Abstract

The backscattering of a laser beam from a particulate medium is studied by means of the radiative transfer equation, taking into account the effect of multiple scattering. An analytical expression for the backscattered power is obtained. From this the widely used single scattered lidar equation is recovered as an asymptotic form. Numerical results are discussed for the case of laser power returned from an advective fog.

© 1983 Optical Society of America

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References

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  1. A. I. Carswell, Can. J. Phys. 61, 378 (1983).
    [CrossRef]
  2. K. E. Kunkel, J. A. Weinmann, J. Atmos. Sci. 33, 1772 (1976).
    [CrossRef]
  3. C. M. R. Platt, J. Atmos. Sci. 38, 156 (1981).
    [CrossRef]
  4. E. D. Hinkley, Ed., Laser Monitoring of the Atmosphere (Springer, Berlin, 1976), Chap. 4.
    [CrossRef]
  5. J. D. Klett, Appl. Opt. 20, 211 (1981).
    [CrossRef] [PubMed]
  6. W. G. Tam, A. Zardecki, Appl. Opt. 21, 2405 (1982).
    [CrossRef] [PubMed]
  7. A. Zardecki, W. G. Tam, Appl. Opt. 21, 2413 (1982).
    [CrossRef] [PubMed]
  8. R. L. Fante, IEEE Trans. Antennas Propag. AP-21, 750 (1973).
    [CrossRef]
  9. R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
    [CrossRef]
  10. D. Arnush, J. Opt. Soc. Am. 62, 1109 (1972).
    [CrossRef]
  11. L. B. Stotts, J. Opt. Soc. Am. 67, 815 (1977).
    [CrossRef]
  12. W. G. Tam, A. Zardecki, J. Opt. Soc. Am. 69, 68 (1979).
    [CrossRef]
  13. W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
    [CrossRef]
  14. A. Zardecki, W. G. Tam, Can. J. Phys. 57, 1301 (1979).
    [CrossRef]
  15. D. A. De Wolf, IEEE Trans. Antennas Propag. AP-20, 805 (1972).
    [CrossRef]
  16. V. Zuev, “Propagation of Visible and Infrared Waves in the Atmosphere,” TTF-707 (NASA, 1372), p. 94.

1983 (1)

A. I. Carswell, Can. J. Phys. 61, 378 (1983).
[CrossRef]

1982 (2)

1981 (2)

C. M. R. Platt, J. Atmos. Sci. 38, 156 (1981).
[CrossRef]

J. D. Klett, Appl. Opt. 20, 211 (1981).
[CrossRef] [PubMed]

1979 (3)

W. G. Tam, A. Zardecki, J. Opt. Soc. Am. 69, 68 (1979).
[CrossRef]

W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
[CrossRef]

A. Zardecki, W. G. Tam, Can. J. Phys. 57, 1301 (1979).
[CrossRef]

1977 (1)

1976 (1)

K. E. Kunkel, J. A. Weinmann, J. Atmos. Sci. 33, 1772 (1976).
[CrossRef]

1974 (1)

1973 (1)

R. L. Fante, IEEE Trans. Antennas Propag. AP-21, 750 (1973).
[CrossRef]

1972 (2)

D. Arnush, J. Opt. Soc. Am. 62, 1109 (1972).
[CrossRef]

D. A. De Wolf, IEEE Trans. Antennas Propag. AP-20, 805 (1972).
[CrossRef]

Arnush, D.

Carswell, A. I.

A. I. Carswell, Can. J. Phys. 61, 378 (1983).
[CrossRef]

De Wolf, D. A.

D. A. De Wolf, IEEE Trans. Antennas Propag. AP-20, 805 (1972).
[CrossRef]

Fante, R. L.

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

R. L. Fante, IEEE Trans. Antennas Propag. AP-21, 750 (1973).
[CrossRef]

Klett, J. D.

Kunkel, K. E.

K. E. Kunkel, J. A. Weinmann, J. Atmos. Sci. 33, 1772 (1976).
[CrossRef]

Platt, C. M. R.

C. M. R. Platt, J. Atmos. Sci. 38, 156 (1981).
[CrossRef]

Stotts, L. B.

Tam, W. G.

Weinmann, J. A.

K. E. Kunkel, J. A. Weinmann, J. Atmos. Sci. 33, 1772 (1976).
[CrossRef]

Zardecki, A.

Zuev, V.

V. Zuev, “Propagation of Visible and Infrared Waves in the Atmosphere,” TTF-707 (NASA, 1372), p. 94.

Appl. Opt. (3)

Can. J. Phys. (2)

A. I. Carswell, Can. J. Phys. 61, 378 (1983).
[CrossRef]

A. Zardecki, W. G. Tam, Can. J. Phys. 57, 1301 (1979).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

D. A. De Wolf, IEEE Trans. Antennas Propag. AP-20, 805 (1972).
[CrossRef]

R. L. Fante, IEEE Trans. Antennas Propag. AP-21, 750 (1973).
[CrossRef]

J. Atmos. Sci. (2)

K. E. Kunkel, J. A. Weinmann, J. Atmos. Sci. 33, 1772 (1976).
[CrossRef]

C. M. R. Platt, J. Atmos. Sci. 38, 156 (1981).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Acta (1)

W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
[CrossRef]

Other (2)

E. D. Hinkley, Ed., Laser Monitoring of the Atmosphere (Springer, Berlin, 1976), Chap. 4.
[CrossRef]

V. Zuev, “Propagation of Visible and Infrared Waves in the Atmosphere,” TTF-707 (NASA, 1372), p. 94.

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Figures (4)

Fig. 1
Fig. 1

Geometrical configuration of the experimental situation.

Fig. 2
Fig. 2

Relative backscattered power vs optical depth in an advective fog with a receiver field of view of 0.2 mrad. Dashed line is single backscattered power; solid line is multiple backscattered power.

Fig. 3
Fig. 3

Relative backscattered power vs optical depth in an advective fog with a receiver field of view of 1 mrad. Dashed line is single backscattered power; solid line is multiple backscattered power.

Fig. 4
Fig. 4

Relative backscattered power vs optical depth in an advective fog with a receiver field of view of 2 mrad. Dashed line is single backscattered power; solid line is multiple backscattered power.

Equations (42)

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( s · r + σ ) I ( s , r ) = d s p ( s , s ) I ( s , r ) ,
p ( s , s ) = p ( + ) ( s , s ) + p ( ) ( s , s ) ,
p ( + ) ( s , s ) = { p ( s , s ) for s · s 0 , 0 otherwise ,
p ( ) ( s , s ) = { p ( s , s ) for s · s < 0 , 0 otherwise .
I ( s , r ) = I ( + ) ( s , r ) + I ( ) ( s , r ) ,
( s · r + σ ) [ I ( + ) ( s , r ) + I ( ) ( s , r ) ] = d s [ p ( + ) ( s , s ) + p ( ) ( s , s ) ] × [ I ( + ) ( s , r ) + I ( ) ( s , r ) ] .
( s · r + σ ) I ( + ) ( s , r ) = d s p ( + ) ( s , s ) I ( + ) ( s , r ) .
( s · r + σ ) I ( ) ( s , r ) = d s p ( + ) ( s , s ) I ( ) ( s , r ) + d s p ( ) ( s , s ) I ( + ) ( s , r ) + d s p ( ) ( s , s ) I ( ) ( s , r ) .
I ( + ) ( ϕ , ρ , z = 0 ) exp [ ( γ ρ / z 0 ) 2 ] δ ( ϕ ρ z 0 ) ,
I ( + ) ( ϕ , ρ , z = 0 ) = z 0 2 δ ( ρ z 0 ϕ ) exp ( γ 2 ϕ 2 ) ,
( ϕ · / ρ + / z + σ ) I ( + ) ( ϕ , ρ , z ) = d ϕ p ( + ) ( ϕ ϕ ) I ( + ) ( ϕ , ρ , z ) .
p ( + ) ( ϕ ) = ( σ s + α 2 / π ) exp ( α 2 ϕ 2 ) ,
σ s + = d s p ( s , s ) = d s p ( + ) ( s , s ) , s · s 0 .
σ s = d s p ( s , s ) = d s p ( ) ( s , s ) , s · s < 0 .
σ s = σ s + + σ s .
I ( + ) ( ϕ , ρ , z ) = I 0 ( + ) ( ϕ , ρ , z ) + m = 1 I m ( + ) ( ϕ , ρ , z ) .
I 0 ( + ) ( ϕ , ρ , z ) = z 0 2 exp ( σ z ) exp ( γ 2 ϕ 2 ) δ [ ρ ( z + z 0 ) ϕ ] ,
I m ( + ) ( ϕ , ρ , z ) = exp ( σ z ) ( 2 π ) 4 π z 0 2 γ 2 ( σ z + z ) m m ! 0 1 d s 1 0 1 d s m × d ξ d ξ exp [ i ( ξ · ϕ + ζ · ρ ) ] × exp { ξ 2 ( m 4 α 2 + 1 4 γ 2 ) 2 ξ · ζ [ z 1 ( m ) 4 α 2 + ( z + z 0 ) 4 γ 2 ] ζ 2 [ z 2 ( m ) 4 α 2 + ( z + z 0 2 ) 4 γ 2 ] } ,
z 1 ( m ) = z i = 1 m ( 1 s i ) and z 2 ( m ) = z 2 i = 1 m ( 1 s i ) 2 .
( s · r + σ ) I ( ) ( s , r ) = d s p ( + ) ( s , s ) I ( ) ( s , r ) + d s p ( ) ( s , s ) I ( + ) ( s , r ) .
d s p ( + ) ( s , s ) I ( ) ( s , r ) σ s + I ( ) ( s , r ) .
[ s · r + ( σ σ s + ) ] I ( ) ( s , r ) = d s p ( ) ( s , s ) I ( + ) ( s , r ) .
[ ϕ · z + ( σ σ s + ) ] I ( ) ( ϕ , ρ , z ) = d ϕ p ( ) ( π ϕ , ϕ ) I ( + ) ( ϕ , ρ , z ) ,
I ( ) ( ϕ , ρ , z ) = 0 Z d z exp { [ ( σ σ s ) + ϕ · ] ( z z ) } × d ϕ p ( ) ( π ϕ ϕ ) I ( + ) ( ϕ , ρ , z ) .
I ( ) ( ϕ , ρ , z ) = 0 Z d z exp { [ ( σ σ s + ) + ϕ · ] ( z z ) } × p ( ) ( π ϕ ) d ϕ d ϕ I ( + ) ( ϕ , ρ , z ) .
I ( ) ( ϕ , ρ , z = 0 ) = 0 Z d z exp { z [ ( σ σ s + ) + ϕ · ] } × p ( ) ( π ϕ ) d ϕ I ( + ) ( ϕ , ρ , z ) .
N ( + ) ( ρ , z ) = d ϕ I ( + ) ( ϕ , ρ , z ) .
I ( ) ( ϕ , ρ , z = 0 ) = 0 Z d z exp [ z ( σ σ s + ) ] p ( ) ( π ϕ ) × N ( + ) ( ρ z ϕ , z ) .
I ( ) ( ϕ , ρ , z = z 0 ) = I ( ) ( ϕ , ρ z 0 ϕ , z = 0 ) .
p ( ) ( ψ , R , z 0 ) = ϕ ψ d ϕ ρ R d ρ I ( ) ( ϕ , ρ z 0 ϕ , z = 0 ) .
N ( + ) ( ρ , z ) = N 0 ( + ) ( ρ , z ) + m = 1 N m ( + ) ( ρ , z ) ,
N 0 ( + ) ( ρ , z ) = z 0 2 ( z + z 0 ) 2 exp ( σ z ) exp [ γ 2 ρ 2 ( z + z 0 ) 2 ] ,
N m ( + ) ( ρ , z ) = exp ( σ z ) z 0 2 4 γ 2 ( σ s + z ) m m ! 0 1 d s 1 0 1 d s m × [ z 2 ( m ) 4 α 2 + ( z + z 0 ) 2 4 γ 2 ] 1 × exp { ρ 2 4 [ z 2 ( m ) 4 α 2 + ( z + z 0 ) 2 4 γ 2 ] } .
p ( ) ( ψ , R , z 0 ) = P 0 ( ) ( ψ , R , z 0 ) + m = 1 P m ( ) ( ψ , R , z 0 ) .
P n ( ) ( ψ , R , z 0 ) = 0 Z d z exp [ z ( σ σ z + ) ] p ( ) ( π ) × ϕ ψ d ϕ ρ R d ρ N n ( + ) [ ρ ( z 0 + z ) ϕ , z ] ,
N ( + ) ( ρ , z ) = z 0 2 exp ( σ z ) ( z + z 0 ) 2 ,
p ( ) ( ψ , R , z 0 ) = 0 Z d z exp ( 2 σ z ) p ( ) ( π ) ϕ ψ d ϕ × ρ R d ρ z 0 2 ( z + z 0 ) 2 = π 2 R 2 z 0 2 ψ 2 p ( ) ( π ) × 0 Z d z exp ( 2 σ z ) ( z + z 0 ) 2 .
p 0 ( ) ( ψ , R , z 0 ) = 0 Z d z exp [ z ( σ σ s + ) ] p ( ) ( π ) f 0 ( ψ , R , z ) ,
p m ( ) ( ψ , R , z 0 ) = 0 Z d z exp [ z ( σ σ s + ) ] p ( ) ( π ) × 0 1 d s 1 0 1 d s m f m ( ψ , R , z ) ,
f n ( ψ , R , z ) = π 2 z 0 2 C n γ 2 exp ( σ z ) × { [ 1 exp ( C n ψ 2 ) ] [ 1 exp ( C n R 2 ) ] + [ 1 C n ψ 2 exp ( C n ψ 2 ) exp ( C n ψ 2 ) ] × [ 1 C n R 2 exp ( C n R 2 ) exp ( C n R 2 ) ] } ,
C n = ( z + z 0 ) 2 γ 2 [ ( 1 δ n , 0 ) z 2 ( n ) α 2 ] 1 , C n = ( z + z 0 ) 2 C n .
f ( r ) = 1 Γ ( μ + 1 ) μ μ + 1 ( r / a ) μ a 1 exp ( μ r / a ) ,

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