With a view to elucidating the effect of a well-known mathematical inconsistency in Kirchhoff’s diffraction theory, a comparison is made of the predictions relating to the field diffracted at an aperture, based on Kirchhoff’s theory (UK) and on formulas due to Rayleigh and Sommerfeld (UR). It is shown that, when the incident wave is plane or spherical, the difference δ = UK-UR represents a boundary wave, i.e., a wave which may be thought of as originating at each point of the edge of the aperture. It is shown further that, when the linear dimensions of the aperture are large compared with the wavelength, the boundary values of δ in the plane of the aperture change very rapidly and almost periodically from point to point, with the mean period close to the wavelength of the incident radiation. This result is shown to imply that if the linear dimensions of the aperture are large compared with the wavelength, the two theories predict essentially the same behavior for the diffracted field in the far zone, at moderate angles of diffraction.
E. WOLFE and W. MARCHAND, "Comparison of the Kirchhoff and the Rayleigh—Sommerfeld Theories of Diffraction at an Aperture," J. Opt. Soc. Am. 54, 587-594 (1964)