Abstract

Theoretical formulations are given for an approximate method based on the solution of the radiative transfer equation in the small angle approximation. The method is approximate in the sense that an approximation is made in addition to the small angle approximation. Numerical results were obtained for multiple scattering effects as functions of the detector field of view as well as the size of the detector's aperture for three different values of the optical depth τ (= 1.0, 4.0, and 10.0). Three cases of aperture size were considered, namely, equal to, smaller, or larger than the laser beam diameter. The contrast between the on-axis intensity and the received power for the last three cases is clearly evident.

© 1983 Optical Society of America

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References

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  1. G. K. Goedecke, in EOSAEL 80, Vol. 1, Technical Documentation, L. D. Duncan, Ed., Atmospheric Sciences Laboratory Report ASL-TR-0072 (Jan.1976).
  2. E. A. Bucher, Appl. Opt. 12, 2391 (1973).
    [CrossRef] [PubMed]
  3. G. N. Plass, G. Kattawar, Appl. Opt. 7, 415 (1968).
    [CrossRef] [PubMed]
  4. W. G. Tam, A. Zardecki, J. Opt. Soc. Am. 69, 68 (1979).
    [CrossRef]
  5. A. Zardecki, W. G. Tam, Can. J. Phys. 57, 1301 (1979).
    [CrossRef]
  6. R. E. Jensen, J. Opt. Soc. Am. 70, 1557 (1980).
    [CrossRef]
  7. M. A. Box, A. Deepak, J. Opt. Soc. Am. 71, 1534 (1981).
    [CrossRef]
  8. A. Deepak, U. O. Farrukh, A. Zardecki, Appl. Opt. 21, 439 (1982).
    [CrossRef] [PubMed]
  9. A. Deepak, A. Zardecki, U. O. Farrukh, M. A. Box in Proceedings, Workshop on Atmospheric Aerosols, Baltimore, Md., 1979 (Spectrum Press, Hampton, Va., 1981).
  10. A. Ishimaru, Proc. IEEE 65, 1030 (1977).
    [CrossRef]
  11. A. Ishimaru, Wave Propagation and Scattering in Pandom Media (Academic, New York, 1978).
  12. R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 90 (1953).
    [CrossRef]
  13. R. O. Gumprecht, C. M. Sliepcevich, J. Phys. Chem. 57, 95 (1953).
    [CrossRef]
  14. V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Atmos. Ocean. Phys. 3, 724 (1967).
  15. A. Deepak, O. H. Vaughan, Appl. Opt. 17, 374 (1978).
    [CrossRef] [PubMed]
  16. A. Deepak, M. A. Box, Appl. Opt. 17, 2900 (1978).
    [CrossRef] [PubMed]
  17. A. Deepak, M. A. Box, Appl. Opt. 17, 3169 (1978).
    [CrossRef] [PubMed]
  18. L. S. Dolin, Izv. Vyssh. Vchebn. Zaued. Radiofiz. 9, 61 (1966).
  19. R. L. Fante, IEEE Trans. Antennas Propag. AP-21, 750 (1973).
    [CrossRef]
  20. R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
    [CrossRef]
  21. R. L. Fante, Proc. IEEE 62, 1400 (1974).
    [CrossRef]
  22. R. L. Fante, Proc. IEEE 68, 1424 (1980).
    [CrossRef]
  23. W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
    [CrossRef]
  24. D. Diermendjian, Electromagnetic Scattering by Spherical Polydispersions (American Elsevier, New York, 1969).
  25. W. G. Tam, A. Zardecki, Appl. Opt. 21, 2405 (1982).
    [CrossRef] [PubMed]

1982 (2)

1981 (1)

1980 (2)

1979 (3)

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

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]

1978 (3)

1977 (1)

A. Ishimaru, Proc. IEEE 65, 1030 (1977).
[CrossRef]

1974 (2)

1973 (2)

E. A. Bucher, Appl. Opt. 12, 2391 (1973).
[CrossRef] [PubMed]

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

1968 (1)

1967 (1)

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Atmos. Ocean. Phys. 3, 724 (1967).

1966 (1)

L. S. Dolin, Izv. Vyssh. Vchebn. Zaued. Radiofiz. 9, 61 (1966).

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]

Box, M. A.

M. A. Box, A. Deepak, J. Opt. Soc. Am. 71, 1534 (1981).
[CrossRef]

A. Deepak, M. A. Box, Appl. Opt. 17, 2900 (1978).
[CrossRef] [PubMed]

A. Deepak, M. A. Box, Appl. Opt. 17, 3169 (1978).
[CrossRef] [PubMed]

A. Deepak, A. Zardecki, U. O. Farrukh, M. A. Box in Proceedings, Workshop on Atmospheric Aerosols, Baltimore, Md., 1979 (Spectrum Press, Hampton, Va., 1981).

Bucher, E. A.

Deepak, A.

Diermendjian, D.

D. Diermendjian, Electromagnetic Scattering by Spherical Polydispersions (American Elsevier, New York, 1969).

Dolin, L. S.

L. S. Dolin, Izv. Vyssh. Vchebn. Zaued. Radiofiz. 9, 61 (1966).

Fante, R. L.

R. L. Fante, Proc. IEEE 68, 1424 (1980).
[CrossRef]

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

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

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

Farrukh, U. O.

A. Deepak, U. O. Farrukh, A. Zardecki, Appl. Opt. 21, 439 (1982).
[CrossRef] [PubMed]

A. Deepak, A. Zardecki, U. O. Farrukh, M. A. Box in Proceedings, Workshop on Atmospheric Aerosols, Baltimore, Md., 1979 (Spectrum Press, Hampton, Va., 1981).

Goedecke, G. K.

G. K. Goedecke, in EOSAEL 80, Vol. 1, Technical Documentation, L. D. Duncan, Ed., Atmospheric Sciences Laboratory Report ASL-TR-0072 (Jan.1976).

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]

Ishimaru, A.

A. Ishimaru, Proc. IEEE 65, 1030 (1977).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Pandom Media (Academic, New York, 1978).

Jensen, R. E.

Kabanov, M. V.

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Atmos. Ocean. Phys. 3, 724 (1967).

Kattawar, G.

Plass, G. N.

Savelev, B. A.

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Atmos. Ocean. 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, A. Zardecki, Appl. Opt. 21, 2405 (1982).
[CrossRef] [PubMed]

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]

Vaughan, O. H.

Zardecki, A.

A. Deepak, U. O. Farrukh, A. Zardecki, Appl. Opt. 21, 439 (1982).
[CrossRef] [PubMed]

W. G. Tam, A. Zardecki, Appl. Opt. 21, 2405 (1982).
[CrossRef] [PubMed]

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

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]

A. Deepak, A. Zardecki, U. O. Farrukh, M. A. Box in Proceedings, Workshop on Atmospheric Aerosols, Baltimore, Md., 1979 (Spectrum Press, Hampton, Va., 1981).

Zuev, V. E.

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Atmos. Ocean. Phys. 3, 724 (1967).

Appl. Opt. (7)

Can. J. Phys. (1)

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

IEEE Trans. Antennas Propag. (1)

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

Izv. Atmos. Ocean. Phys. (1)

V. E. Zuev, M. V. Kabanov, B. A. Savelev, Izv. Atmos. Ocean. Phys. 3, 724 (1967).

Izv. Vyssh. Vchebn. Zaued. Radiofiz. (1)

L. S. Dolin, Izv. Vyssh. Vchebn. Zaued. Radiofiz. 9, 61 (1966).

J. Opt. Soc. Am. (4)

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]

Proc. IEEE (3)

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

R. L. Fante, Proc. IEEE 68, 1424 (1980).
[CrossRef]

A. Ishimaru, Proc. IEEE 65, 1030 (1977).
[CrossRef]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Pandom Media (Academic, New York, 1978).

A. Deepak, A. Zardecki, U. O. Farrukh, M. A. Box in Proceedings, Workshop on Atmospheric Aerosols, Baltimore, Md., 1979 (Spectrum Press, Hampton, Va., 1981).

D. Diermendjian, Electromagnetic Scattering by Spherical Polydispersions (American Elsevier, New York, 1969).

G. K. Goedecke, in EOSAEL 80, Vol. 1, Technical Documentation, L. D. Duncan, Ed., Atmospheric Sciences Laboratory Report ASL-TR-0072 (Jan.1976).

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Figures (14)

Fig. 1
Fig. 1

Normalized intensity on the beam axis as function of detector FOV corresponding to α = 46.80 rad−1, RD = 1.0 cm, and τ = 1.0.

Fig. 2
Fig. 2

Normalized intensity on the beam axis as function of detector FOV corresponding to α = 46.80 rad−1, RD = 1.0 cm, and τ = 4.0.

Fig. 3
Fig. 3

Normalized intensity on the beam axis as function of detector FOV corresponding to α = 46.80 rad−1, RD = 1.0 cm, and τ = 10.0.

Fig. 4
Fig. 4

Received power (arbitrary units) on the beam axis as function of detector FOV corresponding to α = 46.80 rad−1, RD = 1.0 cm, and τ = 1.0.

Fig. 5
Fig. 5

Received power (arbitrary units) on the beam axis as function of detector FOV corresponding to α = 46.80 rad−1, RD = 1.0 cm, and τ = 4.0.

Fig. 6
Fig. 6

Received power (arbitrary units) on the beam axis as function of detector FOV corresponding to α = 46.80 rad−1, RD = 1.0 cm, and τ = 10.0.

Fig. 7
Fig. 7

Normalized intensity on the beam axis as function of detector FOV corresponding to α = 289.33 rad−1, RD = 1.0 cm, and τ = 1.0.

Fig. 8
Fig. 8

Normalized intensity on the beam axis as function of detector FOV corresponding to α = 289.33 rad−1, RD = 1.0 cm, and τ = 4.0.

Fig. 9
Fig. 9

Normalized intensity on the beam axis as function of detector FOV corresponding to α = 289.33 rad−1, RD = 1.0 cm, and τ = 10.0.

Fig. 10
Fig. 10

Received power (arbitrary units) on the beam axis as function of detector FOV correspondingly to α = 289.33 rad−1, RD = 1.0 cm, and τ = 4.0. Received power is multiplied by exp(τ).

Fig. 11
Fig. 11

Normalized intensity on the beam axis as function of detector FOV for the same parameters as in Fig. 8 except RD = 0.2 cm.

Fig. 12
Fig. 12

Received power (arbitrary units) on the beam axis as a function of detector FOV for the same parameters as in Fig. 11. The received power is multiplied by exp(τ).

Fig. 13
Fig. 13

Normalized intensity on the beam axis as function of detector FOV for the same parameters as in Fig. 8 except RD = 2.0 cm.

Fig. 14
Fig. 14

Received power (arbitrary units) on the beam axis as a function of detector FOV for the same parameters as in Fig. 13. The received power is multiplied by exp(τ).

Equations (24)

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ϕ · I r + I z + σ I = σ s P ( ϕ ϕ ) I ( ϕ , r , z ) d ϕ
I ( ϕ , r , z ) = I ( 0 ) ( ϕ , r , z ) + I ( s ) ( ϕ , r , z ) ,
I ( s ) ( ϕ , r , z ) = I ( s ) ( ϕ , r , z ) + k ( ϕ ϕ ) k ϕ k I ( s ) ( ϕ , r , z ) + 1 2 k l ( ϕ ϕ ) k ( ϕ ϕ ) l 2 ϕ k ϕ l I ( s ) ( ϕ , r , z ) + ,
ϕ · I ( 0 ) r + I ( 0 ) z + σ I ( 0 ) = 0 ,
ϕ · I ( s ) r σ ω 4 ϕ 2 ϕ 2 I ( s ) + I ( s ) z + σ ( 1 ω ) I ( s ) = σ ω P ( ϕ ϕ ) I ( 0 ) ( ϕ , r , z ) d 2 ϕ ,
ϕ 2 = P ( ϕ ) ϕ 2 d 2 ϕ .
Î ̂ ( ξ , η , z ) = d 2 ϕ d 2 r I ( ϕ , r , z ) exp [ i ( ξ · ϕ + η · r ) ] ,
P ̂ ( ξ ) = d 2 ϕ P ( ϕ ) exp ( i ξ · ϕ ) ,
Î ̂ ( 0 ) ( ξ , η , z ) = Î ̂ ( ξ + η z , η , z = 0 ) exp ( σ z ) ,
Î ̂ ( s ) ( ξ , η , z ) = Î ̂ ( ξ + η z , η , z = 0 ) 0 z P ̂ [ ξ + η ( z z ) ] exp ( σ z ) · exp { z z [ σ ω 4 ϕ 2 | ξ + η ( z z ) | 2 + σ ( 1 ω ) ] d z } d z ,
P ( ϕ ) = α 2 π exp ( α 2 ϕ 2 )
I ( ϕ , r , z = 0 ) = F 0 π 1 γ 2 δ ( 2 ) ( ϕ ) exp ( γ 2 r 2 ) ,
I ( 0 ) ( ϕ , r , z ) = F 0 π 1 γ 2 exp ( σ z ) δ ( 2 ) ( ϕ ) exp ( γ 2 r 2 ) ,
I ( s ) ( ϕ , r , z ) = F 0 σ ω exp ( σ z ) ( 2 π ) 2 0 z d z [ 4 A ( z ) C ( z ) B 2 ( z ) ] 1 · exp [ A ( z ) r 2 B ( z ) ϕ · r + C ( z ) ϕ 2 4 A ( z ) C ( z ) B 2 ( z ) ] exp ( ω σ z ) ,
A ( z ) = 1 + σ ω z 4 α 2 B ( z ) = 2 z + σ ω z 2 4 α 2 C ( z ) = 1 4 γ 2 + z 2 + σ ω z 3 / 3 4 α 2 .
ϕ x = θ cos b , ϕ y = θ sin b .
F ( s ) ( θ D , r , σ z ) = 0 θ D I ( s ) ( θ , r , σ z ) θ d θ ,
I ( s ) ( θ , r , σ z ) = F 0 ω exp ( σ z ) 2 π 0 σ z [ 4 A ( x σ ) C ( x σ ) B 2 ( x σ ) ] 1 · I 0 [ θ r B ( x / σ ) 4 A ( x / σ ) C ( x / σ ) B 2 ( x / σ ) ] · exp [ A ( x / σ ) r 2 + C ( x / σ ) θ 2 4 A ( x / σ ) C ( x / σ ) B 2 ( x / σ ) ] exp ( ω x ) d x .
F ( 0 ) ( θ D , r , σ z ) = F 0 π 1 γ 2 exp ( σ z ) exp ( γ 2 r 2 ) .
I ( ϕ , r , z = 0 ) = F 0 β 2 γ 2 π 2 exp ( β 2 ϕ 2 γ 2 r 2 )
F ( s ) ( θ D , γ r , σ z ) = θ D 2 0 1 I ( s ) ( θ θ D , γ r , σ z ) θ d θ ,
I ( s ) ( θ , γ r , σ z ) = F 0 ω σ z exp ( σ z ) / 2 π × 0 1 { 4 A [ ( x / σ ) z σ ] C [ ( x / σ ) z σ ] B 2 [ ( x / σ ) z σ ] } 1 × I 0 { θ γ r B [ ( x / σ ) z σ ] / γ 4 A [ ( x / σ ) z σ ] C [ ( x / σ ) z σ ] B 2 [ ( x / σ ) z σ ] } × exp { ( γ r ) 2 A [ ( x / σ ) z σ ] / γ 2 + C [ ( x / σ ) z σ ] θ 2 4 A [ ( x / σ ) z σ ] C [ ( x / σ ) z σ ] B 2 [ ( x / σ ) z σ ] } × exp ( ω x σ z ) · d x ,
F ( 0 ) ( θ D , γ r , σ z ) = 2 F 0 β 2 γ 2 exp ( σ z ) θ D 2 / π + 0 1 θ d θ I 0 [ 2 ( γ / σ ) θ D θ ( γ r ) σ z ] × exp { [ ( γ r ) 2 ( γ / σ ) 2 ( σ z ) 2 + β 2 ] θ D 2 θ 2 } .
P ( R D ) = 2 π 0 R D F ( θ , r , τ ) rdr .

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