Abstract

Within the small-angle approximation, a multi-Gaussian phase function model provides a new description of the multiple-scattering process and leads to improved agreement with experimental data. A general formula for the off-axis laser beam irradiance is given explicitly for a two-component Gaussian phase function. The predictions of the theory are compared with both the calculations based on the Mie scattering phase function and the results of measurements. It appears that the best fit of the model to experimental results requires an adjustment of the weight factors of the component phase functions. The solution, written in the form of a multiple-scattering series, allows one to interpret individual terms of the scattering series as scattering events of different orders.

© 1987 Optical Society of America

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References

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  1. E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. 2, 903 (1985).
    [Crossref]
  2. E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “The Effect of a Strongly Inhomogeneous Medium on the Propagation of Light Beams under Multiple Scattering Conditions,” Opt. Acta 32, 717 (1985).
    [Crossref]
  3. W. G. Tam, A. Zardecki, “Off-Axis Propagation of a Laser Beam in Low Visibility Weather Conditions,” Appl. Opt. 19, 2822 (1980).
    [Crossref] [PubMed]
  4. A. Zardecki, S. A. W. Gerstl, “Multiple Scattering Effects in the Off-Axis Propagation of Laser Radiation,” in International Conference, Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 27–30 May 1986.
  5. J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, “Derivation of Phase Functions from Multiply Scattered Sunlight Transmitted Through a Hazy Atmosphere,” J. Atmos. Sci. 32, 577 (1975).
    [Crossref]
  6. D. A. de Wolf, J. K. Pack, “Wave Kinetic Numerical Approach to Propagation of Optical Beams,” J. Opt. Soc. Am. A 3, 532 (1986).
    [Crossref]
  7. W. G. Tam, A. Zardecki, “Multiple Scattering Corrections to the Beer-Lambert Law. 1: Open Detector,” Appl. Opt. 21, 2405 (1982).
    [Crossref] [PubMed]
  8. A. Zardecki, W. G. Tam, “Multiple Scattering Corrections to The Beer-Lambert Law. 2: Detector with a Variable Field of View,” Appl. Opt. 21, 2413 (1982).
    [Crossref] [PubMed]
  9. Y. Kuga, A. Ishimaru, H. W. Chang, L. Tsang, “Comparisons Between the Small-Angle Approximation and the Numerical Solutions for Transfer Theory,” Appl. Opt. 25, 3803 (1986).
    [Crossref] [PubMed]
  10. M. A. Box, A. Deepak, “Limiting Cases of the Small Angle Scattering Approximation Solutions for the Propagation of Laser Beams in Anisotropic Scattering Media,” J. Opt. Soc. Am. 71, 1534 (1981).
    [Crossref]
  11. A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.
  12. S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “Off-Axis Multiple Scattering of a Laser Beam in Turbid Media: Comparison of Theory and Experiment,” Appl. Opt. 26, 779 (1987).
    [Crossref]
  13. P. V. Hobbs, A. Deepak, Clouds: Their Formation, Optical Properties, and Effects (Academic, New York, 1981).
  14. R. B. Smith, J. D. Houston, A. Ulitsky, A. I. Carswell, “Laboratory Measurements of Forward and Backward Scattering of Laser Beams in Water Droplet Clouds,” in International Laser Lidar Conference, Toronto (1986).

1987 (1)

1986 (2)

1985 (2)

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. 2, 903 (1985).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “The Effect of a Strongly Inhomogeneous Medium on the Propagation of Light Beams under Multiple Scattering Conditions,” Opt. Acta 32, 717 (1985).
[Crossref]

1982 (2)

1981 (1)

1980 (1)

1975 (1)

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, “Derivation of Phase Functions from Multiply Scattered Sunlight Transmitted Through a Hazy Atmosphere,” J. Atmos. Sci. 32, 577 (1975).
[Crossref]

Battistelli, E.

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. 2, 903 (1985).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “The Effect of a Strongly Inhomogeneous Medium on the Propagation of Light Beams under Multiple Scattering Conditions,” Opt. Acta 32, 717 (1985).
[Crossref]

Box, M. A.

Browning, S. R.

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, “Derivation of Phase Functions from Multiply Scattered Sunlight Transmitted Through a Hazy Atmosphere,” J. Atmos. Sci. 32, 577 (1975).
[Crossref]

Bruscaglioni, P.

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “The Effect of a Strongly Inhomogeneous Medium on the Propagation of Light Beams under Multiple Scattering Conditions,” Opt. Acta 32, 717 (1985).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. 2, 903 (1985).
[Crossref]

Carswell, A. I.

R. B. Smith, J. D. Houston, A. Ulitsky, A. I. Carswell, “Laboratory Measurements of Forward and Backward Scattering of Laser Beams in Water Droplet Clouds,” in International Laser Lidar Conference, Toronto (1986).

Chang, H. W.

de Wolf, D. A.

Deepak, A.

Elliott, N. E.

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “Off-Axis Multiple Scattering of a Laser Beam in Turbid Media: Comparison of Theory and Experiment,” Appl. Opt. 26, 779 (1987).
[Crossref]

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

Gerstl, S. A. W.

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “Off-Axis Multiple Scattering of a Laser Beam in Turbid Media: Comparison of Theory and Experiment,” Appl. Opt. 26, 779 (1987).
[Crossref]

A. Zardecki, S. A. W. Gerstl, “Multiple Scattering Effects in the Off-Axis Propagation of Laser Radiation,” in International Conference, Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 27–30 May 1986.

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

Herman, B. M.

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, “Derivation of Phase Functions from Multiply Scattered Sunlight Transmitted Through a Hazy Atmosphere,” J. Atmos. Sci. 32, 577 (1975).
[Crossref]

Hobbs, P. V.

P. V. Hobbs, A. Deepak, Clouds: Their Formation, Optical Properties, and Effects (Academic, New York, 1981).

Houston, J. D.

R. B. Smith, J. D. Houston, A. Ulitsky, A. I. Carswell, “Laboratory Measurements of Forward and Backward Scattering of Laser Beams in Water Droplet Clouds,” in International Laser Lidar Conference, Toronto (1986).

Ishimaru, A.

Ismaelli, A.

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “The Effect of a Strongly Inhomogeneous Medium on the Propagation of Light Beams under Multiple Scattering Conditions,” Opt. Acta 32, 717 (1985).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. 2, 903 (1985).
[Crossref]

Kuga, Y.

Pack, J. K.

Smith, R. B.

R. B. Smith, J. D. Houston, A. Ulitsky, A. I. Carswell, “Laboratory Measurements of Forward and Backward Scattering of Laser Beams in Water Droplet Clouds,” in International Laser Lidar Conference, Toronto (1986).

Stokes, G. H.

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “Off-Axis Multiple Scattering of a Laser Beam in Turbid Media: Comparison of Theory and Experiment,” Appl. Opt. 26, 779 (1987).
[Crossref]

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

Stupin, D. A.

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

Stupin, D. M.

Tam, W. G.

Tsang, L.

Twitty, J. T.

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, “Derivation of Phase Functions from Multiply Scattered Sunlight Transmitted Through a Hazy Atmosphere,” J. Atmos. Sci. 32, 577 (1975).
[Crossref]

Ulitsky, A.

R. B. Smith, J. D. Houston, A. Ulitsky, A. I. Carswell, “Laboratory Measurements of Forward and Backward Scattering of Laser Beams in Water Droplet Clouds,” in International Laser Lidar Conference, Toronto (1986).

Unruh, W. P.

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “Off-Axis Multiple Scattering of a Laser Beam in Turbid Media: Comparison of Theory and Experiment,” Appl. Opt. 26, 779 (1987).
[Crossref]

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

Weinman, J. A.

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, “Derivation of Phase Functions from Multiply Scattered Sunlight Transmitted Through a Hazy Atmosphere,” J. Atmos. Sci. 32, 577 (1975).
[Crossref]

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

Zaccanti, G.

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. 2, 903 (1985).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “The Effect of a Strongly Inhomogeneous Medium on the Propagation of Light Beams under Multiple Scattering Conditions,” Opt. Acta 32, 717 (1985).
[Crossref]

Zardecki, A.

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “Off-Axis Multiple Scattering of a Laser Beam in Turbid Media: Comparison of Theory and Experiment,” Appl. Opt. 26, 779 (1987).
[Crossref]

W. G. Tam, A. Zardecki, “Multiple Scattering Corrections to the Beer-Lambert Law. 1: Open Detector,” Appl. Opt. 21, 2405 (1982).
[Crossref] [PubMed]

A. Zardecki, W. G. Tam, “Multiple Scattering Corrections to The Beer-Lambert Law. 2: Detector with a Variable Field of View,” Appl. Opt. 21, 2413 (1982).
[Crossref] [PubMed]

W. G. Tam, A. Zardecki, “Off-Axis Propagation of a Laser Beam in Low Visibility Weather Conditions,” Appl. Opt. 19, 2822 (1980).
[Crossref] [PubMed]

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

A. Zardecki, S. A. W. Gerstl, “Multiple Scattering Effects in the Off-Axis Propagation of Laser Radiation,” in International Conference, Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 27–30 May 1986.

Appl. Opt. (5)

J. Atmos. Sci. (1)

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, “Derivation of Phase Functions from Multiply Scattered Sunlight Transmitted Through a Hazy Atmosphere,” J. Atmos. Sci. 32, 577 (1975).
[Crossref]

J. Opt. Soc. Am. (2)

M. A. Box, A. Deepak, “Limiting Cases of the Small Angle Scattering Approximation Solutions for the Propagation of Laser Beams in Anisotropic Scattering Media,” J. Opt. Soc. Am. 71, 1534 (1981).
[Crossref]

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Use of Two Scaling Relations in the Study of Multiple-Scattering Effects on the Transmittance of Light Beams Through a Turbid Atmosphere,” J. Opt. Soc. Am. 2, 903 (1985).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

E. Battistelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “The Effect of a Strongly Inhomogeneous Medium on the Propagation of Light Beams under Multiple Scattering Conditions,” Opt. Acta 32, 717 (1985).
[Crossref]

Other (4)

A. Zardecki, S. A. W. Gerstl, “Multiple Scattering Effects in the Off-Axis Propagation of Laser Radiation,” in International Conference, Optical and Millimeter Wave Propagation and Scattering in the Atmosphere, Florence, Italy, 27–30 May 1986.

A. Zardecki, S. A. W. Gerstl, W. P. Unruh, G. H. Stokes, D. A. Stupin, N. E. Elliott, J. A. Weinman, “Multiple Scattering of Laser Beams in Dense Hydrosols,” in International Laser Radar Conference, Toronto, Canada, 11–15 Aug. 1986.

P. V. Hobbs, A. Deepak, Clouds: Their Formation, Optical Properties, and Effects (Academic, New York, 1981).

R. B. Smith, J. D. Houston, A. Ulitsky, A. I. Carswell, “Laboratory Measurements of Forward and Backward Scattering of Laser Beams in Water Droplet Clouds,” in International Laser Lidar Conference, Toronto (1986).

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Figures (4)

Fig. 1
Fig. 1

Comparison between the Mie theory phase function (solid line) for 2.26-μm diam polystyrene spheres and the two two-component Gaussian phase functions: κ1 = 0.57, κ2 = 0.43, α1 = 8.46 rad−1, α2 = 2.29 rad−1, dotted line; κ1 = 0.9, κ2 = 0.5, α1 = 8.46 rad−1, α2 = 2.29 rad−1, dashed line.

Fig. 2
Fig. 2

Measured and calculated irradiances for the optical depth of 5.02. The solid line is obtained from Eqs. (19)(21) using the two-component phase function of Fig. 1; the crosses are obtained from Eqs. (7) and (8) using the Mie scattering phase function, also shown in Fig. 1.

Fig. 3
Fig. 3

Measured and calculated irradiances for the optical depth of 5.02. The weights κ1 = 0.9 and κ2 = 0.5 obtained from the best fit to experimental data: α1 = 8.46 rad−1; α2 = 2.29 rad−1.

Fig. 4
Fig. 4

Measured and calculated irradiances for the optical depth of 10.16. The weight and width factors as in Fig. 3.

Tables (1)

Tables Icon

Table I Relative Contributions of Scattering Events (%)

Equations (27)

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( ϕ r + z + σ ) I ( ϕ , r , z ) = σ s P ( ϕ ϕ ) I ( ϕ , r , z ) d ϕ ,
P ( ϕ ) d ϕ = 1 .
I 0 ( ϕ , r ) = β 2 γ 2 π 2 exp ( β 2 ϕ 2 γ 2 r 2 ) ,
N ( r , z ) = I ( ϕ , r , z ) d ϕ .
I 0 ( ξ , η ) = I 0 ( ϕ , r ) exp [ i ( ξ ϕ + η r ) ] d ϕ d r ,
I 0 ( ξ , η ) = exp [ 1 4 ( ξ 2 β 2 + η 2 γ 2 ) ] .
N ( ρ , τ ) = exp ( τ ) 2 π σ 2 0 I 0 ( η τ , σ η ) J 0 ( η σ ρ γ ) exp [ Ω ( τ , η ) ] η d η ,
Ω ( τ , η ) = 4 π ω 0 τ 2 0 1 d z 0 2 P ( s ) J 0 ( s z η τ σ ) s d s .
N ( ρ , τ ) = exp ( τ ) ( 2 π ) 2 σ 2 I 0 ( τ η , σ η ) × exp [ i σ γ η ρ + Ω ( τ , η ) ] d η ,
Ω ( τ , η ) = ω 0 τ 0 1 p ( η τ σ z ) d z .
p ( ξ ) = P ( ϕ ) exp ( i ξ ϕ d ϕ ) .
I 0 ( ϕ , r ) = I 0 δ ( ϕ ) ,
I 0 ( ξ , η ) = ( 2 π ) 2 I 0 δ ( η ) .
I ( ϕ , r , z ) = I 0 ( 2 π ) 2 exp ( σ z ) × exp [ i ξ ϕ + σ s 0 z p ( ξ ) d z ] d ξ .
N ( ρ , τ ) = I 0 exp [ ( 1 ω 0 ) τ ] ,
P ( ϕ ) = 1 π m = 1 M κ m α m 2 exp ( α m 2 ϕ 2 ) .
m = 1 M κ m = 1 .
Ω ( τ , η ) = ω 0 τ m = 1 M κ m 0 1 exp ( η 2 τ 2 4 α m 2 σ 2 z 2 ) d z .
N ( ρ , τ ) = k 1 = 0 k 2 = 0 N k 1 k 2 ( ρ , τ ) ,
N k 1 k 2 ( ρ , τ ) = σ 2 π τ 2 e τ ( κ 1 ω 0 τ ) k 1 k 1 ! ( κ 2 ω 0 τ ) k 2 k 2 ! × 0 1 d x 1 0 1 d x k 1 0 1 d y 1 0 1 d y k 2 1 Λ k 1 k 2 × exp ( σ 2 ρ 2 γ 2 τ 2 Λ k 1 k 2 ) .
Λ k 1 k 2 = 1 α 1 2 j = 0 k 1 x j 2 + 1 α 2 2 j = 0 k 2 y j 2 + 1 β 2 + σ 2 τ 2 γ 2 .
κ 1 α 2 π + κ 2 α 2 π = P ( 0 ) .
k = 2 N k 0
m = 2 N 0 m
m = 2 N k 1
m = 2 N 1 m
κ = 2 m = 2 N km

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