Hal G. Kraus, "Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: paraxial and exact Gaussian laser beams," J. Opt. Soc. Am. A 7, 47-65 (1990)
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 wave-front 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.
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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Planar at waist, changing to a small section of a paraboloid which becomes essentially spherical for r2 ≪ z2
Planar at waist, changing to section of an oblate ellipsoid
Planar at waist, changing to section of oblate ellipsoid that is near spherical
Axial phase shift from nominal wave front to actual wave front
Deviation from paraboloidal/spherical wave front, “Guoy” effect5
Deviation from oblate ellipsoidal wave front, “Guoy” effect
Deviation from oblate ellipsoidal/spherical wave front, “Guoy” effect5
Radial amplitude distribution
Gaussian
modified Gaussian
Gaussian
Axial amplitude variation
C-a constant
C-a constant
C-a constant
Table 2
Effect on Wave-Front Geometry of Setting Gm = 1 in Arctangent Factor of Expression for z(r), Eq. (52), for Typical Laboratory Lasers
Laser Type, Manufacturer
Model Number
Beam Waist w0 (mm)
Wavelength λ (μm)
Beam Divergence θ0 (mrad)
Gw of Eq. (53) for 1/e3 Beam Truncation Level and z = 1.0 m (λ)a
He–Ne, Melles-Griot
05 LLR 141
0.4
0.6328
0.50
1.2296 × 10−8
He–Ne, Melles-Griot
05 LLR 171
0.375
0.5435
0.46
1.9610 × 10−8
He–Ne, Melles-Griot
05 LIP 171
0.44
1.523
1.1
4.6654 × 10−8
CO2, Melles-Griot
05 COR 080
0.8
1.06
6.5
1.0055 × 10−8
Argon ion, Spectra-Physics
N/A
0.325
0.488
0.475
1.4560 × 10−8
In other words, Gw is expressed here in terms of the number of wavelengths.
Table 3
Characteristic Circular-Aperture Fresnel-Number Equations for Huygens–Fresnel–Kirchhoff Aperture-Plane and Wave-Front Diffraction Formulations for Gaussian Laser Beam, with a Characteristic Equation Am2 + Bm + C = 0a
Other parameters: zw = 1.0 m, location of wave front on axis; λ = 0.6328 μm; Fr = 10.0; w0 = 0.4 mm; rf = 1.6aSf; Nb = Nt = 16; h = zw − za; za = z(r = a) Eq. (52).
Other parameters: zw = 1.0 m, location of wave front on axis; λ = 0.6328 μm; Fr = 10.0; w0 = 0.4 mm; rf = 1.6aSf; Nb = Nt = 16; h = zw − za; za = z(r = a), Eq. (52); error in Iza obtained by using Eq. (63), <5.0 × 10−5 percent.
Planar at waist, changing to a small section of a paraboloid which becomes essentially spherical for r2 ≪ z2
Planar at waist, changing to section of an oblate ellipsoid
Planar at waist, changing to section of oblate ellipsoid that is near spherical
Axial phase shift from nominal wave front to actual wave front
Deviation from paraboloidal/spherical wave front, “Guoy” effect5
Deviation from oblate ellipsoidal wave front, “Guoy” effect
Deviation from oblate ellipsoidal/spherical wave front, “Guoy” effect5
Radial amplitude distribution
Gaussian
modified Gaussian
Gaussian
Axial amplitude variation
C-a constant
C-a constant
C-a constant
Table 2
Effect on Wave-Front Geometry of Setting Gm = 1 in Arctangent Factor of Expression for z(r), Eq. (52), for Typical Laboratory Lasers
Laser Type, Manufacturer
Model Number
Beam Waist w0 (mm)
Wavelength λ (μm)
Beam Divergence θ0 (mrad)
Gw of Eq. (53) for 1/e3 Beam Truncation Level and z = 1.0 m (λ)a
He–Ne, Melles-Griot
05 LLR 141
0.4
0.6328
0.50
1.2296 × 10−8
He–Ne, Melles-Griot
05 LLR 171
0.375
0.5435
0.46
1.9610 × 10−8
He–Ne, Melles-Griot
05 LIP 171
0.44
1.523
1.1
4.6654 × 10−8
CO2, Melles-Griot
05 COR 080
0.8
1.06
6.5
1.0055 × 10−8
Argon ion, Spectra-Physics
N/A
0.325
0.488
0.475
1.4560 × 10−8
In other words, Gw is expressed here in terms of the number of wavelengths.
Table 3
Characteristic Circular-Aperture Fresnel-Number Equations for Huygens–Fresnel–Kirchhoff Aperture-Plane and Wave-Front Diffraction Formulations for Gaussian Laser Beam, with a Characteristic Equation Am2 + Bm + C = 0a
Other parameters: zw = 1.0 m, location of wave front on axis; λ = 0.6328 μm; Fr = 10.0; w0 = 0.4 mm; rf = 1.6aSf; Nb = Nt = 16; h = zw − za; za = z(r = a) Eq. (52).
Other parameters: zw = 1.0 m, location of wave front on axis; λ = 0.6328 μm; Fr = 10.0; w0 = 0.4 mm; rf = 1.6aSf; Nb = Nt = 16; h = zw − za; za = z(r = a), Eq. (52); error in Iza obtained by using Eq. (63), <5.0 × 10−5 percent.