Editing Fermat's principle (section)

== Equivalence to Huygens' construction ==
{{primary sources|section|date=January 2025}}

[[File:Huygens-construction-two-iterations.svg|thumb|280px|'''Fig.{{nnbsp}}4''':{{big|&nbsp;}}Two iterations of Huygens' construction. In the first iteration, the later wavefront {{mvar|W&prime;}} is derived from the earlier wavefront {{mvar|W}} by taking the envelope of all the secondary wavefronts (gray arcs) expanding in a given time from all the points (e.g., {{mvar|P}}) on {{mvar|W}}. The arrows show the ray directions.]]

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 {{mvar|W}} be a wavefront at time {{mvar|t}}, and let the surface {{mvar|W&prime;}} be the same wavefront at the later time {{math|''t'' + Δ''t''}} (Fig.{{nnbsp}}4). Let {{mvar|P}} be a general point on {{mvar|W}}. Then, according to Huygens' construction,<ref>[[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], pp.{{nnbsp}}19,{{px2}}50–51,{{px2}}63–65,{{px2}}68,{{px2}}75.</ref>
<ol style="list-style-type:lower-alpha;">
<li> {{mvar|W&prime;}} is the ''[[envelope (mathematics)|envelope]]'' (common tangent surface), on the forward side of {{mvar|W}}, of all the secondary wavefronts each of which would expand in time {{math|Δ''t''}} from a point on {{mvar|W}}, and</li>
<li> if the secondary wavefront expanding from point {{mvar|P}} in time {{math|Δ''t''}} touches the surface {{mvar|W&prime;}} at point {{mvar|P&prime;}}, then ''{{mvar|P}} and {{mvar|P&prime;}} lie on a ray''.</li>
</ol>
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,<ref>[[#SecMem|Fresnel, 1827, tr.&nbsp;Hobson]], p.{{nnbsp}}309.</ref> and may differ from the normal of the secondary wavefront (cf.&nbsp;[[#Fig2|Fig.{{nnbsp}}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.<ref name="De Witte, 1959">[[#deWitte|De Witte, 1959]], p.{{nnbsp}}294, col.{{nnbsp}}2.</ref>

Now let {{mvar|Q}} be a point on {{mvar|W}} close to {{mvar|P}}, and let {{mvar|Q&prime;}} be a point on {{mvar|W&prime;}} close to {{mvar|P&prime;}}. Then, by the construction,
<ol style="list-style-type:lower-roman;">
<li>&nbsp; the time taken for a secondary wavefront from {{mvar|P}} to reach {{mvar|Q&prime;}} has at most a second-order dependence on the displacement {{mvar|P&prime;Q&prime;}}, and</li>
<li> the time taken for a secondary wavefront to reach {{mvar|P&prime;}} from {{mvar|Q}} has at most a second-order dependence on the displacement {{mvar|PQ}}.</li>
</ol>
By (i), the ray path is a path of stationary traversal time from {{mvar|P}} to {{mvar|W&prime;}};<ref>Cf. [[#SecMem|Fresnel, 1827, tr.&nbsp;Hobson]], p.{{nnbsp}}305.</ref> and by (ii), it is a path of stationary traversal time from a point on {{mvar|W}} to {{mvar|P&prime;}}.<ref>Cf. [[#SecMem|Fresnel, 1827, tr.&nbsp;Hobson]], p.{{nnbsp}}296.</ref>

So Huygens' construction implicitly defines a ray path as ''a&nbsp;path of stationary traversal time between successive positions of a wavefront'', the time being reckoned from a ''point-source'' on the earlier wavefront.<ref group=Note>If the time were reckoned from the earlier wavefront as a whole, that time would everywhere be exactly {{mvar|&Delta;t}}, and it would be meaningless to speak of a "stationary" or "least" time.<br/>The "stationary" time will be the ''least'' time provided that the secondary wavefronts are more convex than the primary wavefronts (as in Fig.{{nnbsp}}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.{{tsp}} Cf.&nbsp;[[#BW|Born & Wolf, 2002]], pp.{{nnbsp}}136–7 (meaning of "regular neighbourhood").</ref> 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 affected paths and the affected portions of the wavefronts.<ref group=Note>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. [[#SecMem|Fresnel, 1827, tr.&nbsp;Hobson]], p.{{nnbsp}}305–6.</ref>

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 {{mvar|W}} at time {{mvar|t}}, and which becomes surface {{mvar|W&prime;}} at the later time {{math|''t'' + Δ''t''}}, is emitted from point {{mvar|A}} at time&nbsp;{{math|0}}. Let {{mvar|P}} be a point on {{mvar|W}} (as before), and {{mvar|B}} a point on {{mvar|W&prime;}}. And let {{mvar|A}}, {{mvar|W}}, {{mvar|W&prime;}}, and {{mvar|B}} be given, so that the problem is to find {{mvar|P}}.

If {{mvar|P}} satisfies Huygens' construction, so that the secondary wavefront from {{mvar|P}} is tangential to {{mvar|W&prime;}} at {{mvar|B}}, then {{mvar|PB}} is a path of stationary traversal time from {{mvar|W}} to {{mvar|B}}. Adding the fixed time from {{mvar|A}} to {{mvar|W}}, we find that {{mvar|APB}} is the path of stationary traversal time from {{mvar|A}} to {{mvar|B}} (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 {{mvar|W&prime;}} has a well-defined tangent plane at {{mvar|B}}. Thus Huygens' construction and Fermat's principle are geometrically equivalent.<ref>[[#deWitte|De Witte (1959)]] gives a more sophisticated proof of the same result, using calculus of variations.</ref><ref group=Note>In Huygens' construction, the choice of the envelope of secondary wavefronts on the ''forward'' side of {{mvar|W}} &ndash; that is, the rejection of "backward" or "retrograde" secondary waves &ndash; is also explained by Fermat's principle. For example, in [[#Fig2|Fig.{{nnbsp}}2]], the traversal time of the path {{mvar|APP&prime;P}} (where the last leg "doubles back") is ''not'' stationary with respect to variation of {{mvar|P&prime;}}, but is maximally sensitive to movement of {{mvar|P&prime;}} along the leg {{mvar|PP&prime;}}.</ref>

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".<ref>The quote is from [[#BW|Born & Wolf, 2002]], p.{{nnbsp}}876.</ref> 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.