The derivation of the point spread function in general uses the mathematical theory of diffraction in detail. This derivation uses geometric relations of the incoming and diffracted wave front to establish the diffraction integral, which is equal to the Hankel integral for the Bessel function of the first kind of order one.
Hubert F. A. Tschunko, "Derivation of the point spread function," Appl. Opt. 22, 1413-1413 (1983)