Abstract

The backscattered intensity from a dense distribution of latex microspheres is measured near the retroreflection direction. It is shown that a sharp peak appears in the retroreflection direction when the volume density is above 1%. The angular width of this peak is much smaller than (wavelength)/(particle size) and cannot be explained by Mie theory, double-passage effects, or radiative-transfer theory. When the particle size D is less than the wavelength λw, a small peak appears at the retroreflection direction. When D is 2–4 times greater than λw, the peak becomes large as the density increases. When D is many times greater than λw, the sharp peak at the retroreflection direction is superimposed upon the Mie-scattering pattern. The angular width of the peak is of the order of (a wavelength)/(a mean free path).

© 1984 Optical Society of America

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References

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  1. T. Gehrels, T. Coffeen, D. Owings, “Wavelength dependence of polarization. III. The lunar surface,” Astron. J. 69, 826–852 (1964).
    [CrossRef]
  2. P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,” J. Geophys. Res. 71, 2505–2513 (1966).
    [CrossRef]
  3. W. G. Egan, T. Hilgeman, “Retroreflectance measurements of photometric standards and coatings,” Appl. Opt. 15, 1845–1849 (1976).
    [CrossRef] [PubMed]
  4. D. A. deWolf, “Electromagnetic reflection from an extended turbulent medium: cumulative forward-scatter single-back-scatterer approximation,” IEEE Trans. Antennas Propag. AP-19, 254–262 (1971).
    [CrossRef]
  5. S. Ito, S. Adachi, “Multiple scattering effect on backscattering from a random medium,” IEEE Trans. Antennas Propag. AP-2, 205–208 (1977).
    [CrossRef]
  6. K. C. Yeh, “Mutual coherence functions and intensities of backscattered signals in a turbulent medium,” Radio Sci. 18, 159–165 (1983).
    [CrossRef]
  7. Yu. A. Kravtsov, A. I. Saichev, “Effects of double passage of waves in randomly inhomogeneous media,” Sov. Phys. Usp. 25, 494–508 (1982).
    [CrossRef]
  8. L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  10. A. Ishimaru, Y. Kuga, R. L.-T. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983). Although Eq. (12) in this reference is correct for the calculation of the transmitted-flux density, it is incorrect for the calculation of the backscattered intensity. The correct second term of A(z) in Eq. (12) is the following:-1γ+αt[exp(γz-γd-αtd)-exp(-αtz)].
    [CrossRef]
  11. A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  12. Y. Kuga, “Laser light propagation and scattering in a dense distribution of spherical particles,” Ph.D. Dissertation (University of Washington, Seattle, Wash., 1983).

1984 (1)

1983 (2)

1982 (2)

Yu. 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 a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

1977 (1)

S. Ito, S. Adachi, “Multiple scattering effect on backscattering from a random medium,” IEEE Trans. Antennas Propag. AP-2, 205–208 (1977).
[CrossRef]

1976 (1)

1971 (1)

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

1966 (1)

P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,” J. Geophys. Res. 71, 2505–2513 (1966).
[CrossRef]

1964 (1)

T. Gehrels, T. Coffeen, D. Owings, “Wavelength dependence of polarization. III. The lunar surface,” Astron. J. 69, 826–852 (1964).
[CrossRef]

Adachi, S.

S. Ito, S. Adachi, “Multiple scattering effect on backscattering from a random medium,” IEEE Trans. Antennas Propag. AP-2, 205–208 (1977).
[CrossRef]

Cheung, R. L.-T.

Coffeen, T.

T. Gehrels, T. Coffeen, D. Owings, “Wavelength dependence of polarization. III. The lunar surface,” Astron. J. 69, 826–852 (1964).
[CrossRef]

deWolf, D. A.

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

Egan, W. G.

Gehrels, T.

T. Gehrels, T. Coffeen, D. Owings, “Wavelength dependence of polarization. III. The lunar surface,” Astron. J. 69, 826–852 (1964).
[CrossRef]

Hilgeman, T.

Ishimaru, A.

Ito, S.

S. Ito, S. Adachi, “Multiple scattering effect on backscattering from a random medium,” IEEE Trans. Antennas Propag. AP-2, 205–208 (1977).
[CrossRef]

Kravtsov, Yu. A.

Yu. 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.

Oetking, P.

P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,” J. Geophys. Res. 71, 2505–2513 (1966).
[CrossRef]

Owings, D.

T. Gehrels, T. Coffeen, D. Owings, “Wavelength dependence of polarization. III. The lunar surface,” Astron. J. 69, 826–852 (1964).
[CrossRef]

Saichev, A. I.

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

Shimizu, K.

Tsang, L.

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]

Appl. Opt. (1)

Astron. J. (1)

T. Gehrels, T. Coffeen, D. Owings, “Wavelength dependence of polarization. III. The lunar surface,” Astron. J. 69, 826–852 (1964).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

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

S. Ito, S. Adachi, “Multiple scattering effect on backscattering from a random medium,” IEEE Trans. Antennas Propag. AP-2, 205–208 (1977).
[CrossRef]

J. Geophys. Res. (1)

P. Oetking, “Photometric studies of diffusely reflecting surfaces with applications to the brightness of the moon,” J. Geophys. Res. 71, 2505–2513 (1966).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Radio Sci. (1)

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)

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

Other (2)

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

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

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the experimental apparatus. S.C., spectrophotometer cell; BS, beam splitter; FO, optical-fiber cable; I1, I3, specific intensity in water and air; n1n3, indices of refraction of water (n1 = 1.33), glass (n2 = 1.47), and air (n3 = 1); T1, T2, transmission coefficients at the boundaries. S.C. is tilted about 5° with respect to the axis normal to this page, so that the specular reflection from S.C. does not enter the detector. θ3 is defined such that the clockwise direction is positive.

Fig. 2
Fig. 2

Backscattered intensity versus angle. Particle size is 0.091 μm. A and a are the experimental data and the value obtained by the diffusion theory, respectively, for the volume density 28.95%; B and b, the same for 19.3%; C and c, the same for 9.65%; D and d, the same for 4.825%; E and e, the same for 2.413%; F and f, the same for 1.206%; G and g, the same for 0.603%; H and h, the same for 0.302%.

Fig. 3
Fig. 3

Backscattered intensity versus angle. Particle size is 1.101 μm. A, experimental data for the volume density 9.55%; B, 4.78%; C, 2.39%; D, 1.19%; E, 0.597%; F, 0.299%; G, 0.149%; H, 0.075%.

Fig. 4
Fig. 4

Backscattered intensity versus angle. Particle size is 2.02 um. A and a are the experimental data and the value obtained by the diffusion theory, respectively, for the volume density 9.75%; B and b, the same for 4.88%; C and c, the same for 2.44%; D and d, the same for 1.22%; E and e, the same for 0.609%; F and f, the same for 0.305%; G and g, the same for 0.152%; H and h, the same for 0.076%.

Fig. 5
Fig. 5

Backscattered intensity versus angle. Particle size is 5.7 μm. A and a are the experimental data and the value obtained by the diffusion theory, respectively, for the volume density 38.22%; B and b, the same for 9.55%; C and c, the same for 4.78%; D and d, the same for 2.389%; E and e, the same for 1.194%; F and f, the same for 0.597%; G and g, the same for 0.299%; H is the experimental data for 0.149%; I is the experimental data for 0.075%.

Fig. 6
Fig. 6

Mie-scattering pattern with size distributions. 0° corresponds to the retroreflection direction.

Fig. 7
Fig. 7

Ratio of the intensity at 0° to 0.6° angle versus volume density. A, 0.091-μm particles; B, 1.101 μm; C, 2.02 μm; D, 5.7 μm.

Fig. 8
Fig. 8

Intensity versus square of the angle. Particle size is 5.7 μm. A, Mie solution; B, volume density 0.149%; C, volume density 9.55%; D, volume density 38.22%.

Equations (3)

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I 3 = n 3 3 cos θ 3 n 1 3 cos θ 1 T 1 2 T 2 2 I 1 ,
F ¯ - ( r = 0 , z = 0 ) = - 2.5 π θ 0 2 U d ( r = 0 , z = 0 ) z ^ ,
-1γ+αt[exp(γz-γd-αtd)-exp(-αtz)].

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