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I was told it was Joseph von Fraunhofer who made the first observation that shining sun rays through a slit created faint rainbow fringes to the sides of the outgoing ray. Is there any evidence of this historical claim? Is there anyone earlier who made similar observations? Also, what were the follow-up investigations after Fraunhofer regarding the single-slit?

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Fraunhofer is credited not with discovering the single slit diffraction, but with explaining the far-field diffraction pattern from a single slit. Fresnel explained the near-field pattern before him, in 1815, but he was not the discoverer either.

The discovery of diffraction and the introduction of the term are due to Grimaldi, specifically his Physico-Mathesis de Lumine posthumously published in Bologna in 1665. In the more commonly cited experiment he placed a sharp knife into the cone of light passed into a dark room through a small hole, and observed knife's anomalously wide shadow along with the colored bands which he called seriae lucidae. But in a second experiment he put the cone through a second small hole, and observed what is now called "single slit" diffraction, see Crosswell, Diffraction: the first recorded observation.

Grimaldi's book was little known, but his experiments were popularized in Fabri's Dialogi Physici (1669), which found its way into Newton's hands in 1672, when Collins sent him a copy, see Hall's Beyond the Fringe: Diffraction as seen by Grimaldi, Fabri, Hooke and Newton. Newton read the book, but did not pay much attention to it even after Hooke presented to the Royal Society in 1675 his own observation of "a Penumbra or darker Ring encompassing the lighter Circle" obtained by passing sunlight through a small hole, and noted that "‘it could not proceed from a Penumbra caused by the bigness of the Hole upon the common Principles, that is, from the Supposition of the Rays from every point of the Sun proceeding in strait Lines". Hooke did not mention anything about colors.

Some time between then and the publication of Opticks (1704) Newton did perform detailed experiments extending those of Grimaldi (c. 1692 in Hall's estimation). He renamed diffraction into "inflexion" (Fabri did not use Grimaldi's term, its return is due to Fresnel), seriae lucidae into "fringes", and came up with a theory of light repulsion by matter to reconcile the effect with his corpuscular theory. He did not seem to know about Grimaldi's second experiment, but one observation concerning the knife's shadow led him to a "single slit" variant anyway. And he did observe "three Parallel Fringes or Bands of Colour’d Light", for which he did not really have a good explanation. Stuewer in Critical Analysis of Newton's Work on Diffraction writes:

"This keen observation (which was understood only after the work of Young and Fresnel) led Newton to further experiments which we would now classify as "single slit diffraction" experiments. Thus he placed two knives with their edges opposite each other and gradually decreased the space between them. When they were quite close together Newton observed a bright line in the center of the pattern with three colored fringes on either side of it. At smaller knife separations, these fringes began to spread out... Closing the gap still further, Newton found that beginning at separations somewhat greater than 1/400 inch the fringes began to disappear, first the two outer fringes, then the two middle fringes, and finally the two inner fringes.
[...] Thus Newton - in a beautifully detailed description - clearly reported observations on light bending into the geometric shadow. His explanation for his observations rested on his belief that each of the two knives repelled the rays as they passed by, thereby producing two sets of three exterior fringes".

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