Editing Fermat's principle (section)

=== Laplace, Young, Fresnel, and Lorentz ===

[[File:Laplace, Pierre-Simon, marquis de.jpg|thumb|Pierre-Simon Laplace {{nowrap|(1749–1827)}}]]

On 30 January 1809,{{r|laplace-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:
<blockquote>
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.<ref>Translated by [[#Young1809|Young (1809)]], p.{{nnbsp}}341; Young's italics.</ref>
</blockquote>

[[File:Thomas Young (scientist).jpg|left|thumb|Thomas Young {{nowrap|(1773–1829)}}]]

Laplace's report was the subject of a wide-ranging rebuttal by [[Thomas Young (scientist)|Thomas Young]], who wrote in part:
<blockquote>
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 {{sic|explicitly}} the basis of every determination in the Huygenian theory...&nbsp; Mr.&nbsp;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.<ref>[[#Young1809|Young, 1809]], p.{{nnbsp}}342.</ref>
</blockquote>
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.<ref>On the proof, see [[#Darr12|Darrigol, 2012]], p.{{nnbsp}}190. On the date of the reading (misprinted as 1808 in early sources), see [[#Fra74|Frankel, 1974]], p.{{nnbsp}}234n. The full text (with the misprint) is "Mémoire sur les mouvements de la lumière dans les milieux diaphanes", ''Mémoires de l'Académie des Sciences'', 1st&nbsp;Series, vol.{{nnbsp}}{{serif|X}} (1810), reprinted in ''Oeuvres complètes de Laplace'', vol.{{nnbsp}}12 (Paris, Gauthier-Villars et fils, 1898), [https://archive.org/stream/oeuvrescomplte12lapluoft/#page/267 pp.{{nnbsp}}267–298]. An intermediate version, including the proof but not the appended "Note", appeared as "Sur le mouvement de la lumière dans les milieux diaphanes", ''Mémoires de Physique et de Chimie de la Société d'Arcueil'', vol.{{nnbsp}}2 (1809), [https://books.google.com/books?id=hnJKAAAAYAAJ&pg=PA111 pp.{{nnbsp}}111–142] & [https://books.google.com/books?id=hnJKAAAAYAAJ&pg=PA495 Plate&nbsp;1] (after p.{{nnbsp}}494).</ref>

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&nbsp;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&nbsp;..." (emphasis added), and was therefore bound to sow confusion rather than clarity.

[[File:Augustin Fresnel.jpg|thumb|Augustin-Jean Fresnel {{nowrap|(1788–1827)}}]]

No such confusion subsists in [[Augustin-Jean Fresnel]]'s "Second Memoir" on double refraction ([[#SecMem|Fresnel,&nbsp;1827]]), which addresses Fermat's principle in several places (without naming Fermat), 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. (In the following summary, page numbers refer to [[#SecMem|Alfred W.{{tsp}}Hobson's translation]].)
* For refraction of a plane wave at parallel incidence on one face of an anisotropic crystalline wedge (pp.{{nnbsp}}291–2), in order to find the "first ray arrived" at an observation point beyond the other face of the wedge, it suffices to treat the rays outside the crystal as normal to the wavefronts, and within the crystal to consider only the parallel wavefronts (whatever the ray direction). So in this case, Fresnel does not attempt to trace the complete ray path.<ref group=Note>In the translation, some lines and symbols are missing from the diagram; the corrected diagram may be found in Fresnel's ''Oeuvres Complètes'', vol.{{nnbsp}}2, [https://books.google.com/books?id=g6tzUG7JmoQC&pg=PA547 p.{{nnbsp}}547].</ref>
* Next, Fresnel considers a ray refracted from a point-source ''M'' inside a crystal, through a point ''A'' on the surface, to an observation point ''B'' outside (pp.{{nnbsp}}294–6). The surface passing through ''B'' and given by the "locus of the disturbances which arrive first" is, according to Huygens' construction, normal to "the ray ''AB'' of swiftest arrival". But this construction requires knowledge of the "surface of the wave" (that is, the secondary wavefront) within the crystal.
* Then he considers a plane wavefront propagating in a medium with non-spherical secondary wavefronts, oriented so that the ray path given by Huygens' construction &ndash; from the source of the secondary wavefront to its point of tangency with the subsequent primary wavefront &ndash; is ''not'' normal to the primary wavefronts (p.{{nnbsp}}296). He shows that this path is nevertheless "the path of quickest arrival of the disturbance" from the earlier primary wavefront to the point of tangency.
* In a later heading (p.{{nnbsp}}305) he declares that "The construction of Huygens, which determines the path of swiftest arrival" is applicable to secondary wavefronts of any shape. He then notes that when we apply Huygens' construction to refraction into a crystal with a two-sheeted secondary wavefront, and draw the lines from the two points of tangency to the center of the secondary wavefront, "we shall have the directions of the two paths of swiftest arrival, and consequently of the ordinary and of the extraordinary ray."
* Under the heading "Definition of the word ''Ray''" (p.{{nnbsp}}309), he concludes that this term must be applied to the line which joins the center of the secondary wave to a point on its surface, whatever the inclination of this line to the surface.
* As a "new consideration" (pp.{{nnbsp}}310–11), he notes that if a plane wavefront is passed through a small hole centered on point ''E'', then the direction ''ED'' of maximum intensity of the resulting beam will be that in which the secondary wave starting from ''E'' will "arrive there the first", and the secondary wavefronts from opposite sides of the hole (equidistant from ''E'') will "arrive at ''D'' in the same time" as each other. This direction is ''not'' assumed to be normal to any wavefront.
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,{{r|lorentz-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 [[Luminiferous aether|aether]] and [[Aether drag hypothesis|aether drag]].

Lorentz's work was cited in 1959 by Adriaan J.&nbsp;de Witte, who then offered his own argument, which "although in essence the same, is believed to be more cogent and more general". De&nbsp;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&nbsp;Witte also noted that "The matter seems to have escaped treatment in textbooks."<ref>[[#deWitte|De Witte, 1959]], esp. pp.{{nnbsp}}293n,{{nnbsp}}298.</ref>