Editing Fresnel equations (section)

== History ==
{{further|Augustin-Jean Fresnel}}

In 1808, [[Étienne-Louis Malus]] discovered that when a ray of light was reflected off a non-metallic surface at the appropriate angle, it behaved like ''one'' of the two rays emerging from a [[birefringence|doubly-refractive]] calcite crystal.<ref>Darrigol, 2012, pp.{{tsp}}191–2.</ref> He later coined the term ''polarization'' to describe this behavior.&nbsp; In 1815, the dependence of the polarizing angle on the refractive index was determined experimentally by [[David Brewster]].<ref name=brewster-1815b>D. Brewster, [http://rstl.royalsocietypublishing.org/content/105/125.full.pdf "On the laws which regulate the polarisation of light by reflexion from transparent bodies"], ''Philosophical Transactions of the Royal Society'', vol.{{tsp}}105, pp.{{tsp}}125–59, read 16&nbsp;March 1815.</ref> But the ''reason'' for that dependence was such a deep mystery that in late 1817, [[Thomas Young (scientist)|Thomas&nbsp;Young]] was moved to write:
{{Blockquote|{{bracket|T}}he great difficulty of all, which is to assign a sufficient reason for the reflection or nonreflection of a polarised ray, will probably long remain, to mortify the vanity of an ambitious philosophy, completely unresolved by any theory.<ref>T. Young, "Chromatics" (written Sep–Oct 1817), ''Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica'', vol.{{tsp}}3 (first half, issued February 1818), pp.{{tsp}}141–63, [https://archive.org/stream/gri_33125011196801#page/n264/mode/1up concluding sentence].</ref>}}
In 1821, however, [[Augustin-Jean Fresnel]] derived results equivalent to his sine and tangent laws (above), by modeling light waves as [[S-wave|transverse elastic waves]] with vibrations perpendicular to what had previously been called the [[plane of polarization]]. Fresnel promptly confirmed by experiment that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water; in particular, the equations gave the correct polarization at Brewster's angle.<ref>Buchwald, 1989, pp.{{tsp}}390–91; Fresnel, 1866, pp.{{tsp}}646–8.</ref> The experimental confirmation was reported in a "postscript" to the work in which Fresnel first revealed his theory that light waves, including "unpolarized" waves, were ''purely'' transverse.<ref name=fresnel-1821a>A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et&nbsp;seq., ''Annales de Chimie et de Physique'', vol.{{nbsp}}17, pp.{{nbsp}}102–11 (May 1821), 167–96 (June 1821), 312–15 ("Postscript", July 1821); reprinted in Fresnel, 1866, pp.{{nbsp}}609–48; translated as "On the calculation of the tints that polarization develops in crystalline plates, &{{nbsp}}postscript", {{Zenodo|4058004}} / {{doi|10.5281/zenodo.4058004}}, 2021.</ref>

Details of Fresnel's derivation, including the modern forms of the sine law and tangent law, were given later, in a memoir read to the [[French Academy of Sciences]] in January 1823.<ref name=fresnel-1823a>A. Fresnel, "Mémoire sur la loi des modifications que la réflexion imprime à la lumière polarisée" ("Memoir on the law of the modifications that reflection impresses on polarized light"), read 7&nbsp;January 1823; reprinted in Fresnel, 1866, pp.{{tsp}}767–99 (full text, published 1831), pp.{{tsp}}753–62 (extract, published 1823). See especially pp.{{tsp}}773 (sine law), 757 (tangent law), 760–61 and 792–6 (angles of total internal reflection for given phase differences).</ref> That derivation combined conservation of energy with continuity of the ''tangential'' vibration at the interface, but failed to allow for any condition on the ''normal'' component of vibration.<ref>Buchwald, 1989, pp.{{tsp}}391–3; Whittaker, 1910, pp.{{tsp}}133–5.</ref> The first derivation from ''electromagnetic'' principles was given by [[Hendrik Lorentz]] in 1875.<ref>Buchwald, 1989, p.{{hsp}}392.</ref>

In the same memoir of January 1823,<ref name=fresnel-1823a /> Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients ({{math|''r''<sub>s</sub>}} and {{math|''r''<sub>p</sub>}}) gave complex values with unit magnitudes. Noting that the magnitude, as usual, represented the ratio of peak amplitudes, he guessed that the [[argument (complex analysis)|argument]] represented the phase shift, and verified the hypothesis experimentally.<ref>Lloyd, 1834, pp.{{tsp}}369–70; Buchwald, 1989, pp.{{tsp}}393–4,{{tsp}}453; Fresnel, 1866, pp.{{tsp}}781–96.</ref> The verification involved
* calculating the angle of incidence that would introduce a total phase difference of 90° between the s and p components, for various numbers of total internal reflections at that angle (generally there were two solutions),
* subjecting light to that number of total internal reflections at that angle of incidence, with an initial linear polarization at 45° to the plane of incidence, and
* checking that the final polarization was [[circular polarization|circular]].<ref>Fresnel, 1866, pp.{{tsp}}760–61,{{tsp}}792–6; Whewell, 1857, p.{{tsp}}359.</ref>
Thus he finally had a quantitative theory for what we now call the ''Fresnel rhomb'' — a device that he had been using in experiments, in one form or another, since 1817 (see ''[[Fresnel rhomb#History|Fresnel rhomb §{{tsp}}History]]'').

The success of the complex reflection coefficient inspired [[James MacCullagh]] and [[Augustin-Louis Cauchy]], beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a [[refractive index#Complex refractive index|complex refractive index]].<ref>Whittaker, 1910, pp.{{tsp}}177–179.</ref>

Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir{{hsp}}<ref name=fresnel-1822z>A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe" ("Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis"), read 9&nbsp;December 1822; printed in Fresnel, 1866, pp.{{tsp}}731–751 (full text), pp.{{tsp}}719–729 (''extrait'', first published in ''Bulletin de la Société philomathique'' for 1822, pp.&nbsp;191–8).</ref> in which he introduced the needed terms ''[[linear polarization]]'', ''[[circular polarization]]'', and ''[[elliptical polarization]]'',<ref>Buchwald, 1989, pp.{{tsp}}230–231; Fresnel, 1866, p.{{hsp}}744.</ref> and in which he explained [[optical rotation]] as a species of [[birefringence]]: linearly-polarized light can be resolved into two circularly-polarized components rotating in opposite directions, and if these propagate at different speeds, the phase difference between them — hence the orientation of their linearly-polarized resultant — will vary continuously with distance.<ref>Buchwald, 1989, p.{{hsp}}442; Fresnel, 1866, pp.{{tsp}}737–739,{{tsp}}749.&nbsp; Cf.&nbsp;Whewell, 1857, pp.{{tsp}}356–358; Jenkins & White, 1976, pp.{{tsp}}589–590.</ref>

Thus Fresnel's interpretation of the complex values of his reflection coefficients marked the confluence of several streams of his research and, arguably, the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis (see ''[[Augustin-Jean Fresnel]]'').