Abstract

In a dense distribution of particles, the propagation characteristics of the coherent field are strongly affected by the pair-correlated distributions of scatterers. This paper presents an optical experimental study to show that, when the particle density is greater than about 0.1%, the attenuation constant departs markedly from the formula based on an uncorrected scatter assumption. It decreases sharply when ka < 1, whereas it shows a slight increase when ka ≫ 1. Experimental data are shown for the volume densities ranging from 10−3 to 40% and ka ranging from 0.529 to 82.793. Comparisons are given with some theoretical calculations.

© 1982 Optical Society of America

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References

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  1. J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
    [CrossRef]
  2. V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. 64, 1710–1719 (1978).
    [CrossRef]
  3. V. Twersky, “Propagation in pair-correlated distributions of small spaced lossy scatterers,” J. Opt. Soc. Am. 69, 1567–1572 (1979).
    [CrossRef]
  4. G. S. Brown, “Coherent wave propagation through a sparse concentration of particles,” Radio Sci. 15, 705–710 (1980).
    [CrossRef]
  5. P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981).
  6. V. N. Bringi, V. V. Varadan, and V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. (to be published).
  7. L. Tsang, J. A. Kong, and T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. 71, 552–558 (1982).
    [CrossRef]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 39.
  10. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 28.
  11. A. Ishimaru and R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Ann. Telecommun. 35, 373–379 (1980).
  12. H. F. Nelson, “Radiative scattering cross sections: comparison of experiment and theory,” Appl. Opt. 20, 500–504 (1981).
    [CrossRef] [PubMed]
  13. T. P. Wallace and J. P. Kratohvill, “Comments on the comparison of scattering of coherent and incoherent light by polydispersed spheres with Mie theory,” Appl. Opt. 8, 824–826 (1969).
    [CrossRef] [PubMed]

1982 (1)

L. Tsang, J. A. Kong, and T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. 71, 552–558 (1982).
[CrossRef]

1981 (1)

1980 (2)

A. Ishimaru and R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Ann. Telecommun. 35, 373–379 (1980).

G. S. Brown, “Coherent wave propagation through a sparse concentration of particles,” Radio Sci. 15, 705–710 (1980).
[CrossRef]

1979 (1)

1978 (1)

V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. 64, 1710–1719 (1978).
[CrossRef]

1969 (1)

1964 (1)

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Bringi, V. N.

V. N. Bringi, V. V. Varadan, and V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. (to be published).

Brown, G. S.

G. S. Brown, “Coherent wave propagation through a sparse concentration of particles,” Radio Sci. 15, 705–710 (1980).
[CrossRef]

Cheung, R. L.-T.

A. Ishimaru and R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Ann. Telecommun. 35, 373–379 (1980).

Chow, P. L.

P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981).

Habashy, T.

L. Tsang, J. A. Kong, and T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. 71, 552–558 (1982).
[CrossRef]

Ishimaru, A.

A. Ishimaru and R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Ann. Telecommun. 35, 373–379 (1980).

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

Keller, J. B.

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Kohler, W. E.

P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981).

Kong, J. A.

L. Tsang, J. A. Kong, and T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. 71, 552–558 (1982).
[CrossRef]

Kratohvill, J. P.

Nelson, H. F.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 28.

Papanicolaou, G. C.

P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981).

Tsang, L.

L. Tsang, J. A. Kong, and T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. 71, 552–558 (1982).
[CrossRef]

Twersky, V.

V. Twersky, “Propagation in pair-correlated distributions of small spaced lossy scatterers,” J. Opt. Soc. Am. 69, 1567–1572 (1979).
[CrossRef]

V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. 64, 1710–1719 (1978).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 39.

Varadan, V. K.

V. N. Bringi, V. V. Varadan, and V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. (to be published).

Varadan, V. V.

V. N. Bringi, V. V. Varadan, and V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. (to be published).

Wallace, T. P.

Ann. Telecommun. (1)

A. Ishimaru and R. L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Ann. Telecommun. 35, 373–379 (1980).

Appl. Opt. (2)

J. Acoust. Soc. Am. (2)

V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. 64, 1710–1719 (1978).
[CrossRef]

L. Tsang, J. A. Kong, and T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. 71, 552–558 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. Symp. Appl. Math. (1)

J. B. Keller, “Stochastic equations and wave propagation in random media,” Proc. Symp. Appl. Math. 16, 145–170 (1964).
[CrossRef]

Radio Sci. (1)

G. S. Brown, “Coherent wave propagation through a sparse concentration of particles,” Radio Sci. 15, 705–710 (1980).
[CrossRef]

Other (5)

P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981).

V. N. Bringi, V. V. Varadan, and V. K. Varadan, “Coherent wave attenuation by a random distribution of particles,” Radio Sci. (to be published).

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 39.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 28.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup: S.C., sample cell; P, polarizer; PH1, pinhole 1 with diameter 3 mm; PH2, pinhole 2 with diameter 25 μm; Le, 10× microscope objective lens; P.D., photodiode; L1, 97 mm; and L2, 57 mm.

Fig. 2
Fig. 2

The measured ratio γ = α/αo. Twersky’s formula is γT = αTo. γ1 is the calculated ratio αeo for 0.091, 0.109, and 0.481 μm and γ2 is the same ratio for 1.101 μm: ○, particle size 0.091 μm; ka = 0.529; Δ, 0.109 μm (0.681); +, 0.481 μm (3.518); and X, 1.101 μm (7.280). Dashed curves are αho calculated from Eq. (11): A, 0.109 μm; B, 0.481 μm; and C, 1.101 μm.

Fig. 3
Fig. 3

The measured ratio γ for □ 2.02 μm (ka = 13.662); ∇, 5.7 μm (36.238); and ◊,11.9 μm (82.793). γ3 is the calculated ratio αeo for 2.02-, 5.7-, and 11.9-μm particles. Dashed curves are αho from Eq. (11): D, 2.02 μm and E, 5.7 μm.

Fig. 4
Fig. 4

Normalized α, αo, and αT: ○, 0.091 μm; Δ, 0.109 μm; +, 0.481 μm; X, 1.101 μm; □, 2.02 μm; ∇, 5.7 μm; and ◊,11.9 μm.

Tables (1)

Tables Icon

Table 1 Characteristics of Latex Particlesa

Equations (11)

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E = E + E f ,
K υ = k + 2 π ρ f ( î , î ) · ê i k ,
K e = [ k 2 + 4 π ρ f ( î , î ) ê i ] 1 / 2 .
I c I o = exp ( α L ) ,
α o = 2 Im K υ = ρ σ t ,
σ t = 4 π k Im [ f ( î , î ) ] · ê i .
α e = 2 Im ( K e ) .
H w = D p H υ D p H υ + D w ( 1 H υ ) ,
α o = ρ σ t = H υ υ σ t ,
α T = ρ σ a + ρ σ s ( 1 H υ ) 4 ( 1 + 2 H υ ) 2 .
K h 2 = k 2 + 4 π ρ f ( î , î ) [ 4 π ρ f ( î , î ) ] 2 o 2 a e ikr sin k r d r k , α h = 2 Im K h .