Note on the remarks of Mr. Biot relating to colors of thin plates

English translation of A. Fresnel, “Note sur les remarques de M. Biot, publiées dans le cahier précédent”, Annales de Chimie et de Physique, Ser. 2, vol. 17, pp. 393–403 (August 1821), as reprinted in Oeuvres complètes d'Augustin Fresnel, vol. 1 (1866), pp. 601–608, with the corresponding extract from the “Table Analytique” in Oeuvres complètes..., vol. 3 (1870), at pp. 580–81.

In a chromatic-polarization experiment with the principal section of the calcite analyzer in the initial “plane of polarization”, and the axis of the “crystalline plate” at an angle thereto, the observed color of the extraordinary image as a function of path difference should, according to Fresnel's theory, match that of Newton's rings in reflected light. Biot agrees with the match, but contends, using Newton's methodology, that the colors predicted by Fresnel change too smoothly and are insufficiently saturated.  Fresnel dismantles this argument, noting that:

• Contrary to Biot, Newton's “empirical formula” [using his color circle to find the resultant of a combination of spectral colors] is not, and does not purport to be, exact.

• The colors named in “Newton's Table” are not unambiguous, and moreover the mismatch alleged by Biot usually does not extend to the names, but involves only the white content (loss of saturation), for which Newton's “empirical formula” seems particularly unreliable, and which Biot has not measured anyway.

• (§2) Newton's diagram showing what prismatic colors are present for a given thickness of air [treating each color as simply present or absent] is not, and does not purport to be, exact.

• The approximation embodied in this diagram — that the rings in monochromatic light alternate sharply between bright and dark bands, corresponding to equal ranges of thickness — is easily debunked by inspecting the rings with a loupe under a sufficiently bright light.

• (§3) Similarly, in the analogous chromatic-polarization configuration, the extraordinary image is not fully dark at the extremes of the path differences corresponding to the dark rings, but just as bright as the ordinary image, as Fresnel's formulae predict.

(§2, cont.) The confirmation of Fresnel's interference formulae through the location of diffraction minima is more thorough than it may seem, because each minimum is not simply a cancellation of two wavetrains, but the convergence of an infinitude of secondary wavetrains with all possible degrees of reinforcement and cancellation.

Fresnel acknowledges his debt to Biot's experimental work (and Young's priority in discovering that the colors of crystalline plates depend on the path difference between the ordinary and extraordinary waves). He acknowledges the relation between Biot's formulae and his own in the case of a single plate. But he notes that his own formulae are not deducible from Biot's, and that his own theory — unlike Biot's — extends to more complicated cases without additional assumptions.