Abstract

Multiple scattering corrections to the Beer-Lambert law are analyzed by means of a rigorous small-angle solution to the radiative transfer equation. Transmission functions for predicting the received radiant power—a directly measured quantity in contrast to the spectral radiance in the Beer-Lambert law—are derived. Numerical algorithms and results relating to the multiple scattering effects for laser propagation in fog, cloud, and rain are presented.

© 1982 Optical Society of America

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References

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  1. V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 724 (1967).
  2. T. S. Chu, D. C. Hogg, Bell Syst. Tech. J. 47, 725 (1968).
  3. R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
    [CrossRef]
  4. R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
    [CrossRef]
  5. A. Deepak, M. A. Box, Appl. Opt. 17, 2900 (1978).
    [CrossRef] [PubMed]
  6. A. Deepak, M. A. Box, Appl. Opt. 17, 3169 (1978).
    [CrossRef] [PubMed]
  7. W. G. Tam, A. Zardecki, J. Opt. Soc. Am. 69, 68 (1979).
    [CrossRef]
  8. W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
    [CrossRef]
  9. A. Zardecki, W. G. Tam, Can. J. Phys. 57, 1301 (1979).
    [CrossRef]
  10. R. L. Fante, IEEE Trans. Antennas Propag. AP-21, 750 (1973).
    [CrossRef]
  11. R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
    [CrossRef]
  12. W. G. Tam, Appl. Opt. 19, 2090 (1980).
    [CrossRef] [PubMed]
  13. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), Sec. 3.4.
  14. J. S. Marshall, W. Palmer, J. Meteorol. 5, 165 (1948).
    [CrossRef]
  15. R. G. Medhurst, IEEE Trans Antennas Propag. AP-13, 550 (1965).
    [CrossRef]
  16. A. Yariv, Quantum Electronics (Wiley, New York, 1975), Sec. 6.6.

1980 (1)

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]

1978 (2)

1974 (1)

1973 (1)

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

1968 (1)

T. S. Chu, D. C. Hogg, Bell Syst. Tech. J. 47, 725 (1968).

1967 (1)

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 724 (1967).

1965 (1)

R. G. Medhurst, IEEE Trans Antennas Propag. AP-13, 550 (1965).
[CrossRef]

1953 (2)

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[CrossRef]

1948 (1)

J. S. Marshall, W. Palmer, J. Meteorol. 5, 165 (1948).
[CrossRef]

Box, M. A.

Chu, T. S.

T. S. Chu, D. C. Hogg, Bell Syst. Tech. J. 47, 725 (1968).

Deepak, A.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), Sec. 3.4.

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]

Gumprecht, R. O.

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[CrossRef]

Hogg, D. C.

T. S. Chu, D. C. Hogg, Bell Syst. Tech. J. 47, 725 (1968).

Kabanov, M. V.

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 724 (1967).

Marshall, J. S.

J. S. Marshall, W. Palmer, J. Meteorol. 5, 165 (1948).
[CrossRef]

Medhurst, R. G.

R. G. Medhurst, IEEE Trans Antennas Propag. AP-13, 550 (1965).
[CrossRef]

Palmer, W.

J. S. Marshall, W. Palmer, J. Meteorol. 5, 165 (1948).
[CrossRef]

Savelev, B. A.

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 724 (1967).

Sliepcevich, C. M.

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[CrossRef]

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

Tam, W. G.

W. G. Tam, Appl. Opt. 19, 2090 (1980).
[CrossRef] [PubMed]

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

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

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

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Sec. 6.6.

Zardecki, A.

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

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

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

Zuev, V. E.

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 724 (1967).

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

T. S. Chu, D. C. Hogg, Bell Syst. Tech. J. 47, 725 (1968).

Can. J. Phys. (1)

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

IEEE Trans Antennas Propag. (1)

R. G. Medhurst, IEEE Trans Antennas Propag. AP-13, 550 (1965).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

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

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 724 (1967).

J. Meteorol. (1)

J. S. Marshall, W. Palmer, J. Meteorol. 5, 165 (1948).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. Chem. (2)

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
[CrossRef]

R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
[CrossRef]

Opt. Acta (1)

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

Other (2)

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), Sec. 3.4.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Sec. 6.6.

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Figures (12)

Fig. 1
Fig. 1

Correction factor for advective fog with σ = 16.119 km−1,γ = 0.35 cm−1, R = 1 cm, liquid water content = 0.1 g/m3, λ = 1.06 μm, α = 40.46 rad−1.

Fig. 2
Fig. 2

Correction factor for radiational fog with σ = 21.775 km−1, γ = 0.35 cm−1, R = 1 cm, liquid water content = 0.1 g/m3, λ = 1.06 μm, α = 28.20 rad−1.

Fig. 3
Fig. 3

Correction factor for advective fog with σ = 161.119 km−1, γ = 0.35 cm−1, R = 1 cm, liquid water content = 1 g/m3, λ = 1.06 μm, α = 40.46 rad−1.

Fig. 4
Fig. 4

Correction factor for radiational fog with σ = 217.75 km−1, γ = 0.35 cm−1, R = 1 cm, liquid water content = 1 g/m3, λ = 1.06 μm, α = 28.20 rad−1.

Fig. 5
Fig. 5

Correction factor for water cloud C1 with σ = 16.33 km−1, γ = 1.41 cm−1, R = 1 cm, λ = 0.45 μm, α = 47.08 rad−1.

Fig. 6
Fig. 6

Correction factor for rain with rain rate = 12.5 mm/h, σ = 1.36 km−1, γ = 1.41 cm−1, R = 1 cm, λ = 1.06 μm, α = 3032.3 rad−1.

Fig. 7
Fig. 7

Amplification factor for advective fog with σ = 16.119 km−1, γ = 0.35 cm−1, R = 1 cm, liquid water content = 0.1 g/m3, λ = 1.06 μm, α = 40.46 rad−1.

Fig. 8
Fig. 8

Amplification factor for advective fog with σ = 161.79 km−1, γ = 0.35 cm−1, R = 1 cm, liquid water content = 0.1 g/m3, λ = 1.06 μm, α = 40.46 rad−1.

Fig. 9
Fig. 9

Amplification factor for radiational fog with σ = 21.775 km−1, γ = 0.35 cm−1, R = 1 cm, liquid water content = 0.1 g/m3, λ = 1.06 μm, α = 28.20 rad−1.

Fig. 10
Fig. 10

Amplification factor for radiational fog with σ = 217.75 km−1, γ = 0.35 cm−1, R = 1 cm, liquid water content = 1 g/m3, λ = 1.06 μm, α = 28.20 rad−1.

Fig. 11
Fig. 11

Amplification factor for water cloud C1 with σ = 16.33 km−1 γ = 1.41 cm−1, R = 1 cm, λ = 0.45 μm, α = 47.08 rad−1.

Fig. 12
Fig. 12

Amplification factor for rain with rain rate = 12.5 mm/h, σ = 1.36 km−1, γ = 1.41 cm−1, R = 1 cm, λ = 1.06 μm, α = 3032.3 rad−1.

Tables (1)

Tables Icon

Table I Amplification Factor A(z,σ)

Equations (35)

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( ϕ r + z + σ ) I ( ϕ , r , z ) = d ϕ p ( ϕ ϕ ) I ( ϕ , r , z ) ,
d ϕ p ( ϕ ) = σ s .
I ( ϕ , r , z ) = I ( u ) ( ϕ , r , z ) + I ( s ) ( ϕ , r , z )
I 0 ( ϕ , r , z = 0 ) = β 2 γ 2 π 2 exp ( β 2 ϕ 2 γ 2 r 2 )
p ( ϕ ) = σ 2 α 2 π exp ( α 2 ϕ 2 ) ,
I ( u ) ( ϕ , r , z ) = 1 ( 2 π ) 4 d ξ d ζ exp [ i ( ξ ϕ + ζ r ) ] × exp ( σ z ) exp [ ( ξ + z ζ ) 2 4 β 2 ζ 2 4 γ 2 ] , ( 6 )
I ( s ) ( ϕ , r , z ) = 1 ( 2 π ) 4 d ξ d ζ exp [ i ( ξ ϕ + ζ r ) ] exp ( σ z ) × exp [ ( ξ + z ζ ) 2 4 β 2 ζ 2 4 γ 2 ] [ exp Ω ( z ) 1 ] ,
Ω ( z ) = σ s 0 z d z exp { [ ξ + ( z z ) ζ ] 2 4 α 2 } .
I ( s ) ( ϕ , r , z ) = m = 1 I m ( ϕ , r , z ) ,
I m ( ϕ , r , z ) = 1 ( 2 π ) 4 d ξ d ζ exp [ i ( ξ ϕ + ζ r ) ] exp ( σ z ) × σ s m m ! exp [ ( ξ + z ζ ) 2 4 β 2 ζ 2 4 γ 2 ] × 0 z 0 z d z 1 d z m × exp { 1 4 α 2 [ m ξ 2 + 2 ξ ζ Z 1 ( m ) + ζ 2 Z 2 ( m ) ] } ,
Z 1 ( m ) = i = 1 m ( z z i ) , Z 2 ( m ) = i = 1 m ( z z i ) 2 .
N ( r , z ) = d ϕ I ( ϕ , r , z ) = N 0 ( r , z ) + m = 1 N m ( r , z )
N 0 ( r , z ) = 1 π ( z 2 β 2 + 1 γ 2 ) 1 exp [ σ z r 2 ( z 2 β 2 + 1 γ 2 ) 1 ] ,
N m ( r , z ) = σ s m π m ! exp ( σ z ) 0 z 0 z d z 1 d z m [ Λ ( m ) ] 1 exp [ r 2 Λ 1 ( m ) ] ,
Λ ( m ) = [ Z 2 ( m ) α 2 + z 2 β 2 + 1 γ 2 ] .
P ( z , σ ) = 2 π 0 R drrN ( r , z ) = exp ( σ z ) { 1 exp [ R 2 ( z 2 β 2 + 1 γ 2 ) 1 ] } + m = 1 exp ( σ z ) σ s m z m m ! [ 1 S ( m ) ] .
S ( m ) = 0 1 0 1 d s 1 d s m × exp [ R 2 ( z 2 α 2 i = 1 m s i 2 + z 2 β 2 + 1 γ 2 ) 1 ] .
P ( z , σ = 0 ) = 1 exp [ R 2 ( z 2 β 2 + 1 γ 2 ) 1 ] ,
P ( z = 0 , σ ) = 1 exp ( γ 2 R 2 ) .
T 1 ( z , σ ) P ( z , σ ) / P ( z , σ = 0 ) T 2 ( z , σ ) P ( z , σ ) / P ( z = 0 , σ ) .
T 1 ( z , σ ) = exp ( σ z ) { 1 + m = 1 σ s m z m m ! [ 1 S ( m ) ] 1 exp [ R 2 ( z 2 β 2 + 1 γ 2 ) 1 ] } .
T 1 ( z , σ ) = exp ( σ z ) [ 1 + C ( z , σ ) ] .
T 2 ( z , σ ) = exp ( σ z ) A ( z , σ ) ,
A ( z , σ ) = { 1 exp [ R 2 ( z 2 β 2 + 1 γ 2 ) 1 ] + m = 1 σ s m z m m ! [ 1 S ( m ) ] } ÷ [ 1 exp ( γ 2 R 2 ) ] .
A ( z , σ ) = 1 + C ( z , σ ) .
S ( m ) = π m / 2 2 m 1 Γ ( m 2 ) 0 R ( m ) dss m 1 × exp { R 2 [ z 2 ( s 2 α 2 + 1 β 2 ) + 1 γ 2 ] 1 } ,
π m / 2 [ R ( m ) ] m Γ ( m 2 + 1 ) = 2 m .
N ( s ) ( r , z ) = d ϕ I ( s ) ( ϕ , r , z ) = 1 ( 2 π ) 4 d ϕ d ξ d ζ exp [ i ( ξ ϕ + ζ r ) ] × exp [ σ z ( ξ + z ζ ) 2 4 β 2 ζ 2 4 γ 2 ] [ exp Ω ( z ) 1 ] = 1 ( 2 π ) 2 d ζ exp [ i ζ r ] × exp [ σ z ( z 2 4 β 2 + 1 4 γ 2 ) ζ 2 ] [ exp Ω 0 ( z ) 1 ] ,
Ω 0 ( z ) = σ s 0 z d z exp [ ( z z ) 2 ζ 2 / 4 α 2 ] = σ s α ζ π 1 / 2 erf ( z ζ 2 α ) .
N ( s ) ( r , z ) = 1 2 π d ζ ζ exp [ σ z ( z 2 4 β 2 + 1 4 γ 2 ) ζ 2 ] × [ exp Ω 0 ( z ) 1 ] J 0 ( ζ r )
P ( s ) ( z , σ ) = 2 π 0 R drrN ( s ) ( r , z ) = R d ζ exp [ σ z ( z 2 4 β 2 + 1 4 γ 2 ) ζ 2 ] × [ exp Ω 0 ( z ) 1 ] J 1 ( ζ R ) .
T 1 ( z , σ ) = exp ( σ z ) { 1 + R d ζ exp [ ( z 2 4 β 2 + 1 4 γ 2 ) ζ 2 ] [ exp Ω 0 ( z ) 1 ] J 1 ( ζ R ) 1 exp [ R 2 ( z 2 β 2 + 1 γ 2 ) 1 ] } ,
T 2 ( z , σ ) = exp ( σ z ) [ 1 exp ( γ 2 R 2 ) ] { 1 exp [ R 2 ( z 2 β 2 + 1 γ 2 ) 1 ] + R d ζ exp [ ( z 2 β 2 + 1 γ 2 ) ζ 2 4 ] [ exp Ω 0 ( z ) 1 ] J 1 ( ζ R ) } .
B ( m ) = exp [ R 2 σ 2 ( τ 2 α 2 m + τ 2 β 2 + σ 2 γ 2 ) 1 ] exp [ R 2 σ 2 ( τ 2 α 2 + τ 2 β 2 + σ 2 γ 2 ) 1 ] .
B ( m ) = exp [ R 2 ( τ 2 α 2 m σ 2 + τ 2 β 2 1 σ 2 + 1 γ 2 ) 1 ] exp [ R 2 ( τ 2 α 2 1 σ 2 + τ 2 β 2 1 σ 2 + 1 γ 2 ) 1 ] .

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