Abstract

In the small-angle approximation, the exact solution of an equation of radiative transfer is cast in a series form suitable for numerical computation. This is used to calculate the radiance produced by a cw laser beam. For the medium with Gaussian phase function, a convergent numerical procedure is described. Explicit results for the scattered radiance, obtained for a wide range of parameters, display the existence of three well-defined regions on the plots of radiance versus transverse distance. A comparison of the exact theory with two approximate approaches is carried out.

© 1979 Optical Society of America

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References

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  1. R. P. Hemenger, J. Opt Soc. Am. 64, 503 (1974).
    [Crossref]
  2. D. Arnush, J. Opt. Soc. Am. 62, 9 (1972).
    [Crossref]
  3. J. A. Weinman, J. Atmos. Sci. 33, 1763 (1976).
    [Crossref]
  4. L. B. Stotts, J. Opt. Soc. Am. 67, 815 (1977).
    [Crossref]
  5. R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
    [Crossref]
  6. R. L. Fante, Proc. IEEE,  62, 1400 (1974).
    [Crossref]
  7. S. K. Friedlander, Smoke, Dust and Haze. Fundamentals of Aerosol Behavior (Wiley, New York, 1977) Ch. 5.
  8. E. A. Bucher, Appl. Opt. 12, 2391 (1972).
    [Crossref]
  9. R. L. Fante, IEEE AP. 21, 750 (1973).
    [Crossref]
  10. H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
    [Crossref]
  11. W. S. Scott, Rev. Mod. Phys. 35, 231 (1963).
    [Crossref]
  12. P. J. Davis and P. Rabinovitch, Numerical Integration (Blaisdel, Toronto, 1967), p. 141.

1977 (1)

1976 (1)

J. A. Weinman, J. Atmos. Sci. 33, 1763 (1976).
[Crossref]

1974 (3)

R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
[Crossref]

R. L. Fante, Proc. IEEE,  62, 1400 (1974).
[Crossref]

R. P. Hemenger, J. Opt Soc. Am. 64, 503 (1974).
[Crossref]

1973 (1)

R. L. Fante, IEEE AP. 21, 750 (1973).
[Crossref]

1972 (2)

D. Arnush, J. Opt. Soc. Am. 62, 9 (1972).
[Crossref]

E. A. Bucher, Appl. Opt. 12, 2391 (1972).
[Crossref]

1963 (1)

W. S. Scott, Rev. Mod. Phys. 35, 231 (1963).
[Crossref]

1949 (1)

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[Crossref]

Arnush, D.

D. Arnush, J. Opt. Soc. Am. 62, 9 (1972).
[Crossref]

Bucher, E. A.

Davis, P. J.

P. J. Davis and P. Rabinovitch, Numerical Integration (Blaisdel, Toronto, 1967), p. 141.

Fante, R. L.

R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
[Crossref]

R. L. Fante, Proc. IEEE,  62, 1400 (1974).
[Crossref]

R. L. Fante, IEEE AP. 21, 750 (1973).
[Crossref]

Friedlander, S. K.

S. K. Friedlander, Smoke, Dust and Haze. Fundamentals of Aerosol Behavior (Wiley, New York, 1977) Ch. 5.

Hemenger, R. P.

R. P. Hemenger, J. Opt Soc. Am. 64, 503 (1974).
[Crossref]

Rabinovitch, P.

P. J. Davis and P. Rabinovitch, Numerical Integration (Blaisdel, Toronto, 1967), p. 141.

Scott, W. S.

W. S. Scott, Rev. Mod. Phys. 35, 231 (1963).
[Crossref]

Scott, W. T.

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[Crossref]

Snyder, H. S.

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[Crossref]

Stotts, L. B.

Weinman, J. A.

J. A. Weinman, J. Atmos. Sci. 33, 1763 (1976).
[Crossref]

Appl. Opt. (1)

IEEE AP. (1)

R. L. Fante, IEEE AP. 21, 750 (1973).
[Crossref]

J. Atmos. Sci. (1)

J. A. Weinman, J. Atmos. Sci. 33, 1763 (1976).
[Crossref]

J. Opt Soc. Am. (1)

R. P. Hemenger, J. Opt Soc. Am. 64, 503 (1974).
[Crossref]

J. Opt. Soc. Am. (3)

Phys. Rev. (1)

H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
[Crossref]

Proc. IEEE (1)

R. L. Fante, Proc. IEEE,  62, 1400 (1974).
[Crossref]

Rev. Mod. Phys. (1)

W. S. Scott, Rev. Mod. Phys. 35, 231 (1963).
[Crossref]

Other (2)

P. J. Davis and P. Rabinovitch, Numerical Integration (Blaisdel, Toronto, 1967), p. 141.

S. K. Friedlander, Smoke, Dust and Haze. Fundamentals of Aerosol Behavior (Wiley, New York, 1977) Ch. 5.

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Figures (4)

FIG. 1
FIG. 1

Scattered radiance vs transverse distance from the beam axis. Exact solution. The extinction coefficient σ = 1.0 × 10−5 cm−1.

FIG. 2
FIG. 2

Scattered radiance vs transverse distance from the beam axis. Exact solution. The extinction coefficient σ = 4.0 × 10−5 cm−1.

FIG. 3
FIG. 3

Scattered radiance vs transverse distance, as predicted by exact theory and the approximate theory based on Arnush-Stotts approximation.

FIG. 4
FIG. 4

Scattered radiance vs transverse distance, as predicted by exact theory and the approximate theory of Fante.

Equations (10)

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ϕ I r + I z + σ I = p ( ϕ ϕ ) I ( ϕ , r , z ) d 2 ϕ ,
p ( ϕ ) = ( a σ / π ) exp ( a ϕ 2 ) ,
I ( ϕ , r , 0 ) = δ ( r ) exp ( b ϕ 2 ) ,
I ( ϕ , r , z ) = π b e σ z ( 2 π ) 4 d 2 ξ d 2 η e Ω ( z ) × exp ( ( ξ + η z ) 2 4 b ) e i ( ξ ϕ + η r )
Ω ( z ) = σ 0 z exp { 1 4 a | ξ + η ( z z ) | 2 } d z .
I ( ϕ , r , z ) = π b e σ z ( 2 π ) 4 m = 0 σ m m ! I m ( ϕ , r , z ) ,
I 0 ( ϕ , r , z ) = 2 ( 2 π ) 3 b z 2 exp ( b z 2 ( x 2 + y 2 ) ) δ ( ϕ r z )
I m = ( 2 π ) 2 z m 2 exp ( ϕ 2 ( m / a ) + b 1 ) 0 1 × 0 1 d z 1 , d z m × Δ 1 ( m ) × exp { ( m / 4 a ) + b 1 Δ ( m ) [ r z ( 1 1 m j = 1 m z j ) ϕ ] 2 } ,
Δ ( m ) = 1 4 a 2 [ m j = 1 m ( 1 z j ) 2 ( j = 1 m ( 1 z j ) ) 2 ] + 1 4 a b ( m + j = 1 m ( 1 z j ) 2 2 j = 1 m ( 1 z j ) )
Ω A S ( z ) = σ [ z 1 4 a ( ξ 2 z + ξ η z 2 + η 2 z 3 3 ) ]