordinary refraction, for which the rays are normal to the wavefronts, Huygens gave a geometric proof that a ray refracted according to this law takes the path of least time.\nHe would hardly have thought this necessary if he had known that the principle of least time followed directly from the same common-tangent construction by which he had deduced not only the law of ordinary refraction, but also the laws of rectilinear propagation and ordinary reflection (which were also known to follow from Fermat's principle), and a previously unknown law of [extraordinary refraction](https://bliptext.com/articles/birefringence) – the last by means of secondary wavefronts that were [spheroidal](https://bliptext.com/articles/spheroid) rather than spherical, with the result that the rays were generally oblique to the wavefronts.\nIt was as if Huygens had not noticed that his construction implied Fermat's principle, and even as if he thought he had found an exception to that principle.\nManuscript evidence cited by Alan E.Shapiro tends to confirm that Huygens believed the principle of least time to be invalid \"in [double refraction](https://bliptext.com/articles/birefringence), where the rays are not normal to the wave fronts\".\nShapiro further reports that the only three authorities who accepted \"Huygens' principle\" in the 17th and 18th centuries, namely [Philippe de La Hire](https://bliptext.com/articles/philippe-de-la-hire), [Denis Papin](https://bliptext.com/articles/denis-papin), and [Gottfried Wilhelm Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz), did so because it accounted for the extraordinary refraction of \"[Iceland crystal](https://bliptext.com/articles/iceland-spar)\" (calcite) in the same manner as the previously known laws of geometrical optics.\nBut, for the time being, the corresponding extension of Fermat's principle went unnoticed.\n# [Christiaan Huygens (1629–1695) | upload.wikimedia.org/wikipedia/commons/0/03/Christiaan-huygens4.jpg]\n\n## Laplace, Young, Fresnel, and Lorentz\nOn 30 January 1809, [Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace), reporting on the work of his protégé [Étienne-Louis Malus](https://bliptext.com/articles/tienne-louis-malus), claimed that the extraordinary refraction of calcite could be explained under the corpuscular theory of light with the aid of [Maupertuis's principle](https://bliptext.com/articles/maupertuis-s-principle) of least action: that the integral of speed with respect to distance was a minimum.\nThe corpuscular speed that satisfied this principle was proportional to the reciprocal of the ray speed given by the radius of Huygens' spheroid.\nLaplace continued:\nAccording to Huygens, the velocity of the extraordinary ray, in the crystal, is simply expressed by the radius of the spheroid; consequently his hypothesis does not agree with the principle of the least action: but it is remarkable that it agrees with the principle of Fermat, which is, that light passes, from a given point without the crystal, to a given point within it, in the least possible time; for it is easy to see that this principle coincides with that of the least action, if we invert the expression of the velocity.\nLaplace's report was the subject of a wide-ranging rebuttal by [Thomas Young](https://bliptext.com/articles/thomas-young-scientist), who wrote in part:\nThe principle of Fermat, although it was assumed by that mathematician on hypothetical, or even imaginary grounds, is in fact a fundamental law with respect to undulatory motion, and is explicitly [sic] the basis of every determination in the Huygenian theory... Mr. Laplace seems to be unacquainted with this most essential principle of one of the two theories which he compares; for he says, that \"it is remarkable\" that the Huygenian law of extraordinary refraction agrees with the principle of Fermat; which he would scarcely have observed, if he had been aware that the law was an immediate consequence of the principle.\nIn fact Laplace was aware that Fermat's principle follows from Huygens' construction in the case of refraction from an isotropic medium to an anisotropic one; a geometric proof was contained in the long version of Laplace's report, printed in 1810.\nYoung's claim was more general than Laplace's, and likewise upheld Fermat's principle even in the case of extraordinary refraction, in which the rays are generally not perpendicular to the wavefronts.\nUnfortunately, however, the omitted middle sentence of the quoted paragraph by Young began \"The motion of every undulation must necessarily be in a direction perpendicular to its surface ...\"\n(emphasis added), and was therefore bound to sow confusion rather than clarity.\nNo such confusion subsists in [Augustin-Jean Fresnel](https://bliptext.com/articles/augustin-jean-fresnel)'s \"Second Memoir\" on double refraction ([Fresnel, 1827](https://bliptext.com/articles/)), which addresses Fermat's principle in several places (without naming Fermat), proceeding from the special case in which rays are normal to wavefronts, to the general case in which rays are paths of least time or stationary time.\n(In the following summary, page numbers refer to [Alfred W.Hobson's translation](https://bliptext.com/articles/).)\nThus Fresnel showed, even for anisotropic media, that the ray path given by Huygens' construction is the path of least time between successive positions of a plane or diverging wavefront, that the ray velocities are the radii of the secondary \"wave surface\" after unit time, and that a stationary traversal time accounts for the direction of maximum intensity of a beam.\nHowever, establishing the general equivalence between Huygens' construction and Fermat's principle would have required further consideration of Fermat's principle in point-to-point terms.\n[Hendrik Lorentz](https://bliptext.com/articles/hendrik-lorentz), in a paper written in 1886 and republished in 1907, deduced the principle of least time in point-to-point form from Huygens' construction.\nBut the essence of his argument was somewhat obscured by an apparent dependence on [aether](https://bliptext.com/articles/luminiferous-aether) and [aether](https://bliptext.com/articles/luminiferous-aether) drag.\nLorentz's work was cited in 1959 by Adriaan J. de Witte, who then offered his own argument, which \"although in essence the same, is believed to be more cogent and more general\".\nDe Witte's treatment is more original than that description might suggest, although limited to two dimensions; it uses calculus of variations to show that Huygens' construction and Fermat's principle lead to the same [differential equation](https://bliptext.com/articles/differential-equation) for the ray path, and that in the case of Fermat's principle, the converse holds.\nDe Witte also noted that \"The matter seems to have escaped treatment in textbooks.\"\n# [Pierre-Simon Laplace (1749–1827) | upload.wikimedia.org/wikipedia/commons/3/39/Laplace%2C///Pierre-Simon%2C///marquis///de.jpg]\n# [Thomas Young (1773–1829) | upload.wikimedia.org/wikipedia/commons/9/9f/Thomas///Young///%28scientist%29.jpg]\n# [Augustin-Jean Fresnel (1788–1827) | upload.wikimedia.org/wikipedia/commons/0/02/Augustin///Fresnel.jpg]\n\n# In popular culture\nThe short story [Story of Your Life](https://bliptext.com/articles/story-of-your-life) by the speculative fiction writer [Ted Chiang](https://bliptext.com/articles/ted-chiang) contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension.\n[Keith Devlin](https://bliptext.com/articles/keith-devlin)'s The Math Instinct contains a chapter, \"Elvis the Welsh Corgi Who Can Do Calculus\" that discusses the calculus \"embedded\" in some animals as they solve the \"least time\" problem in actual situations.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:37:07.695Z","updated_at":"2025-01-25T17:37:07.695Z","search_vector":"'fermat':1 'principl':3","search_content":" \n \n\n# \n**Fermat's principle**, also known as the **principle of least time**, is the link between [ray optics](https://bliptext.com/articles/geometrical-optics) and [wave optics](https://bliptext.com/articles/physical-optics).\nFermat's principle states that the path taken by a [ray](https://bliptext.com/articles/ray-optics) between two given points is the path that can be traveled in the least time.\nFirst proposed by the French mathematician [Pierre de Fermat](https://bliptext.com/articles/pierre-de-fermat) in 1662, as a means of explaining the [ordinary law of refraction](https://bliptext.com/articles/snell-s-law) of light (Fig.1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature.\nNot until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves.\nIf points A and B are given, a [wavefront](https://bliptext.com/articles/wavefront) expanding from A sweeps all possible ray paths radiating from A, whether they pass through B or not.\nIf 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.\nFermat's principle describes any ray that happens to reach point B; there is no implication that the ray \"knew\" the quickest path or \"intended\" to take that path.\nIn its original \"strong\" form, 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.\nIn order to be true in all cases, this statement must be weakened by replacing the \"least\" time with a time that is \"[stationary](https://bliptext.com/articles/stationary-point)\" with respect to variations of the path – so that a deviation in the path causes, at most, a second-order change in the traversal time.\nTo put it loosely, a ray path is surrounded by close paths that can be traversed in very close times.\nIt [can be shown](https://bliptext.com/articles/) that this technical definition corresponds to more intuitive notions of a ray, such as a [line of sight](https://bliptext.com/articles/line-of-sight-propagation) or the path of a [narrow beam](https://bliptext.com/articles/narrow-beam).\nFor 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.\nWithout this condition, the traversal time would be ambiguous; for example, if the propagation time from P to P′ were reckoned from an arbitrary wavefront W containing P (Fig.2), that time could be made arbitrarily small by suitably angling the wavefront.\nTreating a point on the path as a source is the minimum requirement of [Huygens' principle](https://bliptext.com/articles/huygens-principle), and is part of the [explanation](https://bliptext.com/articles/) of Fermat's principle.\nBut it [can also be shown](https://bliptext.com/articles/) that the geometric construction by which [Huygens](https://bliptext.com/articles/christiaan-huygens) tried to apply his own principle (as distinct from the principle itself) is simply an invocation of Fermat's principle.\nHence all the conclusions that Huygens drew from that construction – including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of \"[Iceland crystal](https://bliptext.com/articles/iceland-spar)\" (calcite) – are also consequences of Fermat's principle.\n# [Fig.1: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 law. | upload.wikimedia.org/wikipedia/commons/d/d9/Fermat///Snellius.svg]\n# [Fig.2:Two points P and P′ on a path from A to B. For the purposes of Fermat's principle, the propagation time from P to P′ is taken as for a point-source at P, not (e.g.) for an arbitrary wavefront W passing through P. The surface Σ (with unit normal n̂ at P′) is the locus of points that a disturbance at P can reach in the same time that it takes to reach P′; in other words, Σ is the secondary wavefront with radius PP′. (The medium is not assumed to be homogeneous or isotropic.) | upload.wikimedia.org/wikipedia/commons/6/6b/Fermat-Huygens///principle.svg]\n\n# Derivation\n\n## Sufficient conditions\nLet us suppose that:\nThen the various propagation paths from A to B will help each other, or interfere constructively, if their traversal times agree within the said tolerance.\nFor a small tolerance (in the limiting case), the permissible range of variations of the path is maximized if the path is such that its traversal time is stationary with respect to the variations, so that a variation of the path causes at most a second-order change in the traversal time.\nThe most obvious example of a stationarity in traversal time is a (local or global) minimum – that is, a path of least time, as in the \"strong\" form of Fermat's principle.\nBut that condition is not essential to the argument.\nHaving established that a path of stationary traversal time is reinforced by a maximally wide corridor of neighboring paths, we still need to explain how this reinforcement corresponds to intuitive notions of a ray.\nBut, for brevity in the explanations, let us first define a ray path as a path of stationary traversal time.\n\n## A ray as a signal path (line of sight)\nIf the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly alter the disturbance reaching B from A – unlike a similar-sized obstruction outside any such corridor, blocking paths that do not reinforce each other.\nThe former obstruction will significantly disrupt the signal reaching B from A, while the latter will not; thus the ray path marks a signal path.\nIf the signal is visible light, the former obstruction will significantly affect the appearance of an object at A as seen by an observer at B, while the latter will not; so the ray path marks a line of sight.\nIn optical experiments, a line of sight is routinely assumed to be a ray path.\n\n## A ray as an energy path (beam)\nIf the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly affect the energy reaching B from A – unlike a similar-sized obstruction outside any such corridor.\nThus the ray path marks an energy path – as does a beam.\nSuppose that a wavefront expanding from point A passes point P, which lies on a ray path from point A to point B.\nBy definition, all points on the wavefront have the same propagation time from A.\nNow let the wavefront be blocked except for a window, centered on P, and small enough to lie within the corridor of paths that reinforce the ray path from A to B.\nThen all points on the unobstructed portion of the wavefront will have, nearly enough, equal propagation times to B, but not to points in other directions, so that B will be in the direction of peak intensity of the beam admitted through the window.\nSo the ray path marks the beam.\nAnd in optical experiments, a beam is routinely considered as a collection of rays or (if it is narrow) as an approximation to a ray (Fig.3).\n# [Fig.3:An experiment demonstrating refraction (and partial reflection) of rays – approximated by, or contained in, narrow beams | upload.wikimedia.org/wikipedia/commons/3/32/Concave///lens.jpg]\n\n## Analogies\nAccording to the \"strong\" form of Fermat's principle, the problem of finding the path of a light ray from point A in a medium of faster propagation, to point B in a medium of slower propagation ([Fig.1](https://bliptext.com/articles/)), is analogous to the problem faced by a [lifeguard](https://bliptext.com/articles/lifeguard) in deciding where to enter the water in order to reach a drowning swimmer as soon as possible, given that the [lifeguard](https://bliptext.com/articles/lifeguard) can run faster than (s)he can swim.\nBut that analogy falls short of explaining the behavior of the light, because the lifeguard can think about the problem (even if only for an instant) whereas the light presumably cannot.\nThe discovery that ants are capable of similar calculations does not bridge the gap between the animate and the inanimate.\nIn contrast, the above assumptions (1) to (3) hold for any wavelike disturbance and explain Fermat's principle in purely [mechanistic](https://bliptext.com/articles/mechanical-philosophy) terms, without any imputation of knowledge or purpose.\nThe principle applies to waves in general, including (e.g.) sound waves in fluids and elastic waves in solids.\nIn a modified form, it even works for [matter waves](https://bliptext.com/articles/matter-wave): in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), the [classical path](https://bliptext.com/articles/classical-limit) of a particle is obtainable by applying Fermat's principle to the associated wave – except that, because the frequency may vary with the path, the stationarity is in the [phase shift](https://bliptext.com/articles/phase-shift) (or number of cycles) and not necessarily in the time.\nFermat's principle is most familiar, however, in the case of visible [light](https://bliptext.com/articles/light): it is the link between [geometrical optics](https://bliptext.com/articles/geometrical-optics), which describes certain optical phenomena in terms of rays, and the wave theory of [light](https://bliptext.com/articles/light), which explains the same phenomena on the hypothesis that [light](https://bliptext.com/articles/light) consists of waves.\n\n# Equivalence to Huygens' construction\nIn this article we distinguish between Huygens' principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens' construction, which is described below.\nLet the surface W be a wavefront at time t, and let the surface W′ be the same wavefront at the later time\nt + Δt\n(Fig.4).\nLet P be a general point on W. Then, according to Huygens' construction,\n \n W′ is the [envelope](https://bliptext.com/articles/envelope-mathematics) (common tangent surface), on the forward side of W, of all the secondary wavefronts each of which would expand in time\nΔt\nfrom a point on W, and \n if the secondary wavefront expanding from point P in time\nΔt\ntouches the surface W′ at point P′, then P and P′ lie on a ray. \n \nThe construction may be repeated in order to find successive positions of the primary wavefront, and successive points on the ray.\nThe ray direction given by this construction is the radial direction of the secondary wavefront, and may differ from the normal of the secondary wavefront (cf. [Fig.2](https://bliptext.com/articles/)), and therefore from the normal of the primary wavefront at the point of tangency.\nHence the ray velocity, in magnitude and direction, is the radial velocity of an infinitesimal secondary wavefront, and is generally a function of location and direction.\nNow let Q be a point on W close to P, and let Q′ be a point on W′ close to P′.\nThen, by the construction,\n \n the time taken for a secondary wavefront from P to reach Q′ has at most a second-order dependence on the displacement P′Q′, and \n the time taken for a secondary wavefront to reach P′ from Q has at most a second-order dependence on the displacement PQ. \n \nBy (i), the ray path is a path of stationary traversal time from P to W′; and by (ii), it is a path of stationary traversal time from a point on W to P′.\nSo Huygens' construction implicitly defines a ray path as a path of stationary traversal time between successive positions of a wavefront, the time being reckoned from a point-source on the earlier wavefront.\nThis conclusion remains valid if the secondary wavefronts are reflected or refracted by surfaces of discontinuity in the properties of the medium, provided that the comparison is restricted to the affected paths and the affected portions of the wavefronts.\nFermat's principle, however, is conventionally expressed in point-to-point terms, not wavefront-to-wavefront terms.\nAccordingly, let us modify the example by supposing that the wavefront which becomes surface W at time t, and which becomes surface W′ at the later time\nt + Δt\n, is emitted from point A at time\n0\n. Let P be a point on W (as before), and B a point on W′.\nAnd let A, W, W′, and B be given, so that the problem is to find P.\nIf P satisfies Huygens' construction, so that the secondary wavefront from P is tangential to W′ at B, then PB is a path of stationary traversal time from W to B. Adding the fixed time from A to W, we find that APB is the path of stationary traversal time from A to B (possibly with a restricted domain of comparison, as noted above), in accordance with Fermat's principle.\nThe argument works just as well in the converse direction, provided that W′ has a well-defined tangent plane at B. Thus Huygens' construction and Fermat's principle are geometrically equivalent.\nThrough this equivalence, Fermat's principle sustains Huygens' construction and thence all the conclusions that Huygens was able to draw from that construction.\nIn short, \"The laws of geometrical optics may be derived from Fermat's principle\".\nWith the exception of the Fermat–Huygens principle itself, these laws are special cases in the sense that they depend on further assumptions about the media.\nTwo of them are mentioned under the next heading.\n# [Fig.4:Two iterations of Huygens' construction. In the first iteration, the later wavefront W′ is derived from the earlier wavefront W by taking the envelope of all the secondary wavefronts (gray arcs) expanding in a given time from all the points (e.g., P) on W. The arrows show the ray directions. | upload.wikimedia.org/wikipedia/commons/8/8b/Huygens-construction-two-iterations.svg]\n\n# Special cases\n\n## Isotropic media: rays normal to wavefronts\nIn an isotropic medium, because the propagation speed is independent of direction, the secondary wavefronts that expand from points on a primary wavefront in a given infinitesimal time are spherical, so that their radii are normal to their common tangent surface at the points of tangency.\nBut their radii mark the ray directions, and their common tangent surface is a general wavefront.\nThus the rays are normal (orthogonal) to the wavefronts.\nBecause much of the teaching of optics concentrates on isotropic media, treating anisotropic media as an optional topic, the assumption that the rays are normal to the wavefronts can become so pervasive that even Fermat's principle is explained under that assumption, although in fact Fermat's principle is more general.\n\n## Homogeneous media: rectilinear propagation\nIn a homogeneous medium (also called a uniform medium), all the secondary wavefronts that expand from a given primary wavefront W in a given time\nΔt\nare [congruent](https://bliptext.com/articles/congruence-geometry) and similarly oriented, so that their envelope W′ may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center (source) moves over W. If P is its center while P′ is its point of tangency with W′, then P′ moves parallel to P, so that the plane tangential to W′ at P′ is parallel to the plane tangential to W at P. Let another ([congruent](https://bliptext.com/articles/congruence-geometry) and similarly orientated) secondary wavefront be centered on P′, moving with P, and let it meet its envelope W″ at point P″.\nThen, by the same reasoning, the plane tangential to W″ at P″ is parallel to the other two planes.\nHence, due to the congruence and similar orientations, the ray directions PP′ and P′P″ are the same (but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical).\nThis construction can be repeated any number of times, giving a straight ray of any length.\nThus a homogeneous medium admits rectilinear rays.\n\n# Modern version\n\n## Formulation in terms of refractive index\nLet a path\nΓ\nextend from point A to point B. Let s be the arc length measured along the path from A, and let t be the time taken to traverse that arc length at the ray speed (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path).\nThen the traversal time of the entire path\nΓ\nis\n(where A and B simply denote the endpoints and are not to be construed as values of t or s).\nThe condition for\nΓ\nto be a ray path is that the first-order change in T due to a change in\nΓ\nis zero; that is,\nNow let us define the optical length of a given path ([optical path length](https://bliptext.com/articles/optical-path-length), OPL) as the distance traversed by a ray in a homogeneous isotropic reference medium (e.g., a vacuum) in the same time that it takes to traverse the given path at the local ray velocity.\nThen, if c denotes the propagation speed in the reference medium (e.g., the speed of light in vacuum), the optical length of a path traversed in time dt is\ndS = c dt\n, and the optical length of a path traversed in time T is\nS = cT\n. So, multiplying equation **(1)** through by c, we obtain\nwhere is the ray index – that is, the [refractive index](https://bliptext.com/articles/refractive-index) calculated on the ray velocity instead of the usual [phase velocity](https://bliptext.com/articles/phase-velocity) (wave-normal velocity).\nFor an infinitesimal path, we have indicating that the optical length is the physical length multiplied by the ray index: the OPL is a notional geometric quantity, from which time has been factored out.\nIn terms of OPL, the condition for\nΓ\nto be a ray path (Fermat's principle) becomes\nThis has the form of [Maupertuis's principle](https://bliptext.com/articles/maupertuis-s-principle) in [classical mechanics](https://bliptext.com/articles/classical-mechanics) (for a single particle), with the ray index in optics taking the role of momentum or velocity in mechanics.\nIn an isotropic medium, for which the ray velocity is also the phase velocity, we may substitute the usual refractive index n for\nnr\n.\n\n## Relation to Hamilton's principle\nIf\nx\n,\ny\n,\nz\nare Cartesian coordinates and an overdot denotes differentiation with respect to s, Fermat's principle **(2)** may be written\nIn the case of an isotropic medium, we may replace\nnr\nwith the normal refractive index\nn(x, y, z)\n, which is simply a [scalar field](https://bliptext.com/articles/scalar-field).\nIf we then define the optical [Lagrangian](https://bliptext.com/articles/lagrangian-mechanics) as\nFermat's principle becomes\nIf the direction of propagation is always such that we can use z instead of s as the parameter of the path (and the overdot to denote differentiation w.r.t. z instead of s), the optical Lagrangian can instead be written\nso that Fermat's principle becomes\nThis has the form of [Hamilton's principle](https://bliptext.com/articles/hamilton-s-principle) in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics.\nThe optical Lagrangian is the function which, when integrated w.r.t. the parameter of the path, yields the OPL; it is the foundation of [Lagrangian and Hamiltonian optics](https://bliptext.com/articles/hamiltonian-optics).\n\n# History\nIf a ray follows a straight line, it obviously takes the path of least length.\n[Hero of Alexandria](https://bliptext.com/articles/hero-of-alexandria), in his [Catoptrics](https://bliptext.com/articles/catoptrics) (1st century CE), showed that the ordinary [law of reflection](https://bliptext.com/articles/specular-reflection) off a plane surface follows from the premise that the total length of the ray path is a minimum.\n[Ibn al-Haytham](https://bliptext.com/articles/ibn-al-haytham), an 11th-century polymath later extended this principle to refraction, hence giving an early version of the Fermat's principle.\n\n## Fermat vs. the Cartesians\nIn 1657, Pierre de Fermat received from [Marin Cureau de la Chambre](https://bliptext.com/articles/marin-cureau-de-la-chambre) a copy of newly published treatise, in which La Chambre noted Hero's principle and complained that it did not work for refraction.\nFermat replied that refraction might be brought into the same framework by supposing that light took the path of least resistance, and that different media offered different resistances.\nHis eventual solution, described in a letter to La Chambre dated 1 January 1662, construed \"resistance\" as inversely proportional to speed, so that light took the path of least time.\nThat premise yielded the ordinary law of refraction, provided that light traveled more slowly in the optically denser medium.\nFermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a [variational principle](https://bliptext.com/articles/variational-principle) or [action principle](https://bliptext.com/articles/action-physics), setting the precedent for the [principle of least action](https://bliptext.com/articles/principle-of-least-action) in classical mechanics and the corresponding principles in other fields (see History of [variational principle](https://bliptext.com/articles/variational-principle)s in physics).\nIt was the more notable because it used the method of [adequality](https://bliptext.com/articles/adequality), which may be understood in retrospect as finding the point where the slope of an infinitesimally short [chord](https://bliptext.com/articles/chord-geometry) is zero, without the intermediate step of finding a general expression for the slope (the [derivative](https://bliptext.com/articles/derivative)).\nIt was also immediately controversial.\nThe ordinary law of refraction was at that time attributed to [René Descartes](https://bliptext.com/articles/ren-descartes) (d.1650), who had tried to explain it by supposing that light was a force that propagated instantaneously, or that light was analogous to a tennis ball that traveled faster in the denser medium, either premise being inconsistent with Fermat's. Descartes' most prominent defender, [Claude Clerselier](https://bliptext.com/articles/claude-clerselier), criticized Fermat for apparently ascribing knowledge and intent to nature, and for failing to explain why nature should prefer to economize on time rather than distance.\nClerselier wrote in part:\n1. The principle that you take as the basis of your demonstration, namely that nature always acts in the shortest and simplest ways, is merely a moral principle and not a physical one; it is not, and cannot be, the cause of any effect in nature .... For otherwise we would attribute knowledge to nature; but here, by \"nature\", we understand only this order and this law established in the world as it is, which acts without foresight, without choice, and by a necessary determination.\n2. This same principle would make nature irresolute ... For I ask you ... when a ray of light must pass from a point in a rare medium to a point in a dense one, is there not reason for nature to hesitate if, by your principle, it must choose the straight line as soon as the bent one, since if the latter proves shorter in time, the former is shorter and simpler in length?\nWho will decide and who will pronounce?\nFermat, being unaware of the mechanistic foundations of his own principle, was not well placed to defend it, except as a purely geometric and [kinematic](https://bliptext.com/articles/kinematics) proposition.\nThe [wave theory of light](https://bliptext.com/articles/wave-theory-of-light), first proposed by [Robert Hooke](https://bliptext.com/articles/robert-hooke) in the year of Fermat's death, and rapidly improved by [Ignace-Gaston Pardies](https://bliptext.com/articles/ignace-gaston-pardies) and (especially) [Christiaan Huygens](https://bliptext.com/articles/christiaan-huygens), contained the necessary foundations; but the recognition of this fact was surprisingly slow.\n# [Pierre de Fermat (1607–1665) | upload.wikimedia.org/wikipedia/commons/b/bd/Pierre///de///Fermat///%28F.///Poilly%29.jpg]\n\n## Huygens's oversight\nIn 1678, Huygens proposed that every point reached by a luminous disturbance becomes a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time.\nHuygens repeatedly referred to the envelope of his secondary wavefronts as the termination of the movement, meaning that the later wavefront was the outer boundary that the disturbance could reach in a given time, which was therefore the minimum time in which each point on the later wavefront could be reached.\nBut he did not argue that the direction of minimum time was that from the secondary source to the point of tangency; instead, he deduced the ray direction from the extent of the common tangent surface corresponding to a given extent of the initial wavefront.\nHis only endorsement of Fermat's principle was limited in scope: having derived the law of ordinary refraction, for which the rays are normal to the wavefronts, Huygens gave a geometric proof that a ray refracted according to this law takes the path of least time.\nHe would hardly have thought this necessary if he had known that the principle of least time followed directly from the same common-tangent construction by which he had deduced not only the law of ordinary refraction, but also the laws of rectilinear propagation and ordinary reflection (which were also known to follow from Fermat's principle), and a previously unknown law of [extraordinary refraction](https://bliptext.com/articles/birefringence) – the last by means of secondary wavefronts that were [spheroidal](https://bliptext.com/articles/spheroid) rather than spherical, with the result that the rays were generally oblique to the wavefronts.\nIt was as if Huygens had not noticed that his construction implied Fermat's principle, and even as if he thought he had found an exception to that principle.\nManuscript evidence cited by Alan E.Shapiro tends to confirm that Huygens believed the principle of least time to be invalid \"in [double refraction](https://bliptext.com/articles/birefringence), where the rays are not normal to the wave fronts\".\nShapiro further reports that the only three authorities who accepted \"Huygens' principle\" in the 17th and 18th centuries, namely [Philippe de La Hire](https://bliptext.com/articles/philippe-de-la-hire), [Denis Papin](https://bliptext.com/articles/denis-papin), and [Gottfried Wilhelm Leibniz](https://bliptext.com/articles/gottfried-wilhelm-leibniz), did so because it accounted for the extraordinary refraction of \"[Iceland crystal](https://bliptext.com/articles/iceland-spar)\" (calcite) in the same manner as the previously known laws of geometrical optics.\nBut, for the time being, the corresponding extension of Fermat's principle went unnoticed.\n# [Christiaan Huygens (1629–1695) | upload.wikimedia.org/wikipedia/commons/0/03/Christiaan-huygens4.jpg]\n\n## Laplace, Young, Fresnel, and Lorentz\nOn 30 January 1809, [Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace), reporting on the work of his protégé [Étienne-Louis Malus](https://bliptext.com/articles/tienne-louis-malus), claimed that the extraordinary refraction of calcite could be explained under the corpuscular theory of light with the aid of [Maupertuis's principle](https://bliptext.com/articles/maupertuis-s-principle) of least action: that the integral of speed with respect to distance was a minimum.\nThe corpuscular speed that satisfied this principle was proportional to the reciprocal of the ray speed given by the radius of Huygens' spheroid.\nLaplace continued:\nAccording to Huygens, the velocity of the extraordinary ray, in the crystal, is simply expressed by the radius of the spheroid; consequently his hypothesis does not agree with the principle of the least action: but it is remarkable that it agrees with the principle of Fermat, which is, that light passes, from a given point without the crystal, to a given point within it, in the least possible time; for it is easy to see that this principle coincides with that of the least action, if we invert the expression of the velocity.\nLaplace's report was the subject of a wide-ranging rebuttal by [Thomas Young](https://bliptext.com/articles/thomas-young-scientist), who wrote in part:\nThe principle of Fermat, although it was assumed by that mathematician on hypothetical, or even imaginary grounds, is in fact a fundamental law with respect to undulatory motion, and is explicitly [sic] the basis of every determination in the Huygenian theory... Mr. Laplace seems to be unacquainted with this most essential principle of one of the two theories which he compares; for he says, that \"it is remarkable\" that the Huygenian law of extraordinary refraction agrees with the principle of Fermat; which he would scarcely have observed, if he had been aware that the law was an immediate consequence of the principle.\nIn fact Laplace was aware that Fermat's principle follows from Huygens' construction in the case of refraction from an isotropic medium to an anisotropic one; a geometric proof was contained in the long version of Laplace's report, printed in 1810.\nYoung's claim was more general than Laplace's, and likewise upheld Fermat's principle even in the case of extraordinary refraction, in which the rays are generally not perpendicular to the wavefronts.\nUnfortunately, however, the omitted middle sentence of the quoted paragraph by Young began \"The motion of every undulation must necessarily be in a direction perpendicular to its surface ...\"\n(emphasis added), and was therefore bound to sow confusion rather than clarity.\nNo such confusion subsists in [Augustin-Jean Fresnel](https://bliptext.com/articles/augustin-jean-fresnel)'s \"Second Memoir\" on double refraction ([Fresnel, 1827](https://bliptext.com/articles/)), which addresses Fermat's principle in several places (without naming Fermat), proceeding from the special case in which rays are normal to wavefronts, to the general case in which rays are paths of least time or stationary time.\n(In the following summary, page numbers refer to [Alfred W.Hobson's translation](https://bliptext.com/articles/).)\nThus Fresnel showed, even for anisotropic media, that the ray path given by Huygens' construction is the path of least time between successive positions of a plane or diverging wavefront, that the ray velocities are the radii of the secondary \"wave surface\" after unit time, and that a stationary traversal time accounts for the direction of maximum intensity of a beam.\nHowever, establishing the general equivalence between Huygens' construction and Fermat's principle would have required further consideration of Fermat's principle in point-to-point terms.\n[Hendrik Lorentz](https://bliptext.com/articles/hendrik-lorentz), in a paper written in 1886 and republished in 1907, deduced the principle of least time in point-to-point form from Huygens' construction.\nBut the essence of his argument was somewhat obscured by an apparent dependence on [aether](https://bliptext.com/articles/luminiferous-aether) and [aether](https://bliptext.com/articles/luminiferous-aether) drag.\nLorentz's work was cited in 1959 by Adriaan J. de Witte, who then offered his own argument, which \"although in essence the same, is believed to be more cogent and more general\".\nDe Witte's treatment is more original than that description might suggest, although limited to two dimensions; it uses calculus of variations to show that Huygens' construction and Fermat's principle lead to the same [differential equation](https://bliptext.com/articles/differential-equation) for the ray path, and that in the case of Fermat's principle, the converse holds.\nDe Witte also noted that \"The matter seems to have escaped treatment in textbooks.\"\n# [Pierre-Simon Laplace (1749–1827) | upload.wikimedia.org/wikipedia/commons/3/39/Laplace%2C///Pierre-Simon%2C///marquis///de.jpg]\n# [Thomas Young (1773–1829) | upload.wikimedia.org/wikipedia/commons/9/9f/Thomas///Young///%28scientist%29.jpg]\n# [Augustin-Jean Fresnel (1788–1827) | upload.wikimedia.org/wikipedia/commons/0/02/Augustin///Fresnel.jpg]\n\n# In popular culture\nThe short story [Story of Your Life](https://bliptext.com/articles/story-of-your-life) by the speculative fiction writer [Ted Chiang](https://bliptext.com/articles/ted-chiang) contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension.\n[Keith Devlin](https://bliptext.com/articles/keith-devlin)'s The Math Instinct contains a chapter, \"Elvis the Welsh Corgi Who Can Do Calculus\" that discusses the calculus \"embedded\" in some animals as they solve the \"least time\" problem in actual situations.","revisionCount":0},"hasMore":false,"nextCursor":null}