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English translation of A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées", Annales de Chimie et de Physique, Ser. 2, vol. 17, pp. 102–11 (May 1821), "IIe note sur la coloration des lames cristallisées", vol. cit., pp. 167–96 (June 1821), and "Addition à la IIe note insérée dans le cahier précédent", vol. cit., pp. 312–15 (July 1821), as reprinted in Oeuvres complètes d'Augustin Fresnel, vol. 1 (1866), pp. 609–48, with the corresponding extract from the "Table Analytique" in Oeuvres complètes..., vol. 3 (1870), at pp. 581–6.
The first part of this paper cites empirical results to show that polarization is analogous to the resolution of forces in a plane, proposes that unpolarized light is a "rapid succession of an infinitude of wavetrains polarized in all azimuths", and proceeds to construct the first essentially correct explanation of chromatic polarization — that is, the wavelength-dependent modification of polarized light by birefringent "crystalline plates", causing colors to appear when the emergent light is viewed through an analyzer.
The second part extends the analysis to two stacked plates, delivering the fatal blow to the "mobile polarization" theory of J.-B. Biot, who had tried to explain the same phenomena in corpuscular terms. Then, under the subheading "Mechanical considerations on the polarization of light", it proposes that light waves are purely transverse, for both polarized and unpolarized light; introduces the concept of transverse elastic waves; redefines the axis and the ordinary and extraordinary modes of propagation in a uniaxial birefringent crystal, in terms of reactions to transverse displacements, concluding that the vibrations of the ordinary waves are perpendicular to what has been called their plane of polarization; explains Malus's law as vector decomposition; explains the non-interference of perpendicularly polarized beams; trivializes the empirical rule on whether to add a half-cycle to the path difference when calculating the hue seen through the analyzer in chromatic polarization; derives a formula for the reflectivity for what we now call the 's' component of light reflected obliquely from a transparent body; and claims empirical confirmation of the formula for two angles of incidence, namely those for which D.F.J. Arago has found that as much light is transmitted as reflected, by one glass plate or two stacked glass plates (in each case, the reflectivity for the other component is calculated from the observed azimuth of the reflected beam when that of the incident beam is 45°).
The third part — a brief postscript — announces a reflectivity formula for what we now call the 'p' component (omitting details of the derivation), and presents experimental confirmation of the ratio of the two predicted reflection coefficients, for a range of angles of incidence on glass and water.