Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam

W. B. Miller; J. C. Ricklin; L. C. Andrews

doi:10.1364/JOSAA.11.002719

Journal of the Optical Society of America A
Vol. 11,
Issue 10,
pp. 2719-2726
(1994)
•https://doi.org/10.1364/JOSAA.11.002719

Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam

W. B. Miller, J. C. Ricklin, and L. C. Andrews

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W. B. Miller, J. C. Ricklin, and L. C. Andrews, "Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam," J. Opt. Soc. Am. A 11, 2719-2726 (1994)

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Abstract

The normalized irradiance variance of a Gaussian beam in the weak-fluctuation regime is examined numerically with the use of various spectral models for refractive index fluctuations. The Kolmogorov, Tatarskii, and modified von Kármán spectral models are chosen as conventional models, while the Hill numerical spectral model and the Andrews analytic approximation to the Hill model are selected to feature the characteristic bump at high wave numbers. The latter two models are known to predict higher scintillation levels than conventional spectral models when the Fresnel zone and the inner scale are of comparable size. Outer scale effects appear minimal near the centerline of the beam but can reduce off-axis scintillation significantly. Inner scale effects are prominent on axis as well as off axis, although they sometimes tend to diminish near the diffractive beam edge. Analytic approximations are developed for the irradiance variance based on the Kolmogorov, modified von Kármán, and Andrews spectral models. These analytic expressions are generally in excellent agreement with numerical results.

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Spectrum	Normalized Irradiance Variance for a Gaussian-Beam Wave: Ω₀ ≠ 0^b
Kolmogorov^c	$4.42 σ_{1}^{2} Λ^{5 / 6} \frac{ρ^{2}}{W^{2}} + 3.86 σ_{1}^{2} {0.40 {[{(1 + 2 Θ)}^{2} + 4 Λ^{2}]}^{5 / 12} cos [\frac{5}{6} {tan}^{- 1} (\frac{1 + 2 Θ}{2 Λ})] - \frac{11}{16} Λ^{5 / 6}}$
Von Kármán^d	$\begin{array}{l} 3.93 σ_{1}^{2} Λ^{5 / 6} [{(\frac{Λ Q_{m}}{1 + 0.52 Λ Q_{m}})}^{1 / 6} - 1.29 {(Λ Q_{0})}^{1 / 6}] \frac{ρ^{2}}{W^{2}} \\ + 3.86 σ_{1}^{2} {0.40 \frac{{[{(1 + 2 Θ)}^{2} + {(2 Λ + 3 / Q_{m})}^{2}]}^{11 / 12}}{{[{(1 + 2 Θ)}^{2} + 4 Λ^{2}]}^{1 / 2}} sin (\frac{11}{6} φ_{1} + φ_{2}) - \frac{6 Λ}{{Q_{m}}^{11 / 6} [{(1 + 2 Θ)}^{2} + 4 Λ^{2}]} - \frac{11}{6} {(\frac{1 + 0.31 Λ Q_{m}}{Q_{m}})}^{5 / 6}} \end{array}$
Andrews^e	$\begin{array}{l} 3.93 σ_{1}^{2} Λ^{5 / 6} {[{(\frac{Λ Q_{l}}{1 + 0.52 Λ Q_{l}})}^{1 / 6} + \frac{0.438 {(Λ Q_{l})}^{1 / 6}}{{(1 + 0.70 Λ Q_{l})}^{2 / 3}} - \frac{0.056 {(Λ Q_{l})}^{1 / 6}}{{(1 + 0.70 Λ Q_{l})}^{3 / 4}}] - 1.29 {(Λ Q_{0})}^{1 / 6}} \frac{ρ^{2}}{W^{2}} \\ + 3.86 σ_{1}^{2} {0.40 \frac{{[{(1 + 2 Θ)}^{2} + {(2 Λ + 3 / Q_{l})}^{2}]}^{11 / 12}}{{[{(1 + 2 Θ)}^{2} + 4 Λ^{2}]}^{1 / 2}} [sin (\frac{11}{6} φ_{3} + φ_{2}) + \frac{2.610}{{[{(1 + 2 Θ)}^{2} {Q_{l}}^{2} + {(3 + 2 Λ Q_{l})}^{2}]}^{1 / 4}} sin (\frac{4}{3} φ_{3} + φ_{2}) \\ - \frac{0.518}{{[{(1 + 2 Θ)}^{2} {Q_{l}}^{2} + {(3 + 2 Λ Q_{l})}^{2}]}^{7 / 24}} sin (\frac{5}{4} φ_{3} + φ_{2})] - \frac{13.401 Λ}{{Q_{l}}^{11 / 16} [{(1 + 2 Θ)}^{2} + 4 Λ^{2}]} - \frac{11}{6} [{(\frac{1 + 0.31 Λ Q_{l}}{Q_{l}})}^{5 / 6} \\ + \frac{1.096 {(1 + 0.27 Λ Q_{l})}^{1 / 3}}{{Q_{l}}^{5 / 6}} - \frac{0.186 {(1 + 0.24 Λ Q_{l})}^{1 / 4}}{{Q_{l}}^{5 / 6}}]} \end{array}$

Spectrum	Normalized Irradiance Variance for a Plane Wave: $\tilde{Θ}$ = 0
Kolmogorov^b	$σ_{1}^{2}$
von Kármán^c	$3.86 σ_{1}^{2} [{(1 + 1 / {Q_{m}}^{2})}^{11 / 12} sin (\frac{11}{6} {tan}^{- 1} Q_{m}) - \frac{11}{6} {Q_{m}}^{- 5 / 6}]$
Andrews^d	$3.86 σ_{1}^{2} {{(1 + 1 / {Q_{l}}^{2})}^{11 / 12} [sin (\frac{11}{6} {tan}^{- 1} Q_{l}) + \frac{1.507}{{(1 + {Q_{l}}^{2})}^{1 / 4}} sin (\frac{4}{3} {tan}^{- 1} Q_{l}) - \frac{0.273}{{(1 + {Q_{l}}^{2})}^{7 / 24}} sin (\frac{5}{4} {tan}^{- 1} Q_{l})] - 3.50 {Q_{l}}^{- 5 / 6}}$
Spectrum	Normalized Irradiance Variance for a Spherical Wave: $\tilde{Θ}$ = 1

Kolmogorov^b	$0.40 σ_{1}^{2}$
von Kármán^c	$3.86 σ_{1}^{2} [0.40 {(1 + 9 / {Q_{m}}^{2})}^{11 / 12} sin (\frac{11}{6} {tan}^{- 1} \frac{Q_{m}}{3}) - \frac{11}{6} {Q_{m}}^{- 5 / 6}]$
Andrews^d	$3.86 σ_{1}^{2} {0.40 {(1 + 9 / {Q_{l}}^{2})}^{11 / 12} [sin (\frac{11}{6} {tan}^{- 1} \frac{Q_{l}}{3}) + \frac{2.610}{{(9 + {Q_{l}}^{2})}^{1 / 4}} sin (\frac{4}{3} {tan}^{- 1} \frac{Q_{l}}{3}) - \frac{0.518}{{(9 + {Q_{l}}^{2})}^{7 / 24}} sin (\frac{5}{4} {tan}^{- 1} \frac{Q_{l}}{3})] - 3.50 {Q_{l}}^{- 5 / 6}}$

Spectrum	Normalized Irradiance Variance for a Gaussian-Beam Wave: Ω₀ ≠ 0^b
Kolmogorov^c	$4.42 σ_{1}^{2} Λ^{5 / 6} \frac{ρ^{2}}{W^{2}} + 3.86 σ_{1}^{2} {0.40 {[{(1 + 2 Θ)}^{2} + 4 Λ^{2}]}^{5 / 12} cos [\frac{5}{6} {tan}^{- 1} (\frac{1 + 2 Θ}{2 Λ})] - \frac{11}{16} Λ^{5 / 6}}$
Von Kármán^d	$\begin{array}{l} 3.93 σ_{1}^{2} Λ^{5 / 6} [{(\frac{Λ Q_{m}}{1 + 0.52 Λ Q_{m}})}^{1 / 6} - 1.29 {(Λ Q_{0})}^{1 / 6}] \frac{ρ^{2}}{W^{2}} \\ + 3.86 σ_{1}^{2} {0.40 \frac{{[{(1 + 2 Θ)}^{2} + {(2 Λ + 3 / Q_{m})}^{2}]}^{11 / 12}}{{[{(1 + 2 Θ)}^{2} + 4 Λ^{2}]}^{1 / 2}} sin (\frac{11}{6} φ_{1} + φ_{2}) - \frac{6 Λ}{{Q_{m}}^{11 / 6} [{(1 + 2 Θ)}^{2} + 4 Λ^{2}]} - \frac{11}{6} {(\frac{1 + 0.31 Λ Q_{m}}{Q_{m}})}^{5 / 6}} \end{array}$
Andrews^e	$\begin{array}{l} 3.93 σ_{1}^{2} Λ^{5 / 6} {[{(\frac{Λ Q_{l}}{1 + 0.52 Λ Q_{l}})}^{1 / 6} + \frac{0.438 {(Λ Q_{l})}^{1 / 6}}{{(1 + 0.70 Λ Q_{l})}^{2 / 3}} - \frac{0.056 {(Λ Q_{l})}^{1 / 6}}{{(1 + 0.70 Λ Q_{l})}^{3 / 4}}] - 1.29 {(Λ Q_{0})}^{1 / 6}} \frac{ρ^{2}}{W^{2}} \\ + 3.86 σ_{1}^{2} {0.40 \frac{{[{(1 + 2 Θ)}^{2} + {(2 Λ + 3 / Q_{l})}^{2}]}^{11 / 12}}{{[{(1 + 2 Θ)}^{2} + 4 Λ^{2}]}^{1 / 2}} [sin (\frac{11}{6} φ_{3} + φ_{2}) + \frac{2.610}{{[{(1 + 2 Θ)}^{2} {Q_{l}}^{2} + {(3 + 2 Λ Q_{l})}^{2}]}^{1 / 4}} sin (\frac{4}{3} φ_{3} + φ_{2}) \\ - \frac{0.518}{{[{(1 + 2 Θ)}^{2} {Q_{l}}^{2} + {(3 + 2 Λ Q_{l})}^{2}]}^{7 / 24}} sin (\frac{5}{4} φ_{3} + φ_{2})] - \frac{13.401 Λ}{{Q_{l}}^{11 / 16} [{(1 + 2 Θ)}^{2} + 4 Λ^{2}]} - \frac{11}{6} [{(\frac{1 + 0.31 Λ Q_{l}}{Q_{l}})}^{5 / 6} \\ + \frac{1.096 {(1 + 0.27 Λ Q_{l})}^{1 / 3}}{{Q_{l}}^{5 / 6}} - \frac{0.186 {(1 + 0.24 Λ Q_{l})}^{1 / 4}}{{Q_{l}}^{5 / 6}}]} \end{array}$

Spectrum	Normalized Irradiance Variance for a Plane Wave: $\tilde{Θ}$ = 0
Kolmogorov^b	$σ_{1}^{2}$
von Kármán^c	$3.86 σ_{1}^{2} [{(1 + 1 / {Q_{m}}^{2})}^{11 / 12} sin (\frac{11}{6} {tan}^{- 1} Q_{m}) - \frac{11}{6} {Q_{m}}^{- 5 / 6}]$
Andrews^d	$3.86 σ_{1}^{2} {{(1 + 1 / {Q_{l}}^{2})}^{11 / 12} [sin (\frac{11}{6} {tan}^{- 1} Q_{l}) + \frac{1.507}{{(1 + {Q_{l}}^{2})}^{1 / 4}} sin (\frac{4}{3} {tan}^{- 1} Q_{l}) - \frac{0.273}{{(1 + {Q_{l}}^{2})}^{7 / 24}} sin (\frac{5}{4} {tan}^{- 1} Q_{l})] - 3.50 {Q_{l}}^{- 5 / 6}}$
Spectrum	Normalized Irradiance Variance for a Spherical Wave: $\tilde{Θ}$ = 1

Kolmogorov^b	$0.40 σ_{1}^{2}$
von Kármán^c	$3.86 σ_{1}^{2} [0.40 {(1 + 9 / {Q_{m}}^{2})}^{11 / 12} sin (\frac{11}{6} {tan}^{- 1} \frac{Q_{m}}{3}) - \frac{11}{6} {Q_{m}}^{- 5 / 6}]$
Andrews^d	$3.86 σ_{1}^{2} {0.40 {(1 + 9 / {Q_{l}}^{2})}^{11 / 12} [sin (\frac{11}{6} {tan}^{- 1} \frac{Q_{l}}{3}) + \frac{2.610}{{(9 + {Q_{l}}^{2})}^{1 / 4}} sin (\frac{4}{3} {tan}^{- 1} \frac{Q_{l}}{3}) - \frac{0.518}{{(9 + {Q_{l}}^{2})}^{7 / 24}} sin (\frac{5}{4} {tan}^{- 1} \frac{Q_{l}}{3})] - 3.50 {Q_{l}}^{- 5 / 6}}$

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Equations (27)

Journal of the Optical Society of America A