page_title: Augustin-Jean Fresnel
section_title: Prize memoir (1818) and sequel
hierarchical_section_title: Augustin-Jean Fresnel / Contributions to physical optics / Diffraction / Prize memoir (1818) and sequel
caption_attribution_description: English: The shadow of a circular obstacle (5.8 mm) in diameter lit by a ~0.5 mm sunlight source, exposing the Arago spot in the middle of a shadow. Distance from the light source to the obstacle 5 ft (~153 cm), distance from the obstacle to the paper screen 6 ft (~183 cm).
context_page_description: Augustin-Jean Fresnel was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular theory, from the late 1830s until the end of the 19th century. He is perhaps better known for inventing the catadioptric Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric stepped lens, first proposed by Count Buffon and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors.
  By expressing Huygens's principle of secondary waves and Young's principle of interference in quantitative terms, and supposing that simple colors consist of sinusoidal waves, Fresnel gave the first satisfactory explanation of diffraction by straight edges, including the first satisfactory wave-based explanation of rectilinear propagation. Part of his argument was a proof that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions.
context_section_description: 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. 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.
  In the fall of 1817, Fresnel, supported by de Prony, obtained a leave of absence from the new head of the Corp des Ponts, Louis Becquey, and returned to Paris. He resumed his engineering duties in the spring of 1818; but from then on he was based in Paris, first on the Canal de l'Ourcq, and then (from May 1819) with the cadastre of the pavements.
  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. His method was similar to the phasor representation, except that the "forces" were plane vectors rather than complex numbers; they could be added, and multiplied by 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, 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 . 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. 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:
    ;   
  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, plus a smaller table of maxima and minima of the resulting intensity.
  In his final "Memoir on the diffraction of light", deposited on 29 July  and bearing the Latin epigraph "Natura simplex et fecunda" ("Nature simple and fertile"), 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. 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." 
  For the experimental testing of his calculations, Fresnel used red light with a wavelength of 638 nm, which h
caption: Shadow cast by a 5.8 mm-diameter obstacle on a screen 183 cm behind, in sunlight passing through a pinhole 153 cm in front. The faint colors of the fringes show the wavelength-dependence of the diffraction pattern. In the center is Poisson's /Arago's spot.
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