Editing Augustin-Jean Fresnel (section)

==== First memoir and supplements (1821–22) ====

Until Fresnel turned his attention to biaxial birefringence, it was assumed that one of the two refractions was ordinary, even in biaxial crystals.<ref>Buchwald, 1989, p.&nbsp;260.</ref> But, in a memoir submitted{{hsp}}<ref group=Note>In Fresnel's collected works (1866–70), a paper is said to have been "presented" ("''présenté''") if it was merely delivered to the Permanent Secretary of the Académie for witnessing or processing (cf. vol.&nbsp;1, p.&nbsp;487; vol.&nbsp;2, pp.&nbsp;261,{{px2}}308). In such cases this article prefers the generic word "submitted", to avoid the impression that the paper had a formal reading.</ref> on 19 November 1821,<ref>Printed in Fresnel, 1866–70, vol.&nbsp;2, pp.&nbsp;261–308.</ref> Fresnel reported two experiments on [[topaz]] showing that ''neither refraction'' was ordinary in the sense of satisfying Snell's law; that is, neither ray was the product of spherical secondary waves.<ref>Silliman, 1967, pp.&nbsp;243–246 (first experiment); Buchwald, 1989, pp.&nbsp;261–267 (both experiments). The first experiment was briefly reported earlier in Fresnel, 1821c.</ref>

The same memoir contained Fresnel's first attempt at the biaxial velocity law. For calcite, if we interchange the equatorial and polar radii of Huygens's oblate spheroid while preserving the polar direction, we obtain a ''prolate'' spheroid touching the sphere at the equator. A&nbsp;plane through the center/origin cuts this prolate spheroid in an ellipse whose major and minor semi-axes give the magnitudes of the extraordinary and ordinary ray velocities in the direction normal to the plane, and (said Fresnel) the directions of their respective vibrations. The direction of the optic axis is the normal to the plane for which the ellipse of intersection reduces to a ''circle''. So, for the biaxial case, Fresnel simply replaced the prolate spheroid with a triaxial [[ellipsoid]],<ref>Buchwald (1989, pp.&nbsp;267–272) and Grattan-Guinness (1990, pp.&nbsp;893–894 call it the "ellipsoid of elasticity".</ref> which was to be sectioned by a plane in the same way. In general there would be ''two'' planes passing through the center of the ellipsoid and cutting it in a circle, and the normals to these planes would give ''two'' optic axes. From the geometry, Fresnel deduced Biot's sine law (with the ray velocities replaced by their reciprocals).<ref>Buchwald, 1989, pp.&nbsp;267–272; Grattan-Guinness, 1990, pp.&nbsp;885–887.</ref>

The ellipsoid indeed gave the correct ray velocities (although the initial experimental verification was only approximate). But it did not give the correct directions of vibration, for the biaxial case or even for the uniaxial case, because the vibrations in Fresnel's model were tangential to the wavefront—which, for an extraordinary ray, is ''not'' generally normal to the ray. This error (which is small if, as in most cases, the birefringence is weak) was corrected in an "extract" that Fresnel read to the Académie a week later, on 26 November. Starting with Huygens's spheroid, Fresnel obtained a 4th-degree surface which, when sectioned by a plane as above, would yield the ''wave-normal velocities'' for a wavefront in that plane, together with their vibration directions. For the biaxial case, he generalized the equation to obtain a surface with three unequal principal dimensions; this he subsequently called the "surface of elasticity". But he retained the earlier ellipsoid as an approximation, from which he deduced Biot's dihedral law.<ref>Buchwald, 1989, pp.&nbsp;274–279.</ref>

Fresnel's initial derivation of the surface of elasticity had been purely geometric, and not deductively rigorous. His first attempt at a ''mechanical'' derivation, contained in a "supplement" dated 13 January 1822, assumed that (i)&nbsp;there were three mutually perpendicular directions in which a displacement produced a reaction in the same direction, (ii)&nbsp;the reaction was otherwise a linear function of the displacement, and (iii)&nbsp;the radius of the surface in any direction was the square root of the component, ''in that direction'', of the reaction to a unit displacement in that direction. The last assumption recognized the requirement that if a wave was to maintain a fixed direction of propagation and a fixed direction of vibration, the reaction must not be outside the plane of those two directions.<ref>Buchwald, 1989, pp.&nbsp;279–280.</ref>

In the same supplement, Fresnel considered how he might find, for the biaxial case, the secondary wavefront that expands from the origin in unit time—that is, the surface that reduces to Huygens's sphere and spheroid in the uniaxial case. He noted that this "wave surface" (''surface de l'onde'')<ref>Literally "surface of the wave"—as in Hobson's translation of Fresnel 1827.</ref> is tangential to all possible plane wavefronts that could have crossed the origin one unit of time ago, and he listed the mathematical conditions that it must satisfy. But he doubted the feasibility of deriving the surface ''from'' those conditions.<ref>Fresnel, 1866–70, vol.&nbsp;2, pp.&nbsp;340,{{tsp}}361–363; Buchwald, 1989, pp.&nbsp;281–283. The derivation of the "wave surface" ''from'' its tangent planes was eventually accomplished by Ampère in 1828 (Lloyd, 1834, pp.&nbsp;386–387; Darrigol, 2012, p.&nbsp;218; Buchwald, 1989, pp.&nbsp;281,{{px2}}457).</ref>

In a "second supplement",<ref>Fresnel, 1866–70, vol.&nbsp;2, pp.&nbsp;369–442.</ref> Fresnel eventually exploited two related facts: (i)&nbsp;the "wave surface" was also the ray-velocity surface, which could be obtained by sectioning the ellipsoid that he had initially mistaken for the surface of elasticity, and (ii)&nbsp;the "wave surface" intersected each plane of symmetry of the ellipsoid in two curves: a circle and an ellipse. Thus he found that the "wave surface" is described by the 4th-degree equation

::<math>r^2\big(a^2x^{2\!}+ b^2y^{2\!}+ c^2z^2\big) - a^2\big(b^{2\!} + c^2\big)x^2 - b^2\big(c^{2\!} + a^2\big)y^2 - c^2\big(a^{2\!} + b^2\big)z^2 + a^2b^2c^2 =\, 0\,,</math>

where <math>\,r^2 = x^{2\!} + y^{2\!} + z^2,\,</math> and <math>\,a,b,c\,</math> are the propagation speeds in directions normal to the coordinate axes for vibrations along the axes (the ray and wave-normal speeds being the same in those special cases).<ref>Buchwald, 1989, pp.&nbsp;283–285; Darrigol, 2012, pp.&nbsp;217–218; Fresnel, 1866–70, vol.&nbsp;2, pp.&nbsp;386–388.</ref> Later commentators{{r|griffin-1842|p=19}} put the equation in the more compact and memorable form

::<math>\frac{x^2}{r^2-a^2} + \frac{y^2}{r^2-b^2} + \frac{z^2}{r^2-c^2} \,=\, 1\,.</math>

Earlier in the "second supplement", Fresnel modeled the medium as an array of point-masses and found that the force-displacement relation was described by a [[symmetric matrix]], confirming the existence of three mutually perpendicular axes on which the displacement produced a parallel force.<ref>Grattan-Guinness, 1990, pp.&nbsp;891–892; Fresnel, 1866–70, vol.&nbsp;2, pp.&nbsp;371–379.</ref> Later in the document, he noted that in a biaxial crystal, unlike a uniaxial crystal, the directions in which there is only one wave-normal velocity are not the same as those in which there is only one ray velocity.<ref>Buchwald, 1989, pp.&nbsp;285–286; Fresnel, 1866–70, vol.&nbsp;2, p.&nbsp;396.</ref> Nowadays we refer to the former directions as the ''optic'' axes or ''binormal'' axes, and the latter as the ''ray'' axes or ''biradial'' axes {{crossreference|(see [[Birefringence]])}}.{{r|lunney-weaire-2006}}

Fresnel's "second supplement" was signed on 31 March 1822 and submitted the next day—less than a year after the publication of his pure-transverse-wave hypothesis, and just less than a year after the demonstration of his prototype eight-panel lighthouse lens {{crossreference|(see [[#Lighthouses and the Fresnel lens|below]])}}.