Editing Augustin-Jean Fresnel (section)

==== Background: Uniaxial and biaxial crystals; Biot's laws ====

When light passes through a slice of calcite cut perpendicular to its optic axis, the difference between the propagation times of the ordinary and extraordinary waves has a second-order dependence on the angle of incidence. If the slice is observed in a highly convergent cone of light, that dependence becomes significant, so that a chromatic-polarization experiment will show a pattern of concentric rings. But most minerals, when observed in this manner, show a more complicated pattern of rings involving two foci and a [[lemniscate]] curve, as if they had ''two'' optic axes.<ref>Jenkins & White, 1976, pp.&nbsp;576–579 (§{{hsp}}27.9, esp. Fig.&nbsp;27M).</ref>{{r|derochette-2004}} The two classes of minerals naturally become known as ''uniaxal'' and ''biaxal''—or, in later literature, ''uniaxial'' and ''biaxial''.

In 1813, Brewster observed the simple concentric pattern in "[[beryl]], [[emerald]], [[ruby]] &c." The same pattern was later observed in calcite by [[William Hyde Wollaston|Wollaston]], Biot, and [[Thomas Johann Seebeck|Seebeck]].&nbsp; Biot, assuming that the concentric pattern was the general case, tried to calculate the colors with his theory of chromatic polarization, and succeeded better for some minerals than for others. In&nbsp;1818, Brewster belatedly explained why: seven of the twelve minerals employed by Biot had the lemniscate pattern, which Brewster had observed as early as 1812; and the minerals with the more complicated rings also had a more complicated law of refraction.<ref>Buchwald, 1989, pp.&nbsp;254–255,{{tsp}}402.</ref>

In a uniform crystal, according to Huygens's theory, the secondary wavefront that expands from the origin in unit time is the ''ray-velocity surface''—that is, the surface whose "distance" from the origin in any direction is the ray velocity in that direction. In calcite, this surface is two-sheeted, consisting of a sphere (for the ordinary wave) and an oblate spheroid (for the extraordinary wave) touching each other at opposite points of a common axis—touching at the north and south poles, if we may use a geographic analogy. But according to Malus's ''corpuscular'' theory of double refraction, the ray velocity was proportional to the reciprocal of that given by Huygens's theory, in which case the velocity law was of the form

::<math>v_o^{2\!}-v_e^2 = k\sin^2\theta \,,</math>

where <math>v_o</math> and <math>v_e</math> were the ordinary and extraordinary ray velocities ''according to the corpuscular theory'', and <math>\theta</math> was the angle between the ray and the optic axis.<ref>Cf. Buchwald, 1989, p.&nbsp;269.</ref> By Malus's definition, the plane of polarization of a ray was the plane of the ray and the optic axis if the ray was ordinary, or the perpendicular plane (containing the ray) if the ray was extraordinary. In Fresnel's model, the direction of vibration was normal to the plane of polarization. Hence, for the sphere (the ordinary wave), the vibration was along the lines of latitude (continuing the geographic analogy); and for the spheroid (the extraordinary wave), the vibration was along the lines of longitude.

On 29 March 1819,<ref>Grattan-Guinness, 1990, p.&nbsp;885.</ref> Biot presented a memoir in which he proposed simple generalizations of Malus's rules for a crystal with ''two'' axes, and reported that both generalizations seemed to be confirmed by experiment. For the velocity law, the squared sine was replaced by the ''product'' of the sines of the angles from the ray to the two axes (''Biot's sine law''). And for the polarization of the ordinary ray, the plane of the ray and the axis was replaced by the plane bisecting the [[dihedral angle]] between the two planes each of which contained the ray and one axis (''Biot's dihedral law'').<ref>Buchwald, 1989, pp.&nbsp;269,{{px2}}418.<!-- On p.269, in connection with the dihedral law, Buchwald cites Biot's "Précis élémentaire...", which is listed in the bibliography as Biot 1817. However, the relevant passage first appears on p.502 of vol.2 of the 1821 edition. --></ref>{{r|biot-1819}} Biot's laws meant that a biaxial crystal with axes at a small angle, cleaved in the plane of those axes, behaved nearly like a uniaxial crystal at near-normal incidence; this was fortunate because [[gypsum]], which had been used in chromatic-polarization experiments, is biaxial.<ref>Cf.{{tsp}} Fresnel, 1822a, tr.&nbsp;Young, in ''Quarterly Journal of Science, Literature, and Art'',{{tsp}} Jul.–{{hsp}}Dec.{{tsp}}1828, at pp.&nbsp;[https://books.google.com/books?id=N69MAAAAYAAJ&pg=PA178 178–179].</ref>