structural:summarization, linguistic:translation
First attempt (1815)

On 12 July 1815, as Fresnel was about to leave Paris, Arago left him a note on a new topic:

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 Grimaldi, the one by Newton, the English treatise by Jordan, and the memoirs of Brougham and Young, which are part of the collection of the Philosophical Transactions.

Fresnel would not have ready access to these works outside Paris, and could not read English. But, in Mathieu—with a point-source of light made by focusing sunlight with a drop of honey, a crude micrometer of his own construction, and supporting apparatus made by a local locksmith—he began his own experiments. 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.

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.

On 23 September he wrote to Arago, beginning "I think I 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." He explained that he had not yet had a chance to acquire the items on his reading lists, with the apparent exception of "Young's book", which he could not understand without his brother's help. 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…" 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 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.

On 10 November, Fresnel sent a supplementary note dealing with Newton's rings and with gratings, 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.

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 Poinsot as the other reviewer). On 8 November, Arago wrote to Fresnel:

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. 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.

Fresnel was troubled, wanting to know more precisely where he had collided with Young. Concerning the curved paths of the "colored bands", Young had noted the hyperbolic paths of the fringes in the 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. He had not mentioned the curved paths of the external fringes of a shadow; but, as he later explained, that was because Newton had already done so. Newton evidently thought the fringes were caustics. Thus Arago erred in his belief that the curved paths of the fringes were fundamentally incompatible with the corpuscular theory.

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.

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. Arago later used a similar argument to explain the colors in the scintillation of stars.

Fresnel's updated memoir was eventually published in the March 1816 issue of Annales de Chimie et de Physique, of which Arago had recently become co-editor. That issue did not actually appear until May. In 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.