In an earlier paper dealing with a consistent formulation of Kirchhoff’s theory, an extension of the theory was proposed which seems to be applicable to treatment of diffraction at very small apertures in black screens. The extension consists of the addition to Kirchhoff’s solution of the effect of waves arising from multiple diffraction at the edge of the aperture. In the present paper, calculations based on this modified theory are presented. Corrections to Kirchhoff’s theory are obtained for the case of a plane wave incident normally on a small circular aperture in a black screen. Appreciable departures from Kirchhoff’s theory are found only when the diameter of the aperture is small compared with the wavelength of the incident wave or when the angles of diffraction are very large. It is also shown that in the asymptotic limit ka → ∞ (k is the wave number, a the radius of the aperture) our results are consistent with those obtained on the basis of Keller’s geometrical theory of diffraction.
E. W. MARCHAND and E. WOLF, "Diffraction at Small Apertures in Black Screens," J. Opt. Soc. Am. 59, 79-85 (1969)