Significance of higher-order multiple scattering for laser beam propagation through hazes, fogs, and clouds

Adarsh Deepak; Usamah O. Farrukh; Andrew Zardecki

doi:10.1364/AO.21.000439

Applied Optics
Vol. 21,
Issue 3,
pp. 439-447
(1982)
•https://doi.org/10.1364/AO.21.000439

Significance of higher-order multiple scattering for laser beam propagation through hazes, fogs, and clouds

Adarsh Deepak, Usamah O. Farrukh, and Andrew Zardecki

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Adarsh Deepak, Usamah O. Farrukh, and Andrew Zardecki, "Significance of higher-order multiple scattering for laser beam propagation through hazes, fogs, and clouds," Appl. Opt. 21, 439-447 (1982)

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Abstract

An approach is outlined for computing the different orders of scattering in any medium that possesses a phase function with a strong forward peak. Computations are done for the case of a Gaussian laser beam incident on such a medium. The formulation adopted does reproduce the natural divergence of general Gaussian beams without the need to assume the presence of point sources or the need to assume perfectly collimated beams within the region of interest. Results are discussed for the case of water cloud particles with a strongly forward-peaked phase function for the incident laser radiation.

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Corrections

Adarsh Deepak, Usamah O. Farrukh, and Andrew Zardecki, "Significance of higher-order multiple scattering for laser beam propagation through hazes, fogs, and clouds: errata," Appl. Opt. 21, 2668-2668 (1982)
https://opg.optica.org/ao/abstract.cfm?uri=ao-21-15-2668

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				α

a(μm)	λ(μm)	x	m′	deg⁻¹	rad⁻¹
2.01	0.450	28.06	1.33	0.29	16.82
5.00	0.425	73.92	1.55	0.71	40.51
7.75	0.425	114.58	1.55	1.00	57.30
10.00	0.425	147.84	1.55	1.35	77.33
15.50	0.425	229.15	1.55	2.15	123.37
20.00	0.425	295.68	1.55	2.60	148.71
26.80	0.450	374.20	1.33	3.58	205.15
31.00	0.425	458.30	1.55	4.12	236.24
38.00	0.425	561.30	1.33	5.05	289.33
40.20	0.450	561.30	1.33	5.05	289.33

		Scattered irradiance using series method [Eq. (13)]

z	r cms	SS irradiance using Eq. (20) (1)	Orders 2 and above (2)	Total (1) + (2)	Total scattered irradiance using Eq. (21)
1000 km	0.0	0.849 (−10)a	0.109 (−12)	0.850 (−10)	0.849 (−10)
	2.0	0.849 (−10)	0.109 (−12)	0.850 (−10)	0.849 (−10)
100 km	0.0	0.849 (−8)	0.109 (−10)	0.850 (−8)	0.849 (−8)
	2.0	0.849 (−8)	0.109 (−10)	0.850 (−8)	0.848 (−8)
10 km	0.0	0.847 (−6)	0.109 (−8)	0.848 (−6)	0.847 (−6)
	2.0	0.831 (−6)	0.109 (−8)	0.832 (−6)	0.830 (−6)
1 km	0.0	0.696 (−4)	0.110 (−6)	0.697 (−4)	0.696 (−4)
	2.0	0.277 (−4)	0.110 (−6)	0.278 (−4)	0.278 (−4)
100 m	0.0	0.120 (−2)	0.109 (−4)	0.121 (−2)	0.121 (−2)
	2.0	0.250 (−3)	0.101 (−4)	0.260 (−3)	0.259 (−3)
10 m	0.0	0.119 (−1)	0.779 (−3)	0.127 (−1)	0.127 (−1)
	2.0	0.226 (−2)	0.526 (−3)	0.279 (−2)	0.279 (−2)
1 m	0.0	0.969 (−1)	—	—	0.129 (0)
	2.0	0.573 (−2)	—	—	0.120 (−1)
0.1 m	0.0	0.227 (0)	—	—	0.385 (0)
	2.0	0.976 (−4)	—	—	0.203 (−3)

τ	Irradiance (W/cm²μm) for mth order of scattering					Total scattered irradiance ∑N_m (W/cm²μm)	Direct attenuated irradiance (W/cm²μm)

	First	Second	Third	Fourth	Fifth
0.1	4.5 × 10⁻⁴	3.16 × 10⁻⁵	1.79 × 10⁻⁶			4.85 × 10⁻⁴	5.77 × 10⁻¹
0.3	3.88 × 10⁻²	2.27 × 10⁻³	1.21 × 10⁻⁴	5.89 × 10⁻⁶		4.12 × 10⁻²	4.74 × 10⁻¹
1.0	6.70 × 10⁻²	1.29 × 10⁻²	2.27 × 10⁻³	3.66 × 10⁻⁴		8.25 × 10⁻²	2.39 × 10⁻¹
3.0	2.83 × 10⁻²	1.63 × 10⁻²	8.62 × 10⁻³	4.17 × 10⁻³	1.82 × 10⁻³	5.91 × 10⁻²	3.37 × 10⁻²
5.0	6.45 × 10⁻³	6.29 × 10⁻³	5.58 × 10⁻³	4.54 × 10⁻³	3.31 × 10⁻³	3.08 × 10⁻²	4.74 × 10⁻³
7.0	1.31 × 10⁻³	1.76 × 10⁻³	2.17 × 10⁻³	2.46 × 10⁻³	2.50 × 10⁻³	1.77 × 10⁻²	6.68 × 10⁻⁴

				α

a(μm)	λ(μm)	x	m′	deg⁻¹	rad⁻¹
2.01	0.450	28.06	1.33	0.29	16.82
5.00	0.425	73.92	1.55	0.71	40.51
7.75	0.425	114.58	1.55	1.00	57.30
10.00	0.425	147.84	1.55	1.35	77.33
15.50	0.425	229.15	1.55	2.15	123.37
20.00	0.425	295.68	1.55	2.60	148.71
26.80	0.450	374.20	1.33	3.58	205.15
31.00	0.425	458.30	1.55	4.12	236.24
38.00	0.425	561.30	1.33	5.05	289.33
40.20	0.450	561.30	1.33	5.05	289.33

		Scattered irradiance using series method [Eq. (13)]

z	r cms	SS irradiance using Eq. (20) (1)	Orders 2 and above (2)	Total (1) + (2)	Total scattered irradiance using Eq. (21)
1000 km	0.0	0.849 (−10)a	0.109 (−12)	0.850 (−10)	0.849 (−10)
	2.0	0.849 (−10)	0.109 (−12)	0.850 (−10)	0.849 (−10)
100 km	0.0	0.849 (−8)	0.109 (−10)	0.850 (−8)	0.849 (−8)
	2.0	0.849 (−8)	0.109 (−10)	0.850 (−8)	0.848 (−8)
10 km	0.0	0.847 (−6)	0.109 (−8)	0.848 (−6)	0.847 (−6)
	2.0	0.831 (−6)	0.109 (−8)	0.832 (−6)	0.830 (−6)
1 km	0.0	0.696 (−4)	0.110 (−6)	0.697 (−4)	0.696 (−4)
	2.0	0.277 (−4)	0.110 (−6)	0.278 (−4)	0.278 (−4)
100 m	0.0	0.120 (−2)	0.109 (−4)	0.121 (−2)	0.121 (−2)
	2.0	0.250 (−3)	0.101 (−4)	0.260 (−3)	0.259 (−3)
10 m	0.0	0.119 (−1)	0.779 (−3)	0.127 (−1)	0.127 (−1)
	2.0	0.226 (−2)	0.526 (−3)	0.279 (−2)	0.279 (−2)
1 m	0.0	0.969 (−1)	—	—	0.129 (0)
	2.0	0.573 (−2)	—	—	0.120 (−1)
0.1 m	0.0	0.227 (0)	—	—	0.385 (0)
	2.0	0.976 (−4)	—	—	0.203 (−3)

Abstract

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Equations (36)

Applied Optics