Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: A comparison of exact evaluations for axial points*

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Abstract
The second Rayleigh–Sommerfeld (RS) diffraction integral, wherein the normal derivative is specified, is evaluated in simple closed form for all axial points when a divergent or convergent spherical wave is incident upon a circular aperture or disk. These evaluations (solutions) are compared with known corresponding solutions of the first RS diffraction integral. These sets of solutions are intercompared with their mean value, i.e., the derived solutions of the Kirchhoff diffraction integral. The three diffraction formulations are shown to be in agreement for incident divergent spherical waves when the source and observation points are equally distant from the aperture or disk. Conversely, for convergent spherical waves, the three formulations are never in exact agreement for focal and observation points located at finite distances from the aperture, though at optical frequencies the relative error at the geometric focal point is vanishingly small. The second RS formulation predicts, in the limit of plane waves incident on a disk, that the axial irradiance is everywhere equal to the incident irradiance, whereas the first RS formulation predicts that the irradiance goes to zero at the back of the disk.

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