The Huygens–Fresnel diffraction integral has been formulated for incident spherical waves with use of the Kirchhoff obliquity factor and the wave front as the surface of integration instead of the aperture plane. Accurate numerical integration calculations were used to investigate very-near-field (a few aperture diameters or less) diffraction for the well-established case of a circular aperture. It is shown that the classical aperture-plane formulation degenerates when the wave front, as truncated at the aperture, has any degree of curvature to it, whereas the wave-front formulation produces accurate results from ∞ up to one aperture diameter behind the aperture plane. It is also shown that the Huygens–Fresnel–Kirchhoff incident-plane-wave-aperture-plane-integration and incident-spherical-wave-wave-front-integration formulations produce equally accurate results for apertures with exit f-numbers as small as 1.
Hal G. Kraus, "Huygens–Fresnel–Kirchhoff wave-front diffraction formulations for spherical waves and Gaussian laser beams: discussion and errata," J. Opt. Soc. Am. A 9, 1132-1134 (1992) https://opg.optica.org/josaa/abstract.cfm?uri=josaa-9-7-1132
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Characteristic Circular-Aperture Fresnel-Number Equations of the Form Am2 + Bm + C for Plane-Wave and Spherical-Wave Huygens–Fresnel–Kirchhoff Diffraction Formulationsa
m is the Fresnel number Fr.
This more general form may be derived easily by using optical-path-length differences,2 and it has no restrictions. Equation (11) results from this characteristic equation when m2 is small so that the first term can be neglected.
Derived using Eqs. (4) and (17), assuming that At = πa2.
Other parameters: z = 0.01 m, λ = 0.6328 × 10−6 m, rf = 1.6a, Nb = Nt 16. From Fig. 1, za = z − ha, θ1 = tan−1(a/za), ha = ρ(1 − cos θ1), ρ = (z2 + a2)1/2.
Other parameters: ρ = 0.01 m, λ = 0.6328 × 10−6 m, rf = 1.6aSf, Nb = Nt = 16. From Fig. 2, z = ρ − h = ρ cos θ1, θ1 = sin−1(a/ρ), h = ρ(1 − cos θ1).
Tables (4)
Table 1
Characteristic Circular-Aperture Fresnel-Number Equations of the Form Am2 + Bm + C for Plane-Wave and Spherical-Wave Huygens–Fresnel–Kirchhoff Diffraction Formulationsa
m is the Fresnel number Fr.
This more general form may be derived easily by using optical-path-length differences,2 and it has no restrictions. Equation (11) results from this characteristic equation when m2 is small so that the first term can be neglected.
Derived using Eqs. (4) and (17), assuming that At = πa2.
Other parameters: z = 0.01 m, λ = 0.6328 × 10−6 m, rf = 1.6a, Nb = Nt 16. From Fig. 1, za = z − ha, θ1 = tan−1(a/za), ha = ρ(1 − cos θ1), ρ = (z2 + a2)1/2.