Editing Fermat's principle (section)

=== 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.