The
#paperOfTheDay is "On the intensity of light in the neighbourhood of a caustic" by George Biddell Airy, 1838.
This is a paper from the time long before anything quantum or relativity, where elementary questions of
#optics were at the forefront of theoretical
#physics . Since light has the character of both a wave and a ray, it is important not only where a ray can point, but also "how long" it is; one will find destructive or constructive interference depending on the difference of path length (in relation to the wave length). In the framework of geometric (=light ray) optics, this amounts to the fact that when light is reflected, a small change of the point of reflection does not change the total length of the path travelled to first order (i.e. light is reflected at such angle that the path length is stationary).
For a point of focus, this relation is true not only infinitesimally, but for the full reflecting surface: All incoming rays sent to a focal point have the same length. Airy studies the intermediate case: Not ALL rays have the same length, but still, the difference vanishes to higher than first order locally. Geometrically, this gives rise to a "caustic", basically a focal point deformed to a focal line. The existence of such line amounts to the reflecting surface satisfying a certain differential equation. Most notably, for the total intensity (in the easiest case), Airy finds the expression
integral from z=0 to z=infinity of cos(z^3 + t*z) dz , where t is a parameter. This function Ai(t) is nowadays known as the Airy function, and it constitutes the basic example of many effects in the theory of
#resurgence and special functions in
#mathematics .
https://www.semanticscholar.org/paper/ON-the-Intensity-of-Light-in-the-neighbourhood-of-a-Airy/8d6349998be20279220e8a3e033de1679a58a916