Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: spherical waves

Abstract

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.

Design, fabrication, and evaluation of a multilevel spiral-phase Fresnel zone plate for optical trapping

“…(1). These values were calculated at z u using the scalar diffraction formula [24,25]. The 1=e 2 radii of the beam along the x and y directions were obtained as w x u ≈ 130 μm and w y u ≈ 624 μm, which is consistent with the values obtained from the Gaussian beam-propagation formula.…”

Analytic expression of the diffraction of a circular aperture

“…In the far field, the diffraction of the circular aperture, which is the Fourier transformation of the circular aperture, can be expressed by a simple formula. However, Fresnel diffraction of the circular aperture is usually given by the integral or the infinite series owing to the existence of the second phase factor [1][2][3][4], except for the case on the propagation axis [5,6]. Therefore, the physical meaning of the diffraction is not evident and its distribution rules are not discussed deeply in these studies.…”

In-plane displacement measurement by using circular grating Talbot interferometer

“…If two gratings are separated by a distance Δ (r'-r) along the direction of propagation then after propagating a distance Δ, the field just before G2 can be evaluated using Huygens-FresnelKirchhoff diffraction [33] and given by u 2Bef ore ðr; ϕÞ ¼ expðikΔÞ iλΔ…”