Fermat's principle


Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In 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 traversed 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" 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. To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. It [|can be shown] that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam.
First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light, 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. 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 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 B; there is no implication that the ray "knew" the quickest path or "intended" to take that path.
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. Without this condition, the traversal time would be ambiguous; for example, if the propagation time from to were reckoned from an arbitrary wavefront W containing , 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 [|explanation] of Fermat's principle. But it [|can also be shown] that the geometric construction by which Huygens tried to apply his own principle is simply an invocation of Fermat's principle. Hence 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" — are also consequences of Fermat's principle.

Derivation

Sufficient conditions

Let us suppose that:
Then the various propagation paths from A to B will help each other if their traversal times agree within the said tolerance. For a small tolerance, 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.
The most obvious example of a stationarity in traversal time is a minimum — that is, a path of least time, as in the "strong" form of Fermat's principle. But that condition is not essential to the argument.
Having 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. But, for brevity in the explanations, let us first define a ray path as a path of stationary traversal time.

A ray as a signal path (line of sight)

If 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. The former obstruction will significantly disrupt the signal reaching B from A, while the latter will not; thus the ray path marks a signal path. If 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.
In optical experiments, a line of sight is routinely assumed to be a ray path.

A ray as an energy path (beam)

If 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. Thus the ray path marks an energy path — as does a beam.
Suppose that a wavefront expanding from point A passes point P, which lies on a ray path from point A to point B. By definition, all points on the wavefront have the same propagation time from A. Now 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. Then 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. So the ray path marks the beam. And in optical experiments, a beam is routinely considered as a collection of rays or as an approximation to a ray.

Analogies

According 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, is analogous to the problem faced by a lifeguard in deciding where to enter the water in order to reach a drowning swimmer as soon as possible, given that the lifeguard can run faster than he can swim. But that analogy falls short of explaining the behavior of the light, because the lifeguard can think about the problem whereas the light presumably cannot. The discovery that ants are capable of similar calculations does not bridge the gap between the animate and the inanimate.
In contrast, the above assumptions to hold for any wavelike disturbance and explain Fermat's principle in purely mechanistic terms, without any imputation of knowledge or purpose.
The principle applies to waves in general, including sound waves in fluids and elastic waves in solids. In a modified form, it even works for matter waves: in quantum mechanics, the classical path 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 and not necessarily in the time.
Fermat's principle is most familiar, however, in the case of visible light: it is the link between geometrical optics, which describes certain optical phenomena in terms of rays, and the wave theory of light, which explains the same phenomena on the hypothesis that light consists of waves.

Equivalence to Huygens' construction

In 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.
Let the surface be a wavefront at time, and let the surface be the same wavefront at the later time . Let be a general point on. Then, according to Huygens' construction,
The construction may be repeated in order to find successive positions of the primary wavefront, and successive points on the ray.
The ray direction given by this construction is the radial direction of the secondary wavefront, and may differ from the normal of the secondary wavefront, and therefore from the normal of the primary wavefront at the point of tangency. Hence 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.
Now let be a point on close to, and let be a point on close to. Then, by the construction,
By, the ray path is a path of stationary traversal time from to ; and by, it is a path of stationary traversal time from a point on to.
So 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. This 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 affect paths and the affected portions of the wavefronts.
Fermat's principle, however, is conventionally expressed in point-to-point terms, not wavefront-to-wavefront terms. Accordingly, let us modify the example by supposing that the wavefront which becomes surface at time, and which becomes surface at the later time, is emitted from point at time . Let be a point on , and a point on. And let and be given, so that the problem is to find.
If satisfies Huygens' construction, so that the secondary wavefront from is tangential to at, then is a path of stationary traversal time from to. Adding the fixed time from to, we find that is the path of stationary traversal time from to , in accordance with Fermat's principle. The argument works just as well in the converse direction, provided that has a well-defined tangent plane at. Thus Huygens' construction and Fermat's principle are geometrically equivalent.
Through this equivalence, Fermat's principle sustains Huygens' construction and thence all the conclusions that Huygens was able to draw from that construction. In short, "The laws of geometrical optics may be derived from Fermat's principle". With 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. Two of them are mentioned under the next heading.

Special cases

Isotropic media: Rays normal to wavefronts

In 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. But their radii mark the ray directions, and their common tangent surface is a general wavefront. Thus the rays are normal to the wavefronts.
Because 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.

Homogeneous media: Rectilinear propagation

In a homogeneous medium, all the secondary wavefronts that expand from a given primary wavefront in a given time are congruent and similarly oriented, so that their envelope may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center moves over. If is its center while is its point of tangency with, then moves parallel to, so that the plane tangential to at is parallel to the plane tangential to at. Let another secondary wavefront be centered on, moving with, and let it meet its envelope at point. Then, by the same reasoning, the plane tangential to at is parallel to the other two planes. Hence, due to the congruence and similar orientations, the ray directions and are the same. This construction can be repeated any number of times, giving a straight ray of any length. Thus a homogeneous medium admits rectilinear rays.

Modern version

Formulation in terms of refractive index

Let a path extend from point to point. Let be the arc length measured along the path from, and let be the time taken to traverse that arc length at the ray speed . Then the traversal time of the entire path is
. The condition for to be a ray path is that the first-order change in due to a change in is zero; that is,
Now let us define the optical length of a given path as the distance traversed by a ray in a homogeneous isotropic reference medium in the same time that it takes to traverse the given path at the local ray velocity. Then, if denotes the propagation speed in the reference medium, the optical length of a path traversed in time is, and the optical length of a path traversed in time is. So, multiplying equation through by, we obtain
where is the ray index — that is, the refractive index calculated on the ray velocity instead of the usual phase velocity. For 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. In terms of OPL, the condition for to be a ray path becomes
This has the form of Maupertuis's principle in classical mechanics, with the ray index in optics taking the role of momentum or velocity in mechanics.
In an isotropic medium, for which the ray velocity is also the phase velocity, we may substitute the usual refractive index for .

Relation to Hamilton's principle

If are Cartesian coordinates and an overdot denotes differentiation with respect to, Fermat's principle may be written
In the case of an isotropic medium, we may replace with the normal refractive index, which is simply a scalar field. If we then define the optical Lagrangian as
Fermat's principle becomes
If the direction of propagation is always such that we can use instead of as the parameter of the path, the optical Lagrangian can instead be written
so that Fermat's principle becomes
This has the form of 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. The 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.

History

Fermat vs. the Cartesians

If a ray follows a straight line, it obviously takes the path of least length. Hero of Alexandria, in his Catoptrics, showed that the ordinary law of reflection off a plane surface follows from the premise that the total length of the ray path is a minimum. In 1657, Pierre de Fermat received from 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.
Fermat 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. His 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. That premise yielded the ordinary law of refraction, provided that light traveled more slowly in the optically denser medium.
Fermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a variational principle or action principle, setting the precedent for the principle of least action in classical mechanics and the corresponding principles in other fields. It was the more notable because it used the method of adequality, which may be understood in retrospect as finding the point where the slope of an infinitesimally short chord is zero, without the intermediate step of finding a general expression for the slope.
It was also immediately controversial. The ordinary law of refraction was at that time attributed to René Descartes, 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, 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. Clerselier wrote in part:

1. 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.
2. 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? Who will decide and who will pronounce?

Fermat, being unaware of the mechanistic foundations of his own principle, was not well placed to defend it, except as a purely geometric and kinematic proposition. The wave theory of light, first proposed by Robert Hooke in the year of Fermat's death, and rapidly improved by Ignace-Gaston Pardies and Christiaan Huygens, contained the necessary foundations; but the recognition of this fact was surprisingly slow.

Huygens's oversight

Huygens 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. But 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. His 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. He 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, and a previously unknown law of extraordinary refraction — the last by means of secondary wavefronts that were spheroidal rather than spherical, with the result that the rays were generally oblique to the wavefronts. It 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. Manuscript evidence cited by Alan E.Shapiro tends to confirm that Huygens believed the principle of least time to be invalid "in double refraction, where the rays are not normal to the wave fronts".
Shapiro further reports that the only three authorities who accepted "Huygens' principle" in the 17th and 18th centuries, namely Philippe de La Hire, Denis Papin, and Gottfried Wilhelm Leibniz, did so because it accounted for the extraordinary refraction of "Iceland crystal" in the same manner as the previously known laws of geometrical optics. But, for the time being, the corresponding extension of Fermat's principle went unnoticed.

Laplace, Young, Fresnel, and Lorentz

On 30 January 1809, Pierre-Simon Laplace, reporting on the work of his protégé Étienne-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 of least action: that the integral of speed with respect to distance was a minimum. The corpuscular speed that satisfied this principle was proportional to the reciprocal of the ray speed given by the radius of Huygens' spheroid. Laplace continued:

According 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.

Laplace's report was the subject of a wide-ranging rebuttal by Thomas Young, who wrote in part:

The 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 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.

In 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.
Young'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. Unfortunately, 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...", and was therefore bound to sow confusion rather than clarity.
No such confusion subsists in Augustin-Jean Fresnel's "Second Memoir" on double refraction, which addresses Fermat's principle in several places, 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.
Thus 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. However, 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.
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. But the essence of his argument was somewhat obscured by an apparent dependence on aether and aether drag.
Lorentz'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." De 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 for the ray path, and that in the case of Fermat's principle, the converse holds. De Witte also noted that "The matter seems to have escaped treatment in textbooks."