Editing Augustin-Jean Fresnel (section)

=== Diffraction ===

==== First attempt (1815) ====

On 12 July 1815, as Fresnel was about to leave Paris, Arago left him a note on a new topic:
{{blockquote|I do not know of any book that contains all the experiments that physicists are doing on the ''diffraction'' of light. M'sieur Fresnel will only be able to get to know this part of the optics by reading the work by [[Francesco Maria Grimaldi|Grimaldi]], the one by Newton, the English treatise by Jordan,{{r|jordan-1799}} and the memoirs of [[Henry Brougham, 1st Baron Brougham and Vaux|Brougham]] and Young, which are part of the collection of the ''[[Philosophical Transactions of the Royal Society|Philosophical Transactions]]''.<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;6n; Kipnis, 1991, p.&nbsp;167; emphasis added.</ref>}}
Fresnel would not have ready access to these works outside Paris, and could not read English.<ref name=readings-english>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;6–7.</ref> But, in Mathieu—with a point-source of light made by focusing sunlight with a drop of honey, a crude [[filar micrometer|micrometer]] of his own construction, and supporting apparatus made by a local locksmith—he began his own experiments.<ref>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;xxxi (micrometer, locksmith {{bracket|''serrurier''}}, supports), 6n (locksmith); Buchwald, 1989, pp.&nbsp;122 (honey drop), 125–126 (micrometer, with diagram); Boutry 1948, p.&nbsp;595 and Levitt, 2013, p.&nbsp;40 (locksmith, honey drop, micrometer); Darrigol 2012, pp.&nbsp;198–199 (locksmith, honey drop).</ref> His technique was novel: whereas earlier investigators had projected the fringes onto a screen, Fresnel soon abandoned the screen and observed the fringes in space, through a lens with the micrometer at its focus, allowing more accurate measurements while requiring less light.<ref>Buchwald, 1989, pp.&nbsp;122,{{hsp}}126; Silliman, 1967, pp.&nbsp;147–149.</ref>

Later in July, after Napoleon's final defeat, Fresnel was reinstated with the advantage of having backed the winning side. He requested a two-month leave of absence, which was readily granted because roadworks were in abeyance.<ref>Levitt, 2013, pp.&nbsp;39,{{px2}}239.</ref>

On 23 September he wrote to Arago, beginning "I&nbsp;think I&nbsp;have found the explanation and the law of colored fringes which one notices in the shadows of bodies illuminated by a luminous point." In the same paragraph, however, Fresnel implicitly acknowledged doubt about the novelty of his work: noting that he would need to incur some expense in order to improve his measurements, he wanted to know "whether this is not useless, and whether the law of diffraction has not already been established by sufficiently exact experiments."{{hsp}}<ref>Kipnis, 1991, p.&nbsp;167; Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;5–6.</ref> He explained that he had not yet had a chance to acquire the items on his reading lists,<ref name=readings-english /> with the apparent exception of "Young's book", which he could not understand without his brother's help.<ref>Darrigol, 2012, p.&nbsp;198. Silliman (1967, p.&nbsp;146) identifies the brother as Fulgence, then in Paris; cf.&nbsp;Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;7n.</ref><ref group=Note>"Young's book", which Fresnel distinguished from the ''Philosophical Transactions'', is presumably ''A&nbsp;Course of Lectures on Natural Philosophy and the Mechanical Arts'' (2&nbsp;volumes, 1807). In [https://archive.org/details/lecturescourseof01younrich vol.&nbsp;1], the relevant illustrations are Plate&nbsp;{{serif|XX}} (facing p.&nbsp;777), including the famous two-source interference pattern (Fig.&nbsp;267), and Plate&nbsp;{{serif|XXX}} (facing p.&nbsp;787), including the hyperbolic paths of the fringes in that pattern (Fig.&nbsp;442) followed by sketches of other diffraction patterns and thin-plate patterns, with no visual hints on their physical causes. In [https://archive.org/details/lecturescourseof02younrich vol.&nbsp;2], which includes the Bakerian lectures from the ''Philosophical Transactions'', Fig.&nbsp;108 (p.&nbsp;632) shows just one case of an undeviated direct ray intersecting a reflected ray.</ref>&nbsp; Not surprisingly, he had retraced many of Young's steps.

In a memoir sent to the institute on 15 October 1815, Fresnel mapped the external and internal fringes in the shadow of a wire. He noticed, like Young before him, that the internal fringes disappeared when the light from one side was blocked, and concluded that "the vibrations of two rays that cross each other under a very small angle can contradict each other…"{{hsp}}<ref>Darrigol, 2012, p.&nbsp;199.</ref> But, whereas Young took the disappearance of the internal fringes as ''confirmation'' of the principle of interference, Fresnel reported that it was the internal fringes that first drew his attention to the principle. To explain the diffraction pattern, Fresnel constructed the internal fringes by considering the intersections of circular wavefronts emitted from the two edges of the obstruction, and the external fringes by considering the intersections between direct waves and waves reflected off the nearer edge. For the external fringes, to obtain tolerable agreement with observation, he had to suppose that the reflected wave was [[Phase reversal|inverted]]; and he noted that the predicted paths of the fringes were hyperbolic. In the part of the memoir that most clearly surpassed Young, Fresnel explained the ordinary laws of reflection and refraction in terms of interference, noting that if two parallel rays were reflected or refracted at other than the prescribed angle, they would no longer have the same phase in a common perpendicular plane, and every vibration would be cancelled by a nearby vibration. He noted that his explanation was valid provided that the surface irregularities were much smaller than the wavelength.<ref>Buchwald, 1989, pp.&nbsp;119,{{tsp}}131–132; Darrigol, 2012, pp.&nbsp;199–201; Kipnis, 1991, pp.&nbsp;175–176.</ref>

On 10 November, Fresnel sent a supplementary note dealing with Newton's rings and with gratings,<ref>Darrigol, 2012, p.&nbsp;201.</ref> including, for the first time, ''transmission'' gratings—although in that case the interfering rays were still assumed to be "inflected", and the experimental verification was inadequate because it used only two threads.<ref>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;48–49; Kipnis, 1991, pp.&nbsp;176–178.</ref>

As Fresnel was not a member of the institute, the fate of his memoir depended heavily on the report of a single member. The reporter for Fresnel's memoir turned out to be Arago (with [[Louis Poinsot|Poinsot]] as the other reviewer).<ref>Frankel, 1976, p.&nbsp;158; Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;9n.</ref> On 8 November, Arago wrote to Fresnel:
{{blockquote|I have been instructed by the Institute to examine your memoir on the diffraction of light; I have studied it carefully, and found many interesting experiments, some of which had already been done by Dr.&nbsp;Thomas Young, who in general regards this phenomenon in a manner rather analogous to the one you have adopted. But what neither he nor anyone had seen before you is that the ''external'' colored bands do not travel in a straight line as one moves away from the opaque body. The results you have achieved in this regard seem to me very important; perhaps they can serve to prove the truth of the undulatory system, so often and so feebly combated by physicists who have not bothered to understand it.<ref>Fresnel, 1866–70, vol.&nbsp;1, p.&nbsp;38; italics added.</ref>}}
Fresnel was troubled, wanting to know more precisely where he had collided with Young.<ref>Buchwald, 1989, pp.&nbsp;137–139.</ref> Concerning the curved paths of the "colored bands", Young had noted the hyperbolic paths of the fringes in the [[Young's interference experiment|two-source interference]] pattern, corresponding roughly to Fresnel's ''internal'' fringes, and had described the hyperbolic fringes that appear ''on the screen'' within rectangular shadows.<ref>Young, 1807, vol.&nbsp;1, p.&nbsp;787 & Figs.&nbsp;442,{{px2}}445; Young, 1855, pp.&nbsp;180–181,{{tsp}}184.</ref> He had not mentioned the curved paths of the ''external'' fringes of a shadow; but, as he later explained,<ref>Young to Arago (in&nbsp;English), 12 January 1817, in Young, 1855, pp.&nbsp;380–384, at p.&nbsp;381; quoted in Silliman, 1967, p.&nbsp;171.</ref> that was because Newton had already done so.<ref>Newton, 1730, p.&nbsp;321, Fig.&nbsp;1, where the straight rays {{serif|DG,{{tsp}}EH,{{tsp}}FI}} contribute to the curved path of a fringe, so that the same fringe is made by different rays at different distances from the obstacle{{tsp}} (cf.&nbsp;Darrigol, 2012, p.&nbsp;101, Fig.&nbsp;3.11 – where, in the caption, "1904" should be "1704" and "{{serif|CFG}}" should be "{{serif|CFI}}").</ref> Newton evidently thought the fringes were ''[[caustic (optics)|caustics]]''. Thus Arago erred in his belief that the curved paths of the fringes were fundamentally incompatible with the corpuscular theory.<ref>Kipnis, 1991, pp.&nbsp;204–205.</ref>

Arago's letter went on to request more data on the external fringes. Fresnel complied, until he exhausted his leave and was assigned to [[Rennes]] in the département of [[Ille-et-Vilaine]]. At this point Arago interceded with [[Gaspard de Prony]], head of the École des Ponts, who wrote to [[Louis-Mathieu Molé]], head of the Corps des Ponts, suggesting that the progress of science and the prestige of the Corps would be enhanced if Fresnel could come to Paris for a time. He arrived in March 1816, and his leave was subsequently extended through the middle of the year.<ref>Silliman, 1967, pp.&nbsp;163–164; Frankel, 1976, p.&nbsp;158; Boutry, 1948, p.&nbsp;597; Levitt, 2013, pp.&nbsp;41–43,{{tsp}}239.</ref>

Meanwhile, in an experiment reported on 26 February 1816, Arago verified Fresnel's prediction that the internal fringes were shifted if the rays on one side of the obstacle passed through a thin glass lamina. Fresnel correctly attributed this phenomenon to the lower wave velocity in the glass.<ref>Silliman, 1967, pp.&nbsp;165–166; Buchwald, 1989, p.&nbsp;137; Kipnis, 1991, pp.&nbsp;178,{{tsp}}207,{{tsp}}213.</ref> Arago later used a similar argument to explain the colors in the scintillation of stars.<ref group=Note>Silliman (1967, p.&nbsp;163) and Frankel (1976, p.&nbsp;156) give the date of Arago's note on scintillation as 1814; but the sequence of events implies 1816, in agreement with Darrigol (2012, pp.&nbsp;201,{{px2}}290).&nbsp; Kipnis (1991, pp.&nbsp;202–203,{{tsp}}206) proves the later date and explains the origin and propagation of the incorrect earlier date.</ref>

Fresnel's updated memoir{{hsp}}<ref>Fresnel, 1816.</ref> was eventually published in the March 1816 issue of ''[[Annales de chimie et de physique|Annales de Chimie et de Physique]]'', of which Arago had recently become co-editor.<ref>Darrigol, 2012, p.&nbsp;201; Frankel, 1976, p.&nbsp;159.</ref> That issue did not actually appear until May.<ref>Kipnis, 1991, pp.&nbsp;166n,{{px2}}214n.</ref> In&nbsp;March, Fresnel already had competition: Biot read a memoir on diffraction by himself and his student [[Claude Pouillet]], containing copious data and arguing that the regularity of diffraction fringes, like the regularity of Newton's rings, must be linked to Newton's "fits". But the new link was not rigorous, and Pouillet himself would become a distinguished early adopter of the wave theory.<ref>Kipnis, 1991, pp.&nbsp;212–214; Frankel, 1976, pp.&nbsp;159–160,{{tsp}}173.</ref>

==== "Efficacious ray", double-mirror experiment (1816) ====

[[File:Young diffraction.svg|thumb|307px|Replica of Young's two-source interference diagram (1807), with sources ''A'' and ''B'' producing minima at ''C'', ''D'', ''E'', and ''F''<ref>Cf. Young, 1807, vol.&nbsp;1, p.&nbsp;777 & Fig.&nbsp;267.</ref>]]

[[File:Fresnel double mirror.png|thumb|307px|Fresnel's double mirror (1816). The mirror segments ''M''<sub>1</sub> and ''M''<sub>2</sub> produce virtual images ''S''<sub>1</sub> and ''S''<sub>2</sub> of the slit ''S''. In the shaded region, the beams from the two virtual images overlap and interfere in the manner of Young (above).]]

On 24 May 1816, Fresnel wrote to Young (in French), acknowledging how little of his own memoir was new.<ref>Darrigol, 2012, p.&nbsp;201; the letter is printed in Young, 1855, pp.&nbsp;376–378, and its conclusion is translated by Silliman (1967, p.&nbsp;170).</ref> But in a "supplement" signed on 14 July and read the next day,<ref>Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;129–170.</ref> Fresnel noted that the internal fringes were more accurately predicted by supposing that the two interfering rays came from some distance ''outside'' the edges of the obstacle. To explain this, he divided the incident wavefront at the obstacle into what we now call ''[[Fresnel zone]]s'', such that the secondary waves from each zone were spread over half a cycle when they arrived at the observation point. The zones on one side of the obstacle largely canceled out in pairs, except the first zone, which was represented by an "efficacious ray". This approach worked for the internal fringes, but the superposition of the efficacious ray and the direct ray did ''not'' work for the ''external'' fringes.<ref>Silliman, 1967, pp.&nbsp;177–179; Darrigol, 2012, pp.&nbsp;201–203.</ref>

The contribution from the "efficacious ray" was thought to be only ''partly'' canceled, for reasons involving the dynamics of the medium: where the wavefront was continuous, symmetry forbade oblique vibrations; but near the obstacle that truncated the wavefront, the asymmetry allowed some sideways vibration towards the geometric shadow. This argument showed that Fresnel had not (yet) fully accepted Huygens's principle, which would have permitted oblique radiation from all portions of the front.<ref>Buchwald, 1989, pp.&nbsp;134–135,{{tsp}}144–145; Silliman, 1967, pp.&nbsp;176–177.</ref>

In the same supplement, Fresnel described his well-known double mirror, comprising two flat mirrors joined at an angle of slightly less than 180°, with which he produced a two-slit interference pattern from two virtual images of the same slit. A conventional double-slit experiment required a preliminary ''single'' slit to ensure that the light falling on the double slit was ''[[coherence (physics)|coherent]]'' (synchronized). In Fresnel's version, the preliminary single slit was retained, and the double slit was replaced by the double mirror—which bore no physical resemblance to the double slit and yet performed the same function. This result (which had been announced by Arago in the March issue of the ''Annales'') made it hard to believe that the two-slit pattern had anything to do with corpuscles being deflected as they passed near the edges of the slits.<ref>Silliman, 1967, pp.&nbsp;173–175; Buchwald, 1989, pp.&nbsp;137–138; Darrigol, 2012, pp.&nbsp;201–2; Boutry, 1948, p.&nbsp;597; Fresnel, 1866–70, vol.&nbsp;1, pp.&nbsp;123–128 (Arago's announcement).</ref>

But 1816 was the "[[Year Without a Summer]]": crops failed; hungry farming families lined the streets of Rennes; the central government organized "charity workhouses" for the needy; and in October, Fresnel was sent back to Ille-et-Vilaine to supervise charity workers in addition to his regular road crew.<ref>Levitt, 2013, p.&nbsp;43; Boutry, 1948, p.&nbsp;599.</ref> According to Arago,
{{blockquote|with Fresnel conscientiousness was always the foremost part of his character, and he constantly performed his duties as an engineer with the most rigorous scrupulousness. The mission to defend the revenues of the state, to obtain for them the best employment possible, appeared to his eyes in the light of a question of honour. The functionary, whatever might be his rank, who submitted to him an ambiguous account, became at once the object of his profound contempt.{{hsp}}… Under such circumstances the habitual gentleness of his manners disappeared…<ref>Arago, 1857, pp.&nbsp;404–405.</ref>}}
Fresnel's letters from December 1816 reveal his consequent anxiety. To Arago he complained of being "tormented by the worries of surveillance, and the need to reprimand…" And to Mérimée he wrote: "I&nbsp;find nothing more tiresome than having to manage other men, and I&nbsp;admit that I&nbsp;have no idea what I'm doing."{{hsp}}<ref>Levitt, 2013, pp.&nbsp;28,{{tsp}}237.</ref>

==== 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.