Abstract

A treatment of the phase function in the radiative transfer equation is discussed for describing optical wave propagation in discrete random media with large particles. Unlike the conventional small-angle approximation, the phase function is normalized so that half of the scattered power is removed from a small angle in the forward direction for large particles with the refractive index not close to unity. With this normalization, an improved small-angle solution of the radiative transfer equation is given for the phase function adopted here. The validity of the proposed theory is confirmed by comparisons with both numerical solutions and experimental data on the attenuation of millimeter and optical waves in rain.

© 1993 Optical Society of America

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References

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  1. L. S. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Radiophys. Quantum Electron. 9, 40–47 (1966).
  2. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  4. W. G. Tam, A. Zardecki, “Laser beam propagation in particulate media,” J. Opt. Soc. Am 69, 68–70 (1979).
    [CrossRef]
  5. A. Zardecki, W. G. Tam, “Pulse propagation in particulate media,” Appl. Opt. 19, 3782–3788 (1980).
    [CrossRef] [PubMed]
  6. S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
    [CrossRef]
  7. M. A. Box, A. Deepak, “Limiting cases of the small-angel scattering approximation solutions for the propagation of laser beams in anisotropic scattering media,” J. Opt. Soc. Am. 71, 1534–1539 (1981).
    [CrossRef]
  8. A. Deepak, U. O. Farrukh, A. Zardecki, “Significance of higher-order multiple scattering for laser beam propagation through hazes, fog, and clouds,” Appl. Opt. 21, 439–447 (1982).
    [CrossRef] [PubMed]
  9. Y. Kuga, A. Ishimaru, H. W. Chang, L. Tsang, “Comparisons between the small-angle approximation and the numerical solution for radiative transfer theory,” Appl. Opt. 25, 3803–3805 (1986).
    [CrossRef] [PubMed]
  10. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  11. S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
    [CrossRef]
  12. S. Ito, T. Oguchi, “Approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
    [CrossRef]
  13. S. Ito, T. Oguchi, “Approximate solutions of the vector radiative transfer equation for linearly polarized light in discrete random media,” J. Opt. Soc. Am. A 6, 1852–1858 (1989).
    [CrossRef]
  14. J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

1989 (1)

1987 (1)

S. Ito, T. Oguchi, “Approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[CrossRef]

1986 (1)

1982 (1)

1981 (1)

1980 (2)

A. Zardecki, W. G. Tam, “Pulse propagation in particulate media,” Appl. Opt. 19, 3782–3788 (1980).
[CrossRef] [PubMed]

S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
[CrossRef]

1979 (1)

W. G. Tam, A. Zardecki, “Laser beam propagation in particulate media,” J. Opt. Soc. Am 69, 68–70 (1979).
[CrossRef]

1976 (1)

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

1966 (1)

L. S. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Radiophys. Quantum Electron. 9, 40–47 (1966).

Awaka, J.

J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

Box, M. A.

Chang, H. W.

Deepak, A.

Dolin, L. S.

L. S. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Radiophys. Quantum Electron. 9, 40–47 (1966).

Echizen’ya, Y.

J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

Farrukh, U. O.

Hong, S. T.

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

Ihara, T.

J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

Ishimaru, A.

Y. Kuga, A. Ishimaru, H. W. Chang, L. Tsang, “Comparisons between the small-angle approximation and the numerical solution for radiative transfer theory,” Appl. Opt. 25, 3803–3805 (1986).
[CrossRef] [PubMed]

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

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

Ito, S.

S. Ito, T. Oguchi, “Approximate solutions of the vector radiative transfer equation for linearly polarized light in discrete random media,” J. Opt. Soc. Am. A 6, 1852–1858 (1989).
[CrossRef]

S. Ito, T. Oguchi, “Approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[CrossRef]

S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
[CrossRef]

Kawai, K.

J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

Kitamura, K.

J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

Kuga, Y.

Oguchi, T.

S. Ito, T. Oguchi, “Approximate solutions of the vector radiative transfer equation for linearly polarized light in discrete random media,” J. Opt. Soc. Am. A 6, 1852–1858 (1989).
[CrossRef]

S. Ito, T. Oguchi, “Approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[CrossRef]

Okamoto, K.

J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

Tam, W. G.

A. Zardecki, W. G. Tam, “Pulse propagation in particulate media,” Appl. Opt. 19, 3782–3788 (1980).
[CrossRef] [PubMed]

W. G. Tam, A. Zardecki, “Laser beam propagation in particulate media,” J. Opt. Soc. Am 69, 68–70 (1979).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

Tsang, L.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Zardecki, A.

Appl. Opt. (3)

J. Opt. Soc. Am (1)

W. G. Tam, A. Zardecki, “Laser beam propagation in particulate media,” J. Opt. Soc. Am 69, 68–70 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Radio Sci. (3)

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

S. Ito, T. Oguchi, “Approximate method for solving the vector radiative transfer equation in discrete random media,” Radio Sci. 22, 873–879 (1987).
[CrossRef]

S. Ito, “On the theory of pulse wave propagation in media of discrete random scatterers,” Radio Sci. 15, 893–901 (1980).
[CrossRef]

Radiophys. Quantum Electron. (1)

L. S. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Radiophys. Quantum Electron. 9, 40–47 (1966).

Other (4)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

J. Awaka, K. Kawai, T. Ihara, K. Kitamura, Y. Echizen’ya, K. Okamoto, “Millimeter and optical wave propagation under snow and other conditions,” in Proceedings of the 1989 International Symposium on Antennas and Propagation Institute of Electronics, Information, and Communication Engineers, Tokyo, 1989), pp. 1041–1044.

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Figures (4)

Fig. 1
Fig. 1

Values of f and fp as a function of the size parameter ka for various values of the refractive index n. The solid curves indicate the ratio f of the power within the small angle to the total power. The dashed curves indicate the ratio fp of the power between 0° and 90° to the total power.

Fig. 2
Fig. 2

Incoherent specific intensities of LHC and RHC waves in the forward direction versus optical depths. Solid curves are the preset approximate solutions, and the dashed curves are the numerical solutions of the original radiative transfer equation.

Fig. 3
Fig. 3

Incoherent specific intensities in the forward direction versus optical depths. Curve A indicates the specific intensity calculated by the current method, and curve B indicates that calculated by the conventional method based on the small-angle approximation.

Fig. 4
Fig. 4

Comparison of theoretical values with experimental results14 on the attenuation (ATT) of millimeter and optical waves in rain. The theoretical values represented by the solid line are calculated for the Marshall–Palmer rain size distribution.

Equations (15)

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( Ω · r + σ t ) I ( r , Ω ) = d Ω Ψ ( Ω , Ω ) I ( r , Ω ) ,
· F ( r ) = σ t ( 1 W 0 ) I ( r ) = σ a I ( r ) ,
F ( r ) = d Ω Ω I ( r , Ω ) ,
I ( r ) = d Ω I ( r , Ω ) ,
· F ( r ) = 0 .
Ω · r z + Ω · ρ , r = ( ρ , z ) ,
4 π Ψ ( Ω , Ω ) d Ω Ψ ( Ω Ω ) d Ω .
( z + Ω · ρ + σ t ) I ( r , Ω ) = d Ω Ψ ( Ω Ω ) I ( r , Ω ) ,
1 σ s Ψ ( Ω ) d Ω = 1 .
f = θ 0 Ψ ( Ω Ω , Ω i ) d Ω / 4 π Ψ ( Ω , Ω i ) d Ω ,
1 σ s Ψ ( Ω ) d Ω = f ,
· F ( r ) = σ t ( 1 f W 0 ) I ( r ) = [ σ a + ( 1 f ) σ s ] I ( r ) .
Ψ ( Ω ) = α 2 f σ s π exp ( α 2 Ω 2 ) , α 2 = 2 . 5 ( k a / π ) 2 ,
I ( z , Ω ) = exp ( σ t z ) [ δ ( Ω ) + α 2 π m = 1 ( f σ s z ) m m ! m exp ( α 2 Ω 2 m ) ] ,
I ( z ) = I ( z , Ω ) d Ω = exp [ ( 1 f W 0 ) σ t z ] ,

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