Behavior of focused beams in atmospheric turbulence: Measurements and comments on the theory*

James A. Dowling; Peter M. Livingston

doi:10.1364/JOSA.63.000846

Journal of the Optical Society of America
Vol. 63,
Issue 7,
pp. 846-858
(1973)
•https://doi.org/10.1364/JOSA.63.000846

Behavior of focused beams in atmospheric turbulence: Measurements and comments on the theory

James A. Dowling and Peter M. Livingston

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James A. Dowling and Peter M. Livingston, "Behavior of focused beams in atmospheric turbulence: Measurements and comments on the theory*," J. Opt. Soc. Am. 63, 846-858 (1973)

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Abstract

Laser-beam propagation through atmospheric turbulence is analyzed theoretically and compared with measurements at λ = 0.63 and 10.6 μm. Calculations based either on irradiance statistics or mutual coherence function (MCF) are analyzed; a general expression for long-term-average beam spread based on the turbulence MCF is obtained. The spread of a laser beam focused over moderate distances can be separated into short- and long-term averages that differ by beam wander, which has been found to be essentially independent of wavelength and adequately described by geometric optics. However, a significant wavelength dependence of short-term-average beam spread is found experimentally. Measurements at 10.6 μm are nearly diffraction limited, whereas corresponding data for 0.63 μm are strongly influenced by variations of the refractive-index structure constant C_n. An empirical formula for long-term-average beam spread is at variance with the wave-number-dependent functional form predicted by MCF calculations. Both the irradiance and MCF approaches go over into the same asymptotic functional dependence for visible wavelengths and long ranges. Angular-beam-spread measurements at 0.63 μm support this conclusion.

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Form of curve fit	Fitted equation	Correlation/regression coefficient	Theoretical prediction	Ref.
θ² vs αC_n^βz^γk₀^δ	θ² = −1.70 × 10⁻⁹ + 1.44 × 10⁻⁹C_n^0.85z^0.62k₀^0.65	0.893
Φ_ir² vs Φ_vis²	Φ_ir² = 3.82 × 10⁻¹⁰ + 0.930Φ_vis²	0.953
Φ² vs C_n²z	Φ² = 1.49 × 10⁻⁹ + 14.1C_n²z	0.785	$Φ^{2} = {11.64 C}_{n}^{2} z b^{- \frac{1}{3}}$	29
θ² vs Φ^z(0.63 μm)	θ² = 8.80 × 10⁻¹⁰ + 2.47Φ²	0.934
Φ² vs C_n²z^5/6k₀^1/6	Φ² = 1.64 × 10⁻⁹ + 3.56C_n²z^5/6k₀^1/6	0.740	Φ² = 21.54C_nz^5/6k₀^1/6	30
θ^′2 vs C_n^2.4z^1.2k₀^0.40	θ^′2 = 4.90 × 10⁻⁹ + 5.67C_n^2.4z^1.2k₀^0.40	0.823	θ^′2 = 15.4C_n^2.4z^1.2k₀^0.40	16
θ^′2 vs αC_n^βz^γk₀^δ	θ^′2 = −4.08 × 10⁻¹⁰ + 2.81 × 10⁻⁶C_n^1.2z^1.04k₀^0.33	0.900	θ^′2 = 16.6C_n^2.4z^1.2k₀^0.40	15
θ^′2 vs C_n²z (0.63 μm)	θ_vis^′2 = −5.44 × 10⁻⁹ + 53.1C_n²z	0.833	$θ^{' 2} = {18.32 C}_{n}^{2} {z l}_{0}^{- \frac{1}{3}}$	13

X	Y	θ_av²	Correlation coefficient		Remarks
X	Y	θ_av²	ρ_ux^a	ρ_uy^a	Remarks
e^−20.9C_n^0.85z^0.62k^0.65	θ²	a₀X + b₀	−7.46 × 10⁻²	0.240	Systematic dependence on θ² left.
e^−20.9C_n^0.85z^0.6k^0.65	θ²	See note b	5.3 × 10⁻²	1.5 × 10⁻²	Remarkable improvement; no residual dependence remains.
C_n^2.4z^1.2k^0.40	θ^′2	a₀X + b₀	0.70	0.640	Considerable remaining systematic behavior in the data.
e^−12.9C_n^1.2z^1.04k^0.33	θ^′	a₀X + b₀	−2.65 × 10⁻²	−8.4 × 10⁻³	Very good fit; the data show no residual dependence.

Parameter	θ²		θ^′2
Parameter	Value	95% confidence limits	Value	95% confidence limits
a₀α	1.44 × 10⁻⁹	±0.008 × 10⁻⁹	2.81 × 10⁻⁶	±0.017 × 10⁻⁶
b₀	−1.70 × 10⁻⁹	⋯	−4.08 × 10⁻¹⁰	⋯
β	0.850	±0.015	1.20	±0.011
γ	0.619	±0.017	1.04	±0.012
δ	0.654	±0.006	0.330	±0.004

Form of curve fit	Fitted equation	Correlation/regression coefficient	Theoretical prediction	Ref.
θ² vs αC_n^βz^γk₀^δ	θ² = −1.70 × 10⁻⁹ + 1.44 × 10⁻⁹C_n^0.85z^0.62k₀^0.65	0.893
Φ_ir² vs Φ_vis²	Φ_ir² = 3.82 × 10⁻¹⁰ + 0.930Φ_vis²	0.953
Φ² vs C_n²z	Φ² = 1.49 × 10⁻⁹ + 14.1C_n²z	0.785	$Φ^{2} = {11.64 C}_{n}^{2} z b^{- \frac{1}{3}}$	29
θ² vs Φ^z(0.63 μm)	θ² = 8.80 × 10⁻¹⁰ + 2.47Φ²	0.934
Φ² vs C_n²z^5/6k₀^1/6	Φ² = 1.64 × 10⁻⁹ + 3.56C_n²z^5/6k₀^1/6	0.740	Φ² = 21.54C_nz^5/6k₀^1/6	30
θ^′2 vs C_n^2.4z^1.2k₀^0.40	θ^′2 = 4.90 × 10⁻⁹ + 5.67C_n^2.4z^1.2k₀^0.40	0.823	θ^′2 = 15.4C_n^2.4z^1.2k₀^0.40	16
θ^′2 vs αC_n^βz^γk₀^δ	θ^′2 = −4.08 × 10⁻¹⁰ + 2.81 × 10⁻⁶C_n^1.2z^1.04k₀^0.33	0.900	θ^′2 = 16.6C_n^2.4z^1.2k₀^0.40	15
θ^′2 vs C_n²z (0.63 μm)	θ_vis^′2 = −5.44 × 10⁻⁹ + 53.1C_n²z	0.833	$θ^{' 2} = {18.32 C}_{n}^{2} {z l}_{0}^{- \frac{1}{3}}$	13

X	Y	θ_av²	Correlation coefficient		Remarks
X	Y	θ_av²	ρ_ux^a	ρ_uy^a	Remarks
e^−20.9C_n^0.85z^0.62k^0.65	θ²	a₀X + b₀	−7.46 × 10⁻²	0.240	Systematic dependence on θ² left.
e^−20.9C_n^0.85z^0.6k^0.65	θ²	See note b	5.3 × 10⁻²	1.5 × 10⁻²	Remarkable improvement; no residual dependence remains.
C_n^2.4z^1.2k^0.40	θ^′2	a₀X + b₀	0.70	0.640	Considerable remaining systematic behavior in the data.
e^−12.9C_n^1.2z^1.04k^0.33	θ^′	a₀X + b₀	−2.65 × 10⁻²	−8.4 × 10⁻³	Very good fit; the data show no residual dependence.

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Journal of the Optical Society of America