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 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" 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 (Fig.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. 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 ''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 (Fig.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 explanation of Fermat's principle. But it can also be shown that the geometric ''construction'' by which Huygens tried to apply his own principle (as distinct from the principle itself) 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" (calcite) — are also consequences of Fermat's principle.

Derivation

Sufficient conditions

Let us suppose that:
# A disturbance propagates sequentially through a medium (a vacuum or some material, not necessarily homogeneous or isotropic), without action at a distance;
# During propagation, the influence of the disturbance at any intermediate point ''P'' upon surrounding points has a non-zero angular spread (as if ''P'' were a source), so that a disturbance originating at any point ''A'' arrives at any other point ''B'' via an infinitude of paths, by which ''B'' receives an infinitude of delayed versions of the disturbance at ''A'';Assumption (2) almost follows from (1) because: (a) to the extent that the disturbance at the intermediate point ''P'' can be represented by a scalar, its influence is omnidirectional; (b) to the extent that it can be represented by a vector in the supposed direction of propagation (as in a longitudinal wave), it has a non-zero component in a range of neighboring directions; and (c) to the extent that it can be represented by a vector ''across'' the supposed direction of propagation (as in a transverse wave), it has a non-zero component ''across'' a range of neighboring directions. Thus there are infinitely many paths from ''A'' to ''B'' because there are infinitely many paths radiating from every intermediate point ''P''. and
# These delayed versions of the disturbance will reinforce each other at ''B'' if they are synchronized within some tolerance.

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 (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.

The 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. But that condition is not essential to the argument.If a ray is reflected off a sufficiently concave surface, the point of reflection is such that the total traversal time is a local maximum, ''provided'' that the paths to and from the point of reflection, considered separately, are required to be possible ray paths. But Fermat's principle imposes no such restriction; and without that restriction it is always possible to vary the overall path so as to increase its traversal time. Thus the stationary traversal time of the ray path is never a local maximum (cf.  Born & Wolf, 1970, p.129n). But, as the case of the concave reflector shows, neither is it necessarily a local minimum. Hence it is ''not'' necessarily an extremum. We must therefore be content to call it a stationarity.

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''More precisely, the energy flux density. 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 (if it is narrow) as an approximation to a ray (Fig.3).

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 ( Fig.1), 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 (s)he can swim. But 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. 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 (1) to (3) 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 (e.g.) 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 (or number of cycles) 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 (Fig.4). Let be a general point on . Then, according to Huygens' construction,

1. is the ''envelope'' (common tangent surface), on the forward side of , of all the secondary wavefronts each of which would expand in time from a point on , and


2. if the secondary wavefront expanding from point in time touches the surface at point , then '' and lie on a ray''.

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 (cf.  Fig.2), 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. De Witte, 1959, p.294, col.2.

Now let be a point on close to , and let be a point on close to . Then, by the construction,

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


2. the time taken for a secondary wavefront to reach from has at most a second-order dependence on the displacement .

By (i), the ray path is a path of stationary traversal time from to ; and by (ii), 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.If the time were reckoned from the earlier wavefront as a whole, that time would everywhere be exactly , and it would be meaningless to speak of a "stationary" or "least" time.
The "stationary" time will be the ''least'' time provided that the secondary wavefronts are more convex than the primary wavefronts (as in Fig.4). That proviso, however, does not always hold. For example, if the primary wavefront, within the range of a secondary wavefront, converges to a focus and starts diverging again, the secondary wavefront will touch the later primary wavefront from the outside instead of the inside. To allow for such complexities, we must be content to say "stationary" time rather than "least" time. Cf.  Born & Wolf, 1970, pp.128–9 (meaning of "regular neighbourhood"). 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.Moreover, using Huygens' construction to determine the law of reflection or refraction is a matter of seeking the path of stationary traversal time between two particular wavefronts; cf. Fresnel, 1827, tr. Hobson, p.305–6.

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 (as before), 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 (possibly with a restricted domain of comparison, as noted above), 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.In Huygens' construction, the choice of the envelope of secondary wavefronts on the ''forward'' side of — that is, the rejection of "backward" or "retrograde" secondary waves — is also explained by Fermat's principle. For example, in Fig.2, the traversal time of the path (where the last leg "doubles back") is ''not'' stationary with respect to variation of , but is maximally sensitive to movement of along the leg .

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 (orthogonal) 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 (also called a ''uniform'' 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 (source) 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 (congruent and similarly orientated) 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 (but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical). 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 $v_$ (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path). Then the traversal time of the entire path is

(where and simply denote the endpoints and are not to be construed as values of or ). The condition for to be a ''ray'' path is that the first-order change in due to a change in is zero; that is,
$\delta T =\, \delta\int_A^B\frac \,=\, 0\,.$

Now let us define the ''optical length'' of a given path (''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. Then, if denotes the propagation speed in the reference medium (e.g., the speed of light in a vacuum), the optical length of a path traversed in time is , and the optical length of a path traversed in time is . So, multiplying equation (1) through by , we obtain
$S \,= \int_A^B\!dS \,= \int_A^B\frac\,ds = \int_A^B\!n_\,ds\,,$
where $\,n_=c/v_\,$ is the ''ray index'' — that is, the refractive index calculated on the ''ray'' velocity instead of the usual phase velocity (wave-normal velocity). For an infinitesimal path, we have $dS=n_ds\,,\,$ 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 (Fermat's principle) becomes

This has the form of Maupertuis's principle in classical mechanics (for a single particle), 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,The ray direction is the direction of constructive interference, which is the direction of the group velocity. However, the "ray velocity" is defined not as the group velocity, but as the phase velocity measured in that direction, so that "the phase velocity is the projection of the ray velocity on to the direction of the wave normal" (the quote is from Born & Wolf, 1970, p.669). In an isotropic medium, by symmetry, the directions of the ray and phase velocities are the same, so that the "projection" reduces to an identity. To put it another way: in an isotropic medium, since the ray and phase velocities have the same direction (by symmetry), and since both velocities follow the phase (by definition), they must also have the same magnitude. 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 (2) may be written
$\begin \delta S &=\,\delta\int_A^B\!n_\,\sqrt\\ &=\,\delta\int_A^B\!n_\,\sqrt~ds \,=\,0\,. \end$
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
$L(x,y,z,\dot,\dot,\dot) \,=\, n(x,y,z)\,\sqrt\,,$
Fermat's principle becomes
$\delta S =\, \delta\int_A^B\!L(x,y,z,\dot,\dot,\dot)\,ds\,=\,0\,.$
If the direction of propagation is always such that we can use instead of as the parameter of the path (and the overdot to denote differentiation w.r.t.  instead of ), the optical Lagrangian can instead be written
$L\big(x(z),y(z),\dot(z),\dot(z),z\big) = n(x,y,z)\,\sqrt\,,$
so that Fermat's principle becomes
$\delta S =\, \delta\int_A^B\! L\big(x(z),y(z),\dot(z),\dot(z),z\big)\,dz\,=\,0.$
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'' (1st century CE), 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.Ibn al-Haytham, writing in Cairo in the 2nd decade of the 11th century, also believed that light took the path of least resistance and that denser media offered more resistance, but he retained a more conventional notion of "resistance". If this notion was to explain refraction, it required the resistance to vary with direction in a manner that was hard to reconcile with reflection. Meanwhile Ibn Sahl had already arrived at the correct law of refraction by a different method; but his law was not propagated ( Mihas, 2006, pp.761–5; Darrigol, 2012, pp.20–21,41).
The problem solved by Fermat is mathematically equivalent to the following: given two points in different media with different densities, minimize the ''density-weighted'' length of the path between the two points. In Louvain, in 1634 (by which time Willebrord Snellius had rediscovered Ibn Sahl's law, and Descartes had derived it but not yet published it), the Jesuit professor Wilhelm Boelmans gave a correct solution to this problem, and set its proof as an exercise for his Jesuit students ( Ziggelaar, 1980).

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 (see ''History of variational principles in physics''). It was the more notable because it used the method of ''adequality

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