Editing Fermat's principle (section)

== Derivation ==

=== Sufficient conditions ===

Let us suppose that:
# A disturbance propagates sequentially through a [[transmission medium|medium]] (a&nbsp;vacuum or some material, not necessarily homogeneous or [[isotropy|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 from 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'';<ref group=Note>Assumption (2) almost follows from (1) because: (a)&nbsp;to the extent that the disturbance at the intermediate point ''P'' can be represented by a [[scalar (physics)|scalar]], its influence is omnidirectional; (b)&nbsp;to the extent that it can be represented by a [[Euclidean vector|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)&nbsp;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''.</ref> 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, or interfere constructively, 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.{{r|lipsons}}

The most obvious example of a stationarity in traversal time is a (local or global) minimum &ndash; 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.<ref group=Note>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.&nbsp;[[#BW|Born & Wolf, 2002]], p.{{nnbsp}}137n). 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.</ref>

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'' &ndash; 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.<ref>See (e.g.) [[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], pp.{{nnbsp}}47,{{px2}}55,{{px2}}58,{{px2}}60,{{px2}}{{nowrap|82–6}}; [[#Opticks|Newton, 1730]], pp.{{nnbsp}}8,{{px2}}18,{{px2}}137,{{px2}}143,{{px2}}166,{{px2}}173.</ref>

=== A ray as an energy path (beam) ===

[[File:Concave lens.jpg|thumb|320px|'''Fig.{{nnbsp}}3''':{{big|&nbsp;}}An experiment demonstrating refraction (and partial reflection) of ''rays'' &ndash; approximated by, or contained in, narrow beams]]

If the corridor of paths reinforcing a ray path from ''A'' to ''B'' is substantially obstructed, this will significantly affect the ''energy''<ref group=Note>More precisely, the [[energy flux density]].</ref> reaching ''B'' from ''A'' &ndash; unlike a similar-sized obstruction outside any such corridor. Thus the ray path marks an ''energy'' path &ndash; 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.<ref>This is the essence of the argument given by Fresnel ([[#SecMem|1827, tr.&nbsp;Hobson]], {{nowrap|pp.{{tsp}}310–11}}).</ref> 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.{{nnbsp}}3).<ref>See (e.g.) [[#Opticks|Newton, 1730]], p.{{nnbsp}}55; [[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], pp.{{nnbsp}}40–41,{{tsp}}56.</ref>

=== 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 ([[#Fig1|Fig.{{nnbsp}}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.{{r|feynman-1988-51}} 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{{r|zyga-2013}} 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 [[mechanical philosophy|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.<ref>[[#deWitte|De Witte, 1959]], p.{{nnbsp}}294.</ref> In a modified form, it even works for [[matter&nbsp;wave]]s: in [[quantum mechanics]], the [[classical limit|classical path]] of a particle is obtainable by applying Fermat's principle to the associated wave &ndash; except that, because the frequency may vary with the path, the stationarity is in the [[phase&nbsp;shift]] (or number of cycles) and not necessarily in the time.{{r|ogborn-taylor-2005|vanHouten-beenakker-1995-272}}

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''.