Editing Augustin-Jean Fresnel (section)

==== Prize memoir (1818) and sequel ====

On 17 March 1817, the Académie des Sciences announced that diffraction would be the topic for the biannual physics ''Grand Prix'' to be awarded in 1819.<ref>Kipnis, 1991, p.&nbsp;218; Buchwald, 2013, p.&nbsp;453; Levitt, 2013, p.&nbsp;44.&nbsp; Frankel (1976, pp.&nbsp;160–161) and Grattan-Guinness (1990, p.&nbsp;867) note that the topic was first ''proposed'' on 10 February 1817. Darrigol alone (2012, p.&nbsp;203) says that the competition was "opened" on 17 March ''1818''. Prizes were offered in odd-numbered years for physics and in even-numbered years for mathematics (Frankel, 1974, p.&nbsp;224n).</ref><!-- Buchwald, 1989, p.&nbsp;169. {{r|watson-2016|p=142}} --> The deadline for entries was set at 1 August 1818 to allow time for replication of experiments. Although the wording of the problem referred to rays and inflection and did not invite wave-based solutions, Arago and Ampère encouraged Fresnel to enter.<ref>Buchwald, 1989, pp.&nbsp;169–171; Frankel, 1976, p.&nbsp;161; Silliman, 1967, pp.&nbsp;183–184; Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;xxxvi–xxxvii.</ref>

In the fall of 1817, Fresnel, supported by de&nbsp;Prony, obtained a leave of absence from the new head of the Corp des Ponts, [[Louis Becquey]], and returned to Paris.<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;xxxv; Levitt, 2013, p.&nbsp;44.</ref> He resumed his engineering duties in the spring of 1818; but from then on he was based in Paris,<ref>Silliman, 2008, p.&nbsp;166; Frankel, 1976, p.&nbsp;159.</ref> first on the [[Canal de l'Ourcq]],<ref>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;xxxv,{{tsp}}xcvi; Boutry, 1948, pp.&nbsp;599,{{px2}}601.&nbsp; Silliman (1967, p.&nbsp;180) gives the starting date as 1 May 1818.</ref> and then (from May 1819) with the [[cadastre]] of the pavements.<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;xcvi; Arago, 1857, p.&nbsp;466.</ref>{{r|ripley-dana-1879|p=486}}

On 15 January 1818, in a different context (revisited below), Fresnel showed that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions.{{r|fresnel-1818jan}} His method was similar to the [[phasor]] representation, except that the "forces" were plane [[Euclidean vector|vectors]] rather than [[complex number]]s; they could be added, and multiplied by [[scalar (physics)|scalars]], but not (yet) multiplied and divided by each other. The explanation was algebraic rather than geometric.

Knowledge of this method was assumed in a preliminary note on diffraction,<ref>Printed in Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;171–181.</ref> dated 19 April 1818 and deposited on 20 April, in which Fresnel outlined the elementary theory of diffraction as found in modern textbooks. He restated Huygens's principle in combination with the [[superposition principle]], saying that the vibration at each point on a wavefront is the sum of the vibrations that would be sent to it at that moment by all the elements of the wavefront in any of its previous positions, all elements acting separately {{crossreference|(see [[Huygens–Fresnel principle]])}}. For a wavefront partly obstructed in a previous position, the summation was to be carried out over the unobstructed portion. In directions other than the normal to the primary wavefront, the secondary waves were weakened due to obliquity, but weakened much more by destructive interference, so that the effect of obliquity alone could be ignored.<ref>Cf. Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;174–175; Buchwald, 1989, pp.&nbsp;157–158.</ref> For diffraction by a straight edge, the intensity as a function of distance from the geometric shadow could then be expressed with sufficient accuracy in terms of what are now called the normalized [[Fresnel integrals]]:

[[File:Fresnel Integrals (Normalised).svg|307px|thumb|<div style="text-align: center;">Normalized Fresnel integrals <span style="color:#00b300;">''C''(''x'')</span>{{hsp}},{{tsp}}<span style="color:#b30000;">''S''(''x'')</span></div>]]

[[File:Fresnelintegral-4.svg|307px|thumb|<div style="text-align: center;">Diffraction fringes near the limit of the geometric shadow of a straight edge. Light intensities were calculated from the values of the normalized integrals <span style="color:#00b300;">''C''(''x'')</span>{{hsp}},{{tsp}}<span style="color:#b30000;">''S''(''x'')</span></div>]]

::<math>C(x) = \!\int_0^x \!\cos\big(\tfrac{1}{2}\pi z^2\big)\,dz</math>{{quad}}<math>S(x) = \!\int_0^x \!\sin\big(\tfrac{1}{2}\pi z^2\big)\,dz\,.</math>

The same note included a table of the integrals, for an upper limit ranging from 0 to 5.1 in steps of 0.1, computed with a mean error of 0.0003,<ref>Buchwald, 1989, p.&nbsp;167; 2013, p.&nbsp;454.</ref> plus a smaller table of maxima and minima of the resulting intensity.

In his final "Memoir on the diffraction of light",<ref>Fresnel, 1818b.</ref> deposited on 29 July{{hsp}}<ref>See Fresnel, 1818b, in ''Mémoires de l'Académie Royale des Sciences…'', vol.&nbsp;{{serif|V}}, p.&nbsp;339n, and in Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;247, note{{tsp}}1.</ref> and bearing the Latin epigraph "''Natura simplex et fecunda''" ("Nature simple and fertile"),<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;247; Crew, 1900, p.&nbsp;79; Levitt, 2013, p.&nbsp;46.</ref> Fresnel slightly expanded the two tables without changing the existing figures, except for a correction to the first minimum of intensity. For completeness, he repeated his solution to "the problem of interference", whereby sinusoidal functions are added like vectors. He acknowledged the directionality of the secondary sources and the variation in their distances from the observation point, chiefly to explain why these things make negligible difference in the context, provided of course that the secondary sources do not radiate in the retrograde direction. Then, applying his theory of interference to the secondary waves, he expressed the intensity of light diffracted by a single straight edge (half-plane) in terms of integrals which involved the dimensions of the problem, but which could be converted to the normalized forms above. With reference to the integrals, he explained the calculation of the maxima and minima of the intensity (external fringes), and noted that the calculated intensity falls very rapidly as one moves into the geometric shadow.<ref>Crew, 1900, pp.&nbsp;101–108 (vector-like representation), 109 (no retrograde radiation), 110–111 (directionality and distance), 118–122 (derivation of integrals), 124–125 (maxima & minima), 129–131 (geometric shadow).</ref> The last result, as Olivier Darrigol says, "amounts to a proof of the rectilinear propagation of light in the wave theory, indeed the first proof that a modern physicist would still accept."{{hsp}}<ref>Darrigol, 2012, pp.&nbsp;204–205.</ref>

For the experimental testing of his calculations, Fresnel used red light with a wavelength of 638{{nbsp}}nm, which he deduced from the diffraction pattern in the simple case in which light incident on a single slit was focused by a cylindrical lens. For a variety of distances from the source to the obstacle and from the obstacle to the field point, he compared the calculated and observed positions of the fringes for diffraction by a half-plane, a slit, and a narrow strip—concentrating on the minima, which were visually sharper than the maxima. For the slit and the strip, he could not use the previously computed table of maxima and minima; for each combination of dimensions, the intensity had to be expressed in terms of sums or differences of Fresnel integrals and calculated from the table of integrals, and the extrema had to be calculated anew.<ref>Crew, 1900, pp.&nbsp;127–128 (wavelength), 129–131 (half-plane), 132–135 (extrema, slit); Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;350–355 (narrow strip).</ref> The agreement between calculation and measurement was better than 1.5% in almost every case.<ref>Buchwald, 1989, pp.&nbsp;179–182.</ref>

Near the end of the memoir, Fresnel summed up the difference between Huygens's use of secondary waves and his own: whereas Huygens says there is light only where the secondary waves exactly agree, Fresnel says there is complete darkness only where the secondary waves exactly cancel out.<ref>Crew, 1900, p.&nbsp;144.</ref>

[[File:Simeon Poisson.jpg|thumb|left|<div style="text-align: center;">Siméon Denis Poisson (1781–1840)</div>]]

The judging committee comprised Laplace, Biot, and [[Siméon Denis Poisson|Poisson]] (all corpuscularists), [[Joseph Louis Gay-Lussac|Gay-Lussac]] (uncommitted), and Arago, who eventually wrote the committee's report.<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;xlii; Worrall, 1989, p.&nbsp;136; Buchwald, 1989, pp.&nbsp;171,{{nbsp}}183; Levitt, 2013, pp.&nbsp;45–46.</ref> Although entries in the competition were supposed to be anonymous to the judges, Fresnel's must have been recognizable by the content.<ref>Levitt, 2013, p.&nbsp;46.</ref> There was only one other entry, of which neither the manuscript nor any record of the author has survived.<ref>Frankel, 1976, p.&nbsp;162. However, Kipnis (1991, pp.&nbsp;222–224) offers evidence that the unsuccessful entrant was [[Honoré Flaugergues]] (1755–1830?) and that the essence of his entry is contained in a "supplement" published in ''Journal de Physique'', vol.&nbsp;89 (September 1819), pp.&nbsp;161–186.</ref> That entry (identified as "no.{{nbsp}}1") was mentioned only in the last paragraph of the judges' report,<ref>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;236–237.</ref> noting that the author had shown ignorance of the relevant earlier works of Young and Fresnel, used insufficiently precise methods of observation, overlooked known phenomena, and made obvious errors. In the words of [[John Worrall (philosopher)|John&nbsp;Worrall]], "The competition facing Fresnel could hardly have been less stiff."{{hsp}}<ref>Worrall, 1989, pp.&nbsp;139–140.</ref> We may infer that the committee had only two options: award the prize to Fresnel&nbsp;("no.&nbsp;2"), or withhold it.<ref>Cf. Worrall, 1989, p.&nbsp;141.</ref>

[[File:A photograph of the Arago spot.png|thumb|Shadow cast by a 5.8{{nbsp}}mm-diameter obstacle on a screen 183{{nbsp}}cm behind, in sunlight passing through a pinhole 153{{nbsp}}cm in front. The faint colors of the fringes show the wavelength-dependence of the diffraction pattern. In the center is Poisson's{{hsp}}/Arago's spot.]]

The committee deliberated into the new year.{{r|watson-2016|p=144}} Then Poisson, exploiting a case in which Fresnel's theory gave easy integrals, predicted that if a circular obstacle were illuminated by a point-source, there should be (according to the theory) a bright spot in the center of the shadow, illuminated as brightly as the exterior. This seems to have been intended as a ''[[reductio ad absurdum]]''. Arago, undeterred, assembled an experiment with an obstacle 2{{nbsp}}mm in diameter—and there, in the center of the shadow, was [[Poisson's spot]].<ref>Darrigol, 2012, p.&nbsp;205; Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;xlii.</ref>

The unanimous{{hsp}}<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;xlii; Worrall, 1989, p.&nbsp;141.</ref> report of the committee,<ref>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;229–246.</ref> read at the meeting of the Académie on 15 March 1819,<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;229, note{{tsp}}1; Grattan-Guinness, 1990, p.&nbsp;867; Levitt, 2013, p.&nbsp;47.</ref> awarded the prize to "the memoir marked no.&nbsp;2, and bearing as epigraph: ''Natura simplex et fecunda''."{{hsp}}<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;237; Worrall, 1989, p.&nbsp;140.</ref> At the same meeting,{{r|academie-pv6|p=427}} after the judgment was delivered, the president of the Académie opened a sealed note accompanying the memoir, revealing the author as Fresnel.<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;230n.</ref> The award was announced at the public meeting of the Académie a week later, on 22 March.{{r|academie-pv6|p=432}}

Arago's verification of Poisson's counter-intuitive prediction passed into folklore as if it had decided the prize.<ref>Worrall, 1989, pp.&nbsp;135–138; Kipnis, 1991, p.&nbsp;220.</ref> That view, however, is not supported by the judges' report, which gave the matter only two sentences in the penultimate paragraph.<ref>Worrall, 1989, pp.&nbsp;143–145. The printed version of the report also refers to a note&nbsp;(E), but this note concerns further investigations that took place ''after'' the prize was decided (Worrall, 1989, pp.&nbsp;145–146; Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;236,{{tsp}}245–246). According to Kipnis (1991, pp.&nbsp;221–222), the real significance of Poisson's spot and its complement (at the center of the disk of light cast by a circular ''aperture'') was that they concerned the ''intensities'' of fringes, whereas Fresnel's measurements had concerned only the ''positions'' of fringes; but, as Kipnis also notes, this issue was pursued only ''after'' the prize was decided.</ref> Neither did Fresnel's triumph immediately convert Laplace, Biot, and Poisson to the wave theory,<ref>Concerning their ''later''{{hsp}} views, see{{hsp}} [[#Reception|§{{px2}}Reception]].</ref> for at least four reasons. First, although the professionalization of science in France had established common standards, it was one thing to acknowledge a piece of research as meeting those standards, and another thing to regard it as conclusive.<ref name=frankel-p176 /> Second, it was possible to interpret Fresnel's integrals as rules for combining ''rays''. Arago even encouraged that interpretation, presumably in order to minimize resistance to Fresnel's ideas.<ref>Buchwald, 1989, pp.&nbsp;183–184; Darrigol, 2012, p.&nbsp;205.</ref> Even Biot began teaching the Huygens-Fresnel principle without committing himself to a wave basis.<ref>Kipnis, 1991, pp.&nbsp;219–220,{{tsp}}224,{{tsp}}232–233; Grattan-Guinness, 1990, p.&nbsp;870.</ref> Third, Fresnel's theory did not adequately explain the mechanism of generation of secondary waves or why they had any significant angular spread; this issue particularly bothered Poisson.<ref>Buchwald, 1989, pp.&nbsp;186–198; Darrigol, 2012, pp.&nbsp;205–206; Kipnis, 1991, p.&nbsp;220.</ref> Fourth, the question that most exercised optical physicists at that time was not diffraction, but polarization—on which Fresnel had been working, but was yet to make his critical breakthrough.