Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: paraxial and exact Gaussian laser beams

Abstract

The Huygens-Fresnel diffraction integral has been formulated for incident Gaussian laser beams by using the Kirchhoff obliquity factor with the wave front instead of the aperture plane as the surface of integration. Accurate numerical-integration calculations were used to investigate the Fresnel field diffraction region for the much-studied case of a circular aperture. It is shown that the classical aperture-plane formulation becomes inaccurate when the wave front, as truncated at the aperture, has any degree of curvature to it, whereas the newly developed wave-front formulation produces accurate results for as much as one aperture diameter behind the aperture plane. The wavefront diffraction integral was developed for both the classical paraxial and the recently developed exact solutions to the scalar wave equation for a Gaussian beam. Detailed comparisons of these two diffraction solutions show that they are essentially identical for the typical laboratory laser. The typical laboratory laser is defined as having a wavelength in the near-infrared-through-visible range, a beam diameter as large as several millimeters, and a beam divergence angle as large as several milliradians.

Huygens–Fresnel–Kirchhoff wave-front diffraction formulations for spherical waves and Gaussian laser beams: discussion and errata

“…The first of these papers was for spherical waves,' and the second was for Gaussian laser beams. 2 The incident waves were diffracted by circular apertures. In each of these formulations an obliquity factor, which was a commonly used, slightly simplified form of the Kirchhoff obliquity factor, was added.…”

Exact nonparaxial beams of the scalar Helmholtz equation

“…6 This fact has also been discussed by several authors and even have expressed the virtual source mode wave in spheroidal coordinates. 7,8,9,10 Whatever the coordinate system used, the field expression reduces, under the paraxial approximation, to Gaussian beam. 6,7,8,9,10,11 The virtual complex source point solutions carry an inherent singularity which makes them inadequate to describe propagating fields near the origin or focal source point.…”

“…7,8,9,10 Whatever the coordinate system used, the field expression reduces, under the paraxial approximation, to Gaussian beam. 6,7,8,9,10,11 The virtual complex source point solutions carry an inherent singularity which makes them inadequate to describe propagating fields near the origin or focal source point. 6,8 To elliminate this problem a non-singular superposition of incoming and outgoing spherical waves has been used, 6,8 however even this non-singular solution has problems since, to realise them physically, infinite energy is required.…”

“…That the solutions discussed here are different from those obtained with the complex source point method (CSPM) expressed in spheroidal coordinates. 7,8,9,10 The zero-order nonparaxial beams obtained with the CSPM have a singularity inherited from its construction from a source point. 12 The oblate spheroidal beams presented here do not present any singularity whenever c = 0.…”

“…12 The oblate spheroidal beams presented here do not present any singularity whenever c = 0. Using the CSPM, 7,8,9,10 it is usually proposed that in the paraxial limit d can take the value of R 0 . From eq.…”