Abstract

An efficient and versatile model that describes the temporal characteristics of scattered radiation when single-scatter conditions prevail is proposed and developed. The model is based on the focal radii property of a prolate spheroid, and for an impulsive source located at one focal point and an observer at the other, it associates all scattering events occurring on a given prolate spheriodal surface with the same transit time. All interactions between the radiation and matter are classified as either elastic scattering or absorption, and expressions are obtained for the intensity and the collected energy as a function of time at the observation point. The model is applied to isotropic, Rayleigh, and Mie-type scattering; the single-scatter phase function for Mie-type scattering is approximated by the Henyey-Greenstein function. For simple source and observer geometries, closed-form expressions are obtained for the intensity in the isotropic and Rayleigh cases. Finally, the model is applied to examples which typify middle ultraviolet radiation propagating in the earth’s atmosphere.

© 1979 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media. Volume 1: Single Scattering and Transport Theory (Academic, New York, 1978) pp. 69–74.
    [Crossref]
  2. P. A. Bello, “A Troposcatter Channel Model,” IEEE Trans. Commun. Tech. 17, 130 (1969).
    [Crossref]
  3. C. S. Gardener and M. A. Plonus, “Optical Pulses in Atmospheric Turbulence,” J. Opt. Soc. Am. 64, 68 (1974).
    [Crossref]
  4. K. M. Case and P. F. Zweifel, Linear Transport Theory, (Addison-Wesley, Reading, Massachusetts, 1967).
  5. R. S. Kennedy and J. H. Shapiro, “Multipath Dispersion in Low Visibility Optical Communication Channels,” Technical Report No. RADC-TR-77-73, Rome Air Development Center, Hanscom Air Force Base, Bedford, Massachusetts, (Jan.1977) (unpublished).
  6. S. Chandrasekhar, Radiative Transfer, (Dover, New York, 1960).
  7. H. C. Hottel and A. F. Sarafim, Radiative Transfer, (McGraw Hill, New York, 1967).
  8. H. C. Van de Huist, Light Scattering by Small Particles, (Wiley, New York1957) pp. 4–6.
  9. G. Arfken, Mathematical Methods for Physicists, (Academic, New York, 1966).
  10. C. Flammer, Spheroidal Wave Functions, (Stanford University, Stanford, California, 1957).
  11. P. M. Morse and H. Feshback, Methods of Theoretical Physics: Part I (McGraw-Hill, New York, 1953).
  12. D. M. Reilly, “Atmospheric Optical Communications in the Middle Ultraviolet,” Masters Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (May1976) (unpublished).
  13. I. Sugai, “Exact Equation for Tropospheric Scatter Common Volume,” Proc. IEEE 53, 1795 (1965).
    [Crossref]
  14. H. G. Booker and J. T. de Bettencourt, “Theory of Radio Trans-
  15. A. J. LaRocca and R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” Report No. 107600-10-T, Environmental Research Institute of Michigan, Ann Arbor, Michigan (June1975) (unpublished).
  16. R. A. McClatchey and J. E. A. Selby, “Atmospheric Attenuation of Laser Radiation from 0.76 − 31.25μ m, Technical Report No: ARCRL-TR-74-0003, Air Force Cambridge Research Laboratories, Bedford, Massachusetts, (Jan.1974) (unpublished).
  17. E. P. Shettle and R. W. Fenn, “Models of the Atmospheric Aerosols and their Optical Properties,” in AGARD Conference Proceedings on Optical Propagation in the Atmosphere, No. 183, Lyngby, Denmark (October1975) (Reprinted NTIS N76-29817);

1974 (1)

1969 (1)

P. A. Bello, “A Troposcatter Channel Model,” IEEE Trans. Commun. Tech. 17, 130 (1969).
[Crossref]

1965 (1)

I. Sugai, “Exact Equation for Tropospheric Scatter Common Volume,” Proc. IEEE 53, 1795 (1965).
[Crossref]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, (Academic, New York, 1966).

Bello, P. A.

P. A. Bello, “A Troposcatter Channel Model,” IEEE Trans. Commun. Tech. 17, 130 (1969).
[Crossref]

Booker, H. G.

H. G. Booker and J. T. de Bettencourt, “Theory of Radio Trans-

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory, (Addison-Wesley, Reading, Massachusetts, 1967).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer, (Dover, New York, 1960).

de Bettencourt, J. T.

H. G. Booker and J. T. de Bettencourt, “Theory of Radio Trans-

Fenn, R. W.

E. P. Shettle and R. W. Fenn, “Models of the Atmospheric Aerosols and their Optical Properties,” in AGARD Conference Proceedings on Optical Propagation in the Atmosphere, No. 183, Lyngby, Denmark (October1975) (Reprinted NTIS N76-29817);

Feshback, H.

P. M. Morse and H. Feshback, Methods of Theoretical Physics: Part I (McGraw-Hill, New York, 1953).

Flammer, C.

C. Flammer, Spheroidal Wave Functions, (Stanford University, Stanford, California, 1957).

Gardener, C. S.

Hottel, H. C.

H. C. Hottel and A. F. Sarafim, Radiative Transfer, (McGraw Hill, New York, 1967).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media. Volume 1: Single Scattering and Transport Theory (Academic, New York, 1978) pp. 69–74.
[Crossref]

Kennedy, R. S.

R. S. Kennedy and J. H. Shapiro, “Multipath Dispersion in Low Visibility Optical Communication Channels,” Technical Report No. RADC-TR-77-73, Rome Air Development Center, Hanscom Air Force Base, Bedford, Massachusetts, (Jan.1977) (unpublished).

LaRocca, A. J.

A. J. LaRocca and R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” Report No. 107600-10-T, Environmental Research Institute of Michigan, Ann Arbor, Michigan (June1975) (unpublished).

McClatchey, R. A.

R. A. McClatchey and J. E. A. Selby, “Atmospheric Attenuation of Laser Radiation from 0.76 − 31.25μ m, Technical Report No: ARCRL-TR-74-0003, Air Force Cambridge Research Laboratories, Bedford, Massachusetts, (Jan.1974) (unpublished).

Morse, P. M.

P. M. Morse and H. Feshback, Methods of Theoretical Physics: Part I (McGraw-Hill, New York, 1953).

Plonus, M. A.

Reilly, D. M.

D. M. Reilly, “Atmospheric Optical Communications in the Middle Ultraviolet,” Masters Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (May1976) (unpublished).

Sarafim, A. F.

H. C. Hottel and A. F. Sarafim, Radiative Transfer, (McGraw Hill, New York, 1967).

Selby, J. E. A.

R. A. McClatchey and J. E. A. Selby, “Atmospheric Attenuation of Laser Radiation from 0.76 − 31.25μ m, Technical Report No: ARCRL-TR-74-0003, Air Force Cambridge Research Laboratories, Bedford, Massachusetts, (Jan.1974) (unpublished).

Shapiro, J. H.

R. S. Kennedy and J. H. Shapiro, “Multipath Dispersion in Low Visibility Optical Communication Channels,” Technical Report No. RADC-TR-77-73, Rome Air Development Center, Hanscom Air Force Base, Bedford, Massachusetts, (Jan.1977) (unpublished).

Shettle, E. P.

E. P. Shettle and R. W. Fenn, “Models of the Atmospheric Aerosols and their Optical Properties,” in AGARD Conference Proceedings on Optical Propagation in the Atmosphere, No. 183, Lyngby, Denmark (October1975) (Reprinted NTIS N76-29817);

Sugai, I.

I. Sugai, “Exact Equation for Tropospheric Scatter Common Volume,” Proc. IEEE 53, 1795 (1965).
[Crossref]

Turner, R. E.

A. J. LaRocca and R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” Report No. 107600-10-T, Environmental Research Institute of Michigan, Ann Arbor, Michigan (June1975) (unpublished).

Van de Huist, H. C.

H. C. Van de Huist, Light Scattering by Small Particles, (Wiley, New York1957) pp. 4–6.

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory, (Addison-Wesley, Reading, Massachusetts, 1967).

IEEE Trans. Commun. Tech. (1)

P. A. Bello, “A Troposcatter Channel Model,” IEEE Trans. Commun. Tech. 17, 130 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

I. Sugai, “Exact Equation for Tropospheric Scatter Common Volume,” Proc. IEEE 53, 1795 (1965).
[Crossref]

Other (14)

H. G. Booker and J. T. de Bettencourt, “Theory of Radio Trans-

A. J. LaRocca and R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” Report No. 107600-10-T, Environmental Research Institute of Michigan, Ann Arbor, Michigan (June1975) (unpublished).

R. A. McClatchey and J. E. A. Selby, “Atmospheric Attenuation of Laser Radiation from 0.76 − 31.25μ m, Technical Report No: ARCRL-TR-74-0003, Air Force Cambridge Research Laboratories, Bedford, Massachusetts, (Jan.1974) (unpublished).

E. P. Shettle and R. W. Fenn, “Models of the Atmospheric Aerosols and their Optical Properties,” in AGARD Conference Proceedings on Optical Propagation in the Atmosphere, No. 183, Lyngby, Denmark (October1975) (Reprinted NTIS N76-29817);

A. Ishimaru, Wave Propagation and Scattering in Random Media. Volume 1: Single Scattering and Transport Theory (Academic, New York, 1978) pp. 69–74.
[Crossref]

K. M. Case and P. F. Zweifel, Linear Transport Theory, (Addison-Wesley, Reading, Massachusetts, 1967).

R. S. Kennedy and J. H. Shapiro, “Multipath Dispersion in Low Visibility Optical Communication Channels,” Technical Report No. RADC-TR-77-73, Rome Air Development Center, Hanscom Air Force Base, Bedford, Massachusetts, (Jan.1977) (unpublished).

S. Chandrasekhar, Radiative Transfer, (Dover, New York, 1960).

H. C. Hottel and A. F. Sarafim, Radiative Transfer, (McGraw Hill, New York, 1967).

H. C. Van de Huist, Light Scattering by Small Particles, (Wiley, New York1957) pp. 4–6.

G. Arfken, Mathematical Methods for Physicists, (Academic, New York, 1966).

C. Flammer, Spheroidal Wave Functions, (Stanford University, Stanford, California, 1957).

P. M. Morse and H. Feshback, Methods of Theoretical Physics: Part I (McGraw-Hill, New York, 1953).

D. M. Reilly, “Atmospheric Optical Communications in the Middle Ultraviolet,” Masters Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (May1976) (unpublished).

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Figures (12)

FIG. 1
FIG. 1

Prolate spheroidal coordinate system and the scattering geometry.

FIG. 2
FIG. 2

Scattering angle θs as a function of η with ξ as a parameter.

FIG. 3
FIG. 3

General off-axis scattering geometry.

FIG. 4
FIG. 4

Geometry for determining limits of η.

FIG. 5
FIG. 5

Single-scatter (scalar) phase functions for isotropic and Rayleigh scattering.

FIG. 6
FIG. 6

Isotropic single scattering: Intensity and collected energy/cm2 as a function of time for an impulsive isotropic source and isotropic observer geometry. E1 = 1 J, Ω1 = 4π Sr, Ω2 = 4π Sr, I = 1 km, v = 3 × 108 m/s, αsi = 1 × 10−6 cm−1.

FIG. 7
FIG. 7

Comparison of isotropic and Rayleigh single scattering: Intensity and collected energy/cm2 as a function of time for an impulsive isotropic source and isotropic observer geometry. E1 = 1 J, Ω1 = 4π Sr, Ω2 = 4π Sr, I = 1 km, v = 3 × 108 mAs, αsr = αsi = 1 × 10−6 cm−1, αarαai = 0.

FIG. 8
FIG. 8

Henyey-Greenstein phase function for selected values of the parameter g.

FIG. 9
FIG. 9

Mie-type scattering: Single-scatter intensity and collected energy/cm2 as a function of time for an impulsive isotropic source and isotropic observer geometry for three selected g values. E1 = 1 J, Ω1 = 4π Sr, Ω2 = 4π Sr, I = 1 km, v = 3 × 108 m/s, αsm = 1 × 10−6 cm−1, αam = 1 × 10−6 cm−1.

FIG. 10
FIG. 10

Temporal dependencies of 2500-Å single-scatter radiation in a 23-km visual range atmosphere with an impulsive isotropic source and isotropic observer geometry. E1 = 1 J, Ω1 = 4π Sr, Ω2 =4π Sr, I = 0.25 km, v = 3 × 108 m/s, αsm = 3.0 × 10−6 cm−1, αam = 0.5 × 10−6 cm−1, g = 0.9, αsr = 3.4 × 10−6 cm−1, αar = 7.9 × 10−6 cm−1.

FIG. 11
FIG. 11

Temporal dependencies of 2500-Å single-scatter radiation in a 5-km visual range atmosphere with an impulsive isotropic source and isotropic observer geometry. E1 = 1 J, Ω1 = 4π Sr, Ω2 = 4π Sr, I = 0.25 km, v = 3 × 108 m/s, αsm = 15 × 10−6 cm−1, αam = 2.4 × 10−6 cm−1, g = 0.9, αsr = 3.4 × 10−6 cm−1, αar = 7.9 × 10−6 cm−1.

FIG. 12
FIG. 12

Temporal dependencies of 2500-Å single-scatter radiation in the 5-km visual range atmosphere with an impulsive source. The source and observer cone angles form a closed scattering volume symmetric about the interfocal axis with θ1 = θ2= 45; i.e. Ω1 = Ω2 = 0.586π Sr. E11 = 1/4π J/Sr, I = 0.25 km, v = 3 × 108 m/s.

Tables (2)

Tables Icon

TABLE I Unscattered and scattered components of the collected energy/ cm2 for the 23-km visual range atmosphere for three different source and observer geometries. E11 = 1/4π J/Sr and I = 0.25 km.

Tables Icon

TABLE II Unscattered and scattered components of the collected energy/cm2 for the 5-km visual-range atmosphere for three different source and observer geometries. E11 = 1/4π J/Sr and I = 0.25 km.

Equations (32)

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1 4 π P ( cos θ s ) d Ω = 1 ,
α s = r 1 r 2 N ( r ) σ s ( r ) d r .
r 1 = ( l / 2 ) ( ξ + η ) ,
r 2 = ( l / 2 ) ( ξ η ) ,
cos θ 1 = ( 1 + ξ η ) / ( ξ + η ) ,
cos θ 2 = ( 1 ξ η ) / ( ξ η ) ,
δ V = ( l 3 / 8 ) ( ξ 2 η 2 ) δ ξ δ η δ ϕ .
θ s = θ 1 + θ 2 = cos 1 [ ( 2 ξ 2 η 2 ) / ( ξ 2 η 2 ) ] .
H p = E 1 e α r 1 / Ω 1 r 1 2 ,
δ R p = E 1 α s 4 π Ω 1 e α r 1 r 1 2 P ( cos θ s ) δ V .
δ H 2 = E 1 α s 4 π Ω 1 e α ( r 1 + r 2 ) r 1 2 r 2 2 P ( cos θ s ) δ V .
δ H 2 = E 1 α s 2 π Ω 1 l e α l ξ ( ξ 2 η 2 ) P ( cos θ s ) δ ξ δ η δ ϕ .
ξ = ( υ / l ) , t > t 0 = l / υ .
d I 2 = { 0 , t < l / υ E 1 V α s 2 π r 1 l 2 e α l ξ ( ξ 2 η 2 ) P ( cos θ s ) d η d ϕ , ξ > 1 .
I 2 ( ξ ) = { 0 , 1 < ξ < ξ min E 1 υ α s e α l ξ 2 π Ω 1 l 2 η 1 ( ξ ) η 2 ( ξ ) P ( cos θ s ) [ ϕ 2 ( ξ , η ) ϕ 1 ( ξ , η ) ] ( ξ 2 η 2 ) d η , ξ min ξ ξ max , 0 , ξ > ξ max
H 2 ( ξ ) = ξ min ξ I 2 ( ζ ) d ζ .
η 1 ( ξ , θ 1 max ) = ξ cos θ 1 max 1 ξ cos θ 1 max ,
η 2 ( ξ , θ 2 max ) = 1 ξ cos θ 2 max ξ cos θ 2 max ,
ξ max = b + ( b 2 1 ) 1 / 2 ,
b = ( 1 + cos θ 1 max cos θ 2 max ) / ( cos θ 1 max + cos θ 2 max ) .
[ I 2 ( ξ ) ] i = E 1 υ α s i 2 π Ω 1 l 2 e α i l ξ × η 1 ( ξ ) η 2 ( ξ ) [ ϕ 2 ( ξ , η ) ϕ 1 ( ξ , η ) ] ( ξ 2 η 2 ) d η ,
[ I 2 ( ξ ) ] i = E 1 υ α s i 2 Ω 1 l 2 e α i l ξ ξ ln [ ( ξ + η 2 ( ξ ) ξ η 2 ( ξ ) ) ( ξ η 1 ( ξ ) ξ + η 1 ( ξ ) ) ] ,
[ I 2 ( ξ ) ] i = E 1 υ α s i 4 π l 2 ξ e α i l ξ ln ( ξ + 1 ξ 1 )
[ I 2 ( t ) ] i = E 1 α s i 4 π l e α i υ t t ln ( t + t 0 t t 0 ) , t > t 0
[ H 2 ( t ) ] i = t 0 t 0 + t [ I 2 i ( t ) ] i d t ,
P ( cos θ s ) = ( 3 / 4 ) ( 1 + cos 2 θ s ) ,
[ I 2 ( ξ ) ] r = 3 E 1 α s r υ 4 Ω 1 l 2 e α r l ξ ξ 2 × [ ( 3 ξ 4 2 ξ 2 + 3 ) 4 ξ 3 ln ( ( ξ + η 2 ) ( ξ η 1 ) ( ξ η 2 ) ( ξ + η 1 ) ) ( ξ 4 + 2 ξ 2 3 ) 2 ξ 2 ( η 2 ( ξ 2 η 1 2 ) η 1 ( ξ 2 η 2 2 ) ) + ( ξ 4 2 ξ 2 + 1 ) 1 ( η 2 ( ξ 2 η 2 2 ) 2 η 1 ( ξ 2 η 1 2 ) 2 ) ] ,
[ I 2 ( ξ ) ] r = = 3 E 1 α s r υ 16 π l 2 e α r l ξ ξ 2 × [ ( 3 ξ 4 2 ξ 2 + 3 ) 2 ξ 3 ln ( ξ + 1 ξ 1 ) ( ξ 2 + 3 ) ξ 2 + 2 ] .
p ( cos θ s ) = ( 1 g 2 ) / ( 1 + g 2 2 g cos θ s ) 3 / 2 ,
[ I 2 ( ξ ) ] m = E 1 υ α s m ( 1 g 2 ) 2 π Ω 1 l 2 e α m l ξ × η 1 ( ξ ) η 2 ( ξ ) [ ϕ 2 ( ξ , η ) ϕ 1 ( ξ , η ) ] ( ξ 2 η 2 ) ( 1 + g 2 2 g cos θ s ) 3 / 2 d η ,
[ I 2 ( ξ ) ] m = E 1 α s m υ Ω 1 l 2 e α m l ξ ( 1 g 2 ) ( g 2 + 2 g + 1 ) 3 / 2 η 1 ( ξ ) η 2 ( ξ ) × ( ξ 2 η 2 ) 1 / 2 { [ ( g 2 + 2 g + 1 ) ξ 2 4 g ] / ( g 2 2 g + 1 ) η 2 } 3 / 2 d η .
[ I 2 ( ξ ) ] comp = [ I 2 ( ξ ) ] r + [ I 2 ( ξ ) ] m .