Abstract

A recent laboratory-controlled optical experiment demonstrates that a sharp peak of small but finite angular width is exhibited in backscattering from a random distribution of discrete scatterers. In this paper the phenomenon is explained by using a second-order multiple-scattering theory of discrete particles. The theory gives an angular width of the order of the attenuation rate divided by the wave number and is in agreement with experimental observations. The relations of the present results to past theories on backscattering enhancements are also discussed.

© 1984 Optical Society of America

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References

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  1. A. W. England, “Thermal microwave emission from a half-space containing scatterers,” Radio Sci. 9, 447–454 (1974).
    [CrossRef]
  2. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1981), Vols. 1 and 2.
  3. A. K. Fung, H. J. Eom, “A theory of wave scattering from and inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
    [CrossRef]
  4. R. T. Shin, J. A. Kong, “Radiative transfer theory for active remote sensing of homogeneous layer containing spherical scatterers,” J. Appl. Phys. 52, 4221–4230 (1981).
    [CrossRef]
  5. Y. Q. Jin, J. A. Kong, “Passive and active sensing of atmospheric precipitation,” Appl. Opt. 22, 2535–2545 (1983).
    [CrossRef] [PubMed]
  6. M. A. Karam, A. K. Fung, “Scattering from randomly oriented circular discs with application to vegetation,” Radio Sci. 18, 557–565 (1983).
    [CrossRef]
  7. L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from a half space of densely distributed dielectric scatterers,” Radio Sci. 18, 1260–1272 (1983).
    [CrossRef]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.
  9. D. A. deWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single backscatter approximation,” IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
    [CrossRef]
  10. A. Ishimaru, K. J. Pinter, “Backscattered pulse shape due to small-angle multiple scattering in random media,” Radio Sci. 15, 87–93 (1980).
    [CrossRef]
  11. Y. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
    [CrossRef]
  12. K. C. Yeh, “Mutual coherence functions and intensities of backscattered signals in a turbulent medium,” Radio Sci. 18, 159–165 (1983).
    [CrossRef]
  13. Y. Kuga, “Experimental and theoretical studies of laser light propagation and scattering in a dense distribution of spherical particles,” Ph.D. Dissertation (University of Washington, Seattle, Wash., 1983).
  14. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  15. Y. N. Barabanenkov, “Wave corrections to the transfer equation for backscattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 88–94 (1973).
  16. M. Zuniga, J. A. Kong, L. Tsang, “Depolarization effects in the active remote sensing of random media,” J. Appl. Phys. 51, 2315–2325 (1980).
    [CrossRef]
  17. If particle positions are independent, the pair-distribution function is equal to 1. The correlation distance l of particle positions is the two-particle separation distance beyond which the pair-distribution function is practically equal to 1. For particle concentrations less than 30%, the maximum correlation distance is three to four times the diameter of the particle. See Ref. 7 and also Y. Waseda, The Structure of Noncrystalline Materials, Liquids and Amorphous Solids (McGraw Hill, New York, 1980).
  18. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  19. A. Ishimaru, Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  20. V. Frisch, “Wave propagation in random medium,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968).

1984 (1)

1983 (4)

K. C. Yeh, “Mutual coherence functions and intensities of backscattered signals in a turbulent medium,” Radio Sci. 18, 159–165 (1983).
[CrossRef]

Y. Q. Jin, J. A. Kong, “Passive and active sensing of atmospheric precipitation,” Appl. Opt. 22, 2535–2545 (1983).
[CrossRef] [PubMed]

M. A. Karam, A. K. Fung, “Scattering from randomly oriented circular discs with application to vegetation,” Radio Sci. 18, 557–565 (1983).
[CrossRef]

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from a half space of densely distributed dielectric scatterers,” Radio Sci. 18, 1260–1272 (1983).
[CrossRef]

1982 (2)

Y. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

1981 (2)

A. K. Fung, H. J. Eom, “A theory of wave scattering from and inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[CrossRef]

R. T. Shin, J. A. Kong, “Radiative transfer theory for active remote sensing of homogeneous layer containing spherical scatterers,” J. Appl. Phys. 52, 4221–4230 (1981).
[CrossRef]

1980 (2)

A. Ishimaru, K. J. Pinter, “Backscattered pulse shape due to small-angle multiple scattering in random media,” Radio Sci. 15, 87–93 (1980).
[CrossRef]

M. Zuniga, J. A. Kong, L. Tsang, “Depolarization effects in the active remote sensing of random media,” J. Appl. Phys. 51, 2315–2325 (1980).
[CrossRef]

1974 (1)

A. W. England, “Thermal microwave emission from a half-space containing scatterers,” Radio Sci. 9, 447–454 (1974).
[CrossRef]

1973 (1)

Y. N. Barabanenkov, “Wave corrections to the transfer equation for backscattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 88–94 (1973).

1971 (1)

D. A. deWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single backscatter approximation,” IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
[CrossRef]

Barabanenkov, Y. N.

Y. N. Barabanenkov, “Wave corrections to the transfer equation for backscattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 88–94 (1973).

deWolf, D. A.

D. A. deWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single backscatter approximation,” IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
[CrossRef]

England, A. W.

A. W. England, “Thermal microwave emission from a half-space containing scatterers,” Radio Sci. 9, 447–454 (1974).
[CrossRef]

Eom, H. J.

A. K. Fung, H. J. Eom, “A theory of wave scattering from and inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[CrossRef]

Frisch, V.

V. Frisch, “Wave propagation in random medium,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968).

Fung, A. K.

M. A. Karam, A. K. Fung, “Scattering from randomly oriented circular discs with application to vegetation,” Radio Sci. 18, 557–565 (1983).
[CrossRef]

A. K. Fung, H. J. Eom, “A theory of wave scattering from and inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[CrossRef]

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1981), Vols. 1 and 2.

Ishimaru, A.

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

A. Ishimaru, K. J. Pinter, “Backscattered pulse shape due to small-angle multiple scattering in random media,” Radio Sci. 15, 87–93 (1980).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Jin, Y. Q.

Karam, M. A.

M. A. Karam, A. K. Fung, “Scattering from randomly oriented circular discs with application to vegetation,” Radio Sci. 18, 557–565 (1983).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from a half space of densely distributed dielectric scatterers,” Radio Sci. 18, 1260–1272 (1983).
[CrossRef]

Y. Q. Jin, J. A. Kong, “Passive and active sensing of atmospheric precipitation,” Appl. Opt. 22, 2535–2545 (1983).
[CrossRef] [PubMed]

R. T. Shin, J. A. Kong, “Radiative transfer theory for active remote sensing of homogeneous layer containing spherical scatterers,” J. Appl. Phys. 52, 4221–4230 (1981).
[CrossRef]

M. Zuniga, J. A. Kong, L. Tsang, “Depolarization effects in the active remote sensing of random media,” J. Appl. Phys. 51, 2315–2325 (1980).
[CrossRef]

Kravtsov, Y. A.

Y. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Kuga, Y.

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

Y. Kuga, “Experimental and theoretical studies of laser light propagation and scattering in a dense distribution of spherical particles,” Ph.D. Dissertation (University of Washington, Seattle, Wash., 1983).

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1981), Vols. 1 and 2.

Pinter, K. J.

A. Ishimaru, K. J. Pinter, “Backscattered pulse shape due to small-angle multiple scattering in random media,” Radio Sci. 15, 87–93 (1980).
[CrossRef]

Saichev, A. I.

Y. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Shin, R. T.

R. T. Shin, J. A. Kong, “Radiative transfer theory for active remote sensing of homogeneous layer containing spherical scatterers,” J. Appl. Phys. 52, 4221–4230 (1981).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from a half space of densely distributed dielectric scatterers,” Radio Sci. 18, 1260–1272 (1983).
[CrossRef]

M. Zuniga, J. A. Kong, L. Tsang, “Depolarization effects in the active remote sensing of random media,” J. Appl. Phys. 51, 2315–2325 (1980).
[CrossRef]

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1981), Vols. 1 and 2.

Waseda, Y.

If particle positions are independent, the pair-distribution function is equal to 1. The correlation distance l of particle positions is the two-particle separation distance beyond which the pair-distribution function is practically equal to 1. For particle concentrations less than 30%, the maximum correlation distance is three to four times the diameter of the particle. See Ref. 7 and also Y. Waseda, The Structure of Noncrystalline Materials, Liquids and Amorphous Solids (McGraw Hill, New York, 1980).

Yeh, K. C.

K. C. Yeh, “Mutual coherence functions and intensities of backscattered signals in a turbulent medium,” Radio Sci. 18, 159–165 (1983).
[CrossRef]

Zuniga, M.

M. Zuniga, J. A. Kong, L. Tsang, “Depolarization effects in the active remote sensing of random media,” J. Appl. Phys. 51, 2315–2325 (1980).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (2)

A. K. Fung, H. J. Eom, “A theory of wave scattering from and inhomogeneous layer with an irregular interface,” IEEE Trans. Antennas Propag. AP-29, 899–910 (1981).
[CrossRef]

D. A. deWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single backscatter approximation,” IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

Y. N. Barabanenkov, “Wave corrections to the transfer equation for backscattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 88–94 (1973).

J. Appl. Phys. (2)

M. Zuniga, J. A. Kong, L. Tsang, “Depolarization effects in the active remote sensing of random media,” J. Appl. Phys. 51, 2315–2325 (1980).
[CrossRef]

R. T. Shin, J. A. Kong, “Radiative transfer theory for active remote sensing of homogeneous layer containing spherical scatterers,” J. Appl. Phys. 52, 4221–4230 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Radio Sci. (5)

A. W. England, “Thermal microwave emission from a half-space containing scatterers,” Radio Sci. 9, 447–454 (1974).
[CrossRef]

M. A. Karam, A. K. Fung, “Scattering from randomly oriented circular discs with application to vegetation,” Radio Sci. 18, 557–565 (1983).
[CrossRef]

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from a half space of densely distributed dielectric scatterers,” Radio Sci. 18, 1260–1272 (1983).
[CrossRef]

A. Ishimaru, K. J. Pinter, “Backscattered pulse shape due to small-angle multiple scattering in random media,” Radio Sci. 15, 87–93 (1980).
[CrossRef]

K. C. Yeh, “Mutual coherence functions and intensities of backscattered signals in a turbulent medium,” Radio Sci. 18, 159–165 (1983).
[CrossRef]

Sov. Phys. Usp. (1)

Y. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
[CrossRef]

Other (6)

Y. Kuga, “Experimental and theoretical studies of laser light propagation and scattering in a dense distribution of spherical particles,” Ph.D. Dissertation (University of Washington, Seattle, Wash., 1983).

If particle positions are independent, the pair-distribution function is equal to 1. The correlation distance l of particle positions is the two-particle separation distance beyond which the pair-distribution function is practically equal to 1. For particle concentrations less than 30%, the maximum correlation distance is three to four times the diameter of the particle. See Ref. 7 and also Y. Waseda, The Structure of Noncrystalline Materials, Liquids and Amorphous Solids (McGraw Hill, New York, 1980).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1981), Vols. 1 and 2.

V. Frisch, “Wave propagation in random medium,” in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, New York, 1968).

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Figures (3)

Fig. 1
Fig. 1

Scattering of waves by a half-space of point scatterers.

Fig. 2
Fig. 2

Path difference between field and its conjugate in double scattering.

Fig. 3
Fig. 3

Bistatic scattering coefficients as a function of θs for different albedos with θi = 5°, θs = 180°, and 2K″/K′ = 3.18 × 10−3.

Equations (27)

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K = k + 2 π n 0 f k = K + i K ,
ψ m ( r ¯ ) = exp ( i K ¯ i r ¯ ) ,
K ¯ i = K i x x ^ - K i z z ^ ,
K i x = k sin θ i ,
K i z = ( K 2 - k 2 sin 2 θ i ) 1 / 2 .
G 01 m ( r ¯ , r ¯ ) = - exp ( i k r ) 4 π r exp ( - i K ¯ s · r ¯ ) ,
K ¯ s = K s x x ^ + K s y y ^ + K s z z ^ ,
K s x = k sin θ s cos ϕ s ,
K s y = k sin θ s sin ϕ s ,
K s z = ( K 2 - k 2 sin 2 θ s ) 1 / 2 ,
G 11 m ( r ¯ , r ¯ ) = - exp ( i K r ¯ - r ¯ ) 4 π r ¯ - r ¯ .
F = n 0 d r ¯ j d r ¯ 1 d r ¯ 2 d r ¯ 3 d r ¯ 4 G 01 m ( r ¯ , r ¯ 1 ) T j ( r ¯ 1 , r ¯ 2 ) ψ m ( r ¯ 2 ) × G 01 m * ( r ¯ , r ¯ 3 ) T j * ( r ¯ 3 , r ¯ 4 ) ψ m * ( r ¯ 4 ) ,
T j ( r ¯ 1 , r ¯ 2 ) = - 4 π f δ ( r ¯ 1 - r ¯ j ) δ ( r ¯ 2 - r ¯ j ) .
F = f 2 n 0 A r 2 1 2 ( K s z + K i z ) ,
K i z = K / μ i ,
K s z = K / μ s ,
F = ω ˜ 4 π A r 2 1 ( 1 μ s + 1 μ i ) ,
L = n 0 2 d r ¯ j d r ¯ l ( 4 π ) 4 f 4 G 01 m ( r ¯ , r ¯ j ) 2 × G 11 m ( r ¯ j , r ¯ l ) 2 ψ m ( r ¯ l ) 2 .
d z A d r ¯ d = 0 2 π d ϕ d 0 d r d r d 2 ( 0 π / 2 d θ d sin θ d × - - r d cos θ d / 2 d z A + π / 2 π d θ d sin θ d - r d cos θ d / 2 d z A ) .
L = n 0 2 f 4 A r 2 2 π 2 K s z + 2 K i z × 0 1 d μ d μ d 4 K μ d + 2 K i z + 2 K s z ( 2 K μ d + 2 K i z ) ( 2 K μ d + 2 K s z ) .
L = ( ω ˜ 4 π ) 2 2 π A r 2 1 ( 1 μ s + 1 μ i ) [ μ i l n ( 1 + 1 μ i ) + μ s l n ( 1 + 1 μ s ) ] .
C = n 0 2 d r ¯ j d r ¯ l ( 4 π ) 4 f 4 G 01 m ( r ¯ , r ¯ j ) G 11 m ( r ¯ j , r ¯ l ) ψ m ( r ¯ l ) × G 01 m * ( r ¯ , r ¯ l ) G 11 m * ( r ¯ l , r ¯ j ) ψ m * ( r ¯ j ) .
C = n 0 2 f 4 A r 2 d z A d r ¯ d exp ( - 2 K r d ) r d 2 × exp ( - i K ¯ d · r ¯ d ) exp [ 2 ( K i z + K s z ) z A ] ,
K ¯ d = K ¯ s + K ¯ i .
C = ( ω ˜ 4 π ) 2 4 π A r 2 1 ( 1 μ i + 1 μ s ) × 0 π / 2 d θ d sin θ d Re ( 1 ( β 2 + γ 2 ) 1 / 2 ) ,
γ = 1 + i ( K s z - K i z ) cos θ d 2 K + cos θ d 2 ( 1 μ i + 1 μ s ) ,
β = sin θ d 2 K [ K d 2 - ( K s z - K i z ) 2 ] 1 / 2 ,

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