Aperture Averaging
Weak-fluctuation aperture-averaging factor for plane and spherical waves (exact integrals for a hard circular or Gaussian aperture, and Andrews' fits), Rytov variance and the all-regime scintillation index versus receiver diameter.
Fresnel zone √(L/k)
1.57 cm
Aperture parameter d² = kD²/4L
10.1
A, plane wave (hard aperture)
0.0484
uniform circular pupil
A, plane wave (Gaussian aperture)
0.0544
Andrews 1992: 0.0564
A, spherical wave (Gaussian aperture)
0.144
Andrews 1992: 0.179
Rytov variance σ_R² / β₀²
0.0199 / 8.03e-3
plane / spherical
Weak σ_I²(D) = A·σ_R²
9.63e-4
plane wave, hard aperture, weak fluctuations only
All-regime σ_I²(D)
1.91e-3
plane wave, Gaussian aperture; weak-to-strong interpolation
Scintillation reduction
13.2 dB
10·log10(1/A), plane wave, hard aperture
Aperture-averaging factor vs receiver diameter
Model
A = σ_I²(D)/σ_I²(0), d² = kD²/(4L)
In weak fluctuations A is computed by numerical integration of the Kolmogorov spectrum: for a Gaussian aperture (plane and spherical wave) and for a hard circular aperture with the Airy filter [2J₁(x)/x]² (plane wave). The hard-aperture integral is checked against its exact small- and large-d² expansions (Mellin transforms, Gradshteyn & Ryzhik 6.574.2). Andrews' interpolation [1 + 1.062 d²]^(−7/6) is within 13 % of the exact plane-wave value for d² ≥ 1 (4 % at 10 cm, 1 km), but his spherical-wave version (0.333) runs 25–37 % high at d² = 1–10 (peak 1.37× near d² ≈ 2.3) and 10–13 % low for d² ≥ 1000. For D ≫ √(L/k), A ∝ d^(−7/3).
The factor applies only for σ_R² < 1. For stronger turbulence use the all-regime σ_I²(D) (plane wave, Andrews & Phillips, Laser Beam Propagation through Random Media, 2nd ed., 2005). That expression is an interpolation whose aperture terms fall as d⁻² instead of d^(−7/3), so for d² ≫ 1 it averages too little (1.8× the weak value at 10 cm, 1 km); in weak turbulence A·σ_R² is the better value. The hard aperture gives 5–14 % lower A (more averaging) than the Gaussian aperture at d² = 1–30, but a higher A at large d²: the ratio crosses 1 at d² ≈ 77 and tends to 1.130 as d² → ∞. The spherical-wave hard aperture is not computed.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Weak fluctuations, constant Cn²: exact Kolmogorov integrals for a hard circular aperture (plane wave) and a Gaussian aperture (plane and spherical); Andrews 1992 fits shown. All-regime σ_I² is an interpolation.
- Andrews L. C. (1992). Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere. J. Opt. Soc. Am. A 9(4), 597. doi:10.1364/josaa.9.000597
- Andrews L. C., Phillips R. L. (2005). Laser Beam Propagation through Random Media, 2nd ed. SPIE. Gaussian-aperture weak-fluctuation integrals and the all-regime aperture-averaged scintillation index.
- Gradshteyn I. S., Ryzhik I. M. (2007). Table of Integrals, Series, and Products, 7th ed. Academic Press. Eq. 6.574.2, the Mellin transform of the Airy filter [2J₁(x)/x]².