Editing Augustin-Jean Fresnel (section)

==== Interference of polarized light, chromatic polarization (1816–21) ====

In July or August 1816, Fresnel discovered that when a birefringent crystal produced two images of a single slit, he could ''not'' obtain the usual two-slit interference pattern, even if he compensated for the different propagation times. A more general experiment, suggested by Arago, found that if the two beams of a double-slit device were separately polarized, the interference pattern appeared and disappeared as the polarization of one beam was rotated, giving full interference for parallel polarizations, but no interference for perpendicular polarizations {{crossreference|(see [[Fresnel–Arago laws]])}}.<ref>Buchwald, 1989, pp.&nbsp;203,{{px2}}205; Darrigol, 2012, p.&nbsp;206; Silliman, 1967, pp.&nbsp;203–205.</ref> These experiments, among others, were eventually reported in a brief memoir published in 1819 and later translated into English.<ref>Arago & Fresnel, 1819.</ref>

In a memoir drafted on 30 August 1816 and revised on 6 October, Fresnel reported an experiment in which he placed two matching thin laminae in a double-slit apparatus—one over each slit, with their optic axes perpendicular—and obtained two interference patterns offset in opposite directions, with perpendicular polarizations. This, in combination with the previous findings, meant that each lamina split the incident light into perpendicularly polarized components with different velocities—just like a normal (thick) birefringent crystal, and contrary to Biot's "mobile polarization" hypothesis.<ref>Darrigol, 2012, p.&nbsp;207; Frankel, 1976, pp.&nbsp;163–164,{{tsp}}182.</ref>

Accordingly, in the same memoir, Fresnel offered his first attempt at a wave theory of chromatic polarization. When polarized light passed through a crystal lamina, it was split into ordinary and extraordinary waves (with intensities described by Malus's law), and these were perpendicularly polarized and therefore did not interfere, so that no colors were produced (yet). But if they then passed through an ''analyzer'' (second polarizer), their polarizations were brought into alignment (with intensities again modified according to Malus's law), and they would interfere.<ref>Darrigol, 2012, p.&nbsp;206.</ref> This explanation, by itself, predicts that if the analyzer is rotated 90°, the ordinary and extraordinary waves simply switch roles, so that if the analyzer takes the form of a calcite crystal, the two images of the lamina should be of the same hue (this issue is revisited below). But in fact, as Arago and Biot had found, they are of complementary colors. To correct the prediction, Fresnel proposed a phase-inversion rule whereby ''one'' of the constituent waves of ''one'' of the two images suffered an additional 180° phase shift on its way through the lamina. This inversion was a weakness in the theory relative to Biot's, as Fresnel acknowledged,<ref>Frankel, 1976, p.&nbsp;164.</ref> although the rule specified which of the two images had the inverted wave.<ref>Buchwald, 1989, p.&nbsp;386.</ref> Moreover, Fresnel could deal only with special cases, because he had not yet solved the problem of superposing sinusoidal functions with arbitrary phase differences due to propagation at different velocities through the lamina.<ref>Buchwald, 1989, pp.&nbsp;216,{{px2}}384.</ref>

He solved that problem in a "supplement" signed on 15 January 1818{{hsp}}{{r|fresnel-1818jan}} (mentioned above). In the same document, he accommodated Malus's law by proposing an underlying law: that if polarized light is incident on a birefringent crystal with its optic axis at an angle ''θ'' to the "plane of polarization", the ordinary and extraordinary vibrations (as functions of time) are scaled by the factors cos{{tsp}}''θ'' and sin{{tsp}}''θ'', respectively. Although modern readers easily interpret these factors in terms of perpendicular components of a ''transverse'' oscillation, Fresnel did not (yet) explain them that way. Hence he still needed the phase-inversion rule. He applied all these principles to a case of chromatic polarization not covered by Biot's formulae, involving ''two'' successive laminae with axes separated by 45°, and obtained predictions that disagreed with Biot's experiments (except in special cases) but agreed with his own.<ref>Buchwald, 1989, pp.&nbsp;333–336; Darrigol, 2012, pp.&nbsp;207–208. (Darrigol gives the date as 1817, but the page numbers in his footnote 95 fit his reference "1818b", not "1817".)</ref>

Fresnel applied the same principles to the standard case of chromatic polarization, in which ''one'' birefringent lamina was sliced parallel to its axis and placed between a polarizer and an analyzer. If the analyzer took the form of a thick calcite crystal with its axis in the plane of polarization, Fresnel predicted that the intensities of the ordinary and extraordinary images of the lamina were respectively proportional to

::<math>I_o = \cos^2i\,\cos^2(i{-}s) + \sin^2i\,\sin^2(i{-}s) + \tfrac{1}{2}\sin 2i\,\sin 2(i{-}s)\cos\phi\,,</math>

::<math>I_e = \cos^2i\,\sin^2(i{-}s) + \sin^2i\,\cos^2(i{-}s) - \tfrac{1}{2}\sin 2i\,\sin 2(i{-}s)\cos\phi\,,</math>

where <math>i</math> is the angle from the initial plane of polarization to the optic axis of the lamina,{{tsp}} <math>s</math> is the angle from the initial plane of polarization to the plane of polarization of the final ordinary image, and <math>\phi</math> is the phase lag of the extraordinary wave relative to the ordinary wave due to the difference in propagation times through the lamina. The terms in <math>\phi</math> are the frequency-dependent terms and explain why the lamina must be ''thin'' in order to produce discernible colors: if the lamina is too thick, <math>\cos\phi</math> will pass through too many cycles as the frequency varies through the visible range, and the eye (which divides the visible spectrum into only [[cone cell|three bands]]) will not be able to resolve the cycles.

From these equations it is easily verified that <math>\,I_o+I_e=1\,</math> for all <math>\phi,</math> so that the colors are complementary. Without the phase-inversion rule, there would be a ''plus'' sign in front of the last term in the second equation, so that the <math>\phi</math>-dependent term would be the same in both equations, implying (incorrectly) that the colors were of the same hue.

These equations were included in an undated note that Fresnel gave to Biot,<ref>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;533–537. On the provenance of the note, see p.&nbsp;523. In the above text, ''φ'' is an abbreviation for Fresnel's {{nowrap|2''π''(''e''{{hsp}}−{{hsp}}''o'')}}, where ''e'' and ''o'' are the numbers of cycles taken by the extraordinary and ordinary waves to travel through the lamina.</ref> to which Biot added a few lines of his own. If we substitute

::<math>U=\cos^2\tfrac{\phi}{2}</math> &nbsp;and&nbsp; <math>A=\sin^2\tfrac{\phi}{2}\,,</math>

then Fresnel's formulae can be rewritten as

::<math> \!I_o = U\cos^2 s + A\cos^2(2i-s)\,,</math>
::<math>   I_e = U\sin^2 s + A\sin^2(2i-s)\,,</math>

which are none other than Biot's empirical formulae of 1812,<ref>Buchwald, 1989, p.&nbsp;97; Frankel, 1976, p.  148.</ref> except that Biot interpreted <math>U</math> and <math>A</math> as the "unaffected" and "affected" selections of the rays incident on the lamina. If Biot's substitutions were accurate, they would imply that his experimental results were more fully explained by Fresnel's theory than by his own.

Arago delayed reporting on Fresnel's works on chromatic polarization until June 1821, when he used them in a broad attack on Biot's theory. In his written response, Biot protested that Arago's attack went beyond the proper scope of a report on the nominated works of Fresnel. But Biot also claimed that the substitutions for <math>U</math> and <math>A,</math> and therefore Fresnel's expressions for <math>I_o</math> and <math>I_e,</math> were empirically wrong because when Fresnel's intensities of spectral colors were mixed according to Newton's rules, the squared cosine and sine functions varied too smoothly to account for the observed sequence of colors. That claim drew a written reply from Fresnel,<ref>Fresnel, 1821b.</ref> who disputed whether the colors changed as abruptly as Biot claimed,<ref>Fresnel, 1821b, §3.</ref> and whether the human eye could judge color with sufficient objectivity for the purpose. On the latter question, Fresnel pointed out that different observers may give different names to the same color. Furthermore, he said, a single observer can only compare colors side by side; and even if they are judged to be the same, the identity is of sensation, not necessarily of composition.<ref>Fresnel, 1821b, §1 & footnotes.</ref> Fresnel's oldest and strongest point—that thin crystals were subject to the same laws as thick ones and did not need or allow a separate theory—Biot left unanswered.&nbsp; Arago and Fresnel were seen to have won the debate.<ref>Buchwald, 1989, pp.&nbsp;237–251; Frankel, 1976, pp.&nbsp;165–168; Darrigol, 2012, pp.&nbsp;208–209.</ref>

Moreover, by this time Fresnel had a new, simpler explanation of his equations on chromatic polarization.