Abstract

Diffusion theory is applied to the transmission of an optical beam through randomly distributed particles, and the theoretical calculations are compared with experimental data for an optical beam at 0.6 μm propagating through latex scatterers of sizes 0.109 and 2.02 μm. It is shown that, for particles small compared with the wavelength, the diffusion theory gives good agreement with experimental data; whereas for particles large compared with the wavelength, the diffusion theory is applicable when the optical depth is greater than about 20. For shorter optical depth, experimental results are also compared with the first-order scattering theory.

© 1983 Optical Society of America

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References

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  1. A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation Through the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).
    [CrossRef]
  2. A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
    [CrossRef]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume I: Single Scattering and Transport Theory; Volume II: Multiple Scattering, Turbulence, Rough Surfaces and Remote Sensing (Academic, New York, 1978).
  4. S. T. Hong and 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]
  5. C. Warde, “Atmospheric optical communication,” Opt. Eng. 20, 62 (1981).
    [CrossRef]
  6. A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 20, 63–70 (1981).
    [CrossRef]
  7. G. C. Mooradian, “Surface and subsurface optical communications in the marine environment,” Opt. Eng. 20, 71–75 (1981).
    [CrossRef]
  8. E. A. Bucher and R. M. Lerner, “Experiments on light pulse communication and propagation through atmospheric clouds,” Appl. Opt. 12, 2401–2414 (1973).
    [CrossRef] [PubMed]
  9. E. Collett, J. T. Foley, and E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,” J. Opt.Soc. Am. 67, 465–467 (1977).
    [CrossRef]
  10. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II.
  11. P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981).
  12. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  13. A. Ishimaru and R.L.-T. Cheung, “Multiple scattering effects on wave propagation due to rain,” Ann. Telecommun. 35, 373–379 (1980).
  14. L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium,” Appl. Opt. 15, 2059–2067 (1976).
    [CrossRef] [PubMed]
  15. A. Ishimaru, “Diffusion of a pulse in densely distributed scattered,” J. Opt.Soc. Am. 68, 1045–1050 (1978).
    [CrossRef]
  16. K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
    [CrossRef]
  17. K. Shimizu, “Remote sensing of microparticles by laser scattering for medical applications,” Ph.D. Thesis (University of Washington, Seattle, Wash., 1979).
  18. A. Ishimaru, “Theoretical and experimental study of transient phenomena in random media,” in Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981), pp. 155–163.
  19. A. Ishimaru and Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  20. Equation (7) can be expressed in other forms. For example, a reviewer noted that the surface integral can be converted into a volume integral.

1982 (1)

1981 (3)

C. Warde, “Atmospheric optical communication,” Opt. Eng. 20, 62 (1981).
[CrossRef]

A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 20, 63–70 (1981).
[CrossRef]

G. C. Mooradian, “Surface and subsurface optical communications in the marine environment,” Opt. Eng. 20, 71–75 (1981).
[CrossRef]

1980 (2)

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

K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
[CrossRef]

1978 (1)

A. Ishimaru, “Diffusion of a pulse in densely distributed scattered,” J. Opt.Soc. Am. 68, 1045–1050 (1978).
[CrossRef]

1977 (2)

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
[CrossRef]

E. Collett, J. T. Foley, and E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,” J. Opt.Soc. Am. 67, 465–467 (1977).
[CrossRef]

1976 (2)

S. T. Hong and 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]

L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

1973 (1)

Bucher, E. A.

Chandrasekhar, S.

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

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).

Collett, E.

E. Collett, J. T. Foley, and E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,” J. Opt.Soc. Am. 67, 465–467 (1977).
[CrossRef]

Foley, J. T.

E. Collett, J. T. Foley, and E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,” J. Opt.Soc. Am. 67, 465–467 (1977).
[CrossRef]

Furutsu, K.

Hong, S. T.

S. T. Hong and 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]

Ishimaru, A.

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

A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 20, 63–70 (1981).
[CrossRef]

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

A. Ishimaru, “Diffusion of a pulse in densely distributed scattered,” J. Opt.Soc. Am. 68, 1045–1050 (1978).
[CrossRef]

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
[CrossRef]

S. T. Hong and 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]

L. Reynolds, C. C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

A. Ishimaru, “Theoretical and experimental study of transient phenomena in random media,” in Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981), pp. 155–163.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume I: Single Scattering and Transport Theory; Volume II: Multiple Scattering, Turbulence, Rough Surfaces and Remote Sensing (Academic, New York, 1978).

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation Through the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).
[CrossRef]

Johnson, C. C.

Kohler, W. E.

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

Kuga, Y.

Lerner, R. M.

Mooradian, G. C.

G. C. Mooradian, “Surface and subsurface optical communications in the marine environment,” Opt. Eng. 20, 71–75 (1981).
[CrossRef]

Papanicolaou, G. C.

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

Reynolds, L.

Shimizu, K.

K. Shimizu, “Remote sensing of microparticles by laser scattering for medical applications,” Ph.D. Thesis (University of Washington, Seattle, Wash., 1979).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II.

Warde, C.

C. Warde, “Atmospheric optical communication,” Opt. Eng. 20, 62 (1981).
[CrossRef]

Wolf, E.

E. Collett, J. T. Foley, and E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,” J. Opt.Soc. Am. 67, 465–467 (1977).
[CrossRef]

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. Opt. Soc. Am. (2)

J. Opt.Soc. Am. (2)

A. Ishimaru, “Diffusion of a pulse in densely distributed scattered,” J. Opt.Soc. Am. 68, 1045–1050 (1978).
[CrossRef]

E. Collett, J. T. Foley, and E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,” J. Opt.Soc. Am. 67, 465–467 (1977).
[CrossRef]

Opt. Eng. (3)

C. Warde, “Atmospheric optical communication,” Opt. Eng. 20, 62 (1981).
[CrossRef]

A. Ishimaru, “Theory of optical propagation in the atmosphere,” Opt. Eng. 20, 63–70 (1981).
[CrossRef]

G. C. Mooradian, “Surface and subsurface optical communications in the marine environment,” Opt. Eng. 20, 71–75 (1981).
[CrossRef]

Proc. IEEE (1)

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
[CrossRef]

Radio Sci. (1)

S. T. Hong and 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]

Other (8)

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation Through the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume I: Single Scattering and Transport Theory; Volume II: Multiple Scattering, Turbulence, Rough Surfaces and Remote Sensing (Academic, New York, 1978).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. I and II.

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

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

Equation (7) can be expressed in other forms. For example, a reviewer noted that the surface integral can be converted into a volume integral.

K. Shimizu, “Remote sensing of microparticles by laser scattering for medical applications,” Ph.D. Thesis (University of Washington, Seattle, Wash., 1979).

A. Ishimaru, “Theoretical and experimental study of transient phenomena in random media,” in Multiple Scattering and Waves in Random Media (North-Holland, Amsterdam, 1981), pp. 155–163.

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Figures (6)

Fig. 1
Fig. 1

ŝ is a unit vector in the direction of scattering at position r. θ is an angle between ŝ and is normal to the surface. r is a distance from the beam center to the observation point. 2W is a diameter of the incident beam.

Fig. 2
Fig. 2

Measured and calculated total transmitted flux density versus distance r. A is the volume density 0.0955%, corresponding to τ = 1.28; B = 0.239%, τ = 3.07; C = 0.478%, τ = 5.86; D = 0.955%, τ = 10.92; E = 2.388%, τ = 23.4; F = 4.777%, τ = 37.6; the field of view (FOV) = cos θ.

Fig. 3
Fig. 3

Same as Fig. 2 except that FOV = 1.79 deg.

Fig. 4
Fig. 4

Measured and calculated total transmitted flux density versus r. a is the volume density 0.00975%, corresponding to τ = 2.43; b = 0.0244%, τ = 6.08; c = 0.0487%, τ = 12.14; d = 0.0975%, τ = 24.28; e = 0.244%, τ = 60.80; f = 0.975%, τ = 243.0.

Fig. 5
Fig. 5

Measured and calculated total transmitted flux density versus optical distance τ. r0, r1, and r2 are measurements and the diffusion approximation at r = 0 mm, r = 5 mm, and r = 10 mm, respectively. r0(FS) is calculated by using the first-order multiple-scattering approximation. Particle size, 0.109 μm.

Fig. 6
Fig. 6

Same as Fig. 5 except that the particle size is 2.02 μm.

Tables (1)

Tables Icon

Table 1 Equivalent Extinction Cross Section σt (eq) at Various Densitiesa

Equations (25)

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F 0 ( r ) = I 0 exp ( r 2 W 2 ) z ̂ .
I ( r , s ̂ ) = I coh ( r , s ̂ ) + I d ( r , s ̂ ) , I coh ( r , s ̂ ) = F 0 ( r ) exp ( α t z ) δ ( s ̂ z ̂ ) ,
I d ( r , s ̂ ) = n = 0 U n ( r ) P n ( cos θ ) = U d ( r ) + 3 4 π F d ( r ) s ̂ + ,
2 U d ( r ) q 2 U d ( r ) = Q 0 ( r )
U d ( r ) h z U d ( r ) + Q 1 ( r ) 2 π = 0 at z = 0 , U d ( r ) + h z U d ( r ) Q 1 ( r ) 2 π = 0 at z = d ,
q 2 = 3 ( 1 W 0 ) ( 1 W 0 μ ¯ ) α t 2 , h = 2 / [ 3 ( 1 W 0 μ ¯ ) α t ] , Q 0 ( r ) = C 0 exp ( r 2 / W 2 ) exp ( α t z ) , Q 1 ( r ) = C 1 exp ( r 2 / W 2 ) exp ( α t z ) , C 0 = 3 4 π W 0 α t 2 ( 1 W 0 μ ¯ + μ ¯ ) , C 1 = W 0 μ ¯ 1 W 0 μ ¯ ,
μ ¯
W 0 = σ s σ t
U d ( r ) = υ G ( r , r ) Q 0 ( r ) d υ 1 2 π h s G ( r , r ) Q 1 ( r ) d s ,
( 2 q 2 ) G ( r , r ) = δ ( r r ) , G ( r , r ) h z G ( r , r ) = 0 at z = 0 , G ( r , r ) + h z G ( r , r ) = 0 at z = d .
U d ( r , z ) = 0 r d r 0 d d z G 1 ( r , r ; z , z ) Q 0 ( r ; z ) 1 2 π h 0 [ G 1 ( r , r ; z , 0 ) Q 1 ( r ; 0 ) G 1 ( r , r ; z , d ) Q 1 ( r ; d ) ] r d r .
G 1 ( r , r ; z , z ) = 0 g ( λ , r ; z , z ) λ J 0 ( λ r ) d λ ,
g ( λ , r ; z , z ) = g 1 ( z ) g 2 ( z ) Δ J 0 ( λ r ) for z > z , g ( λ , r ; z , z ) = g 1 ( z ) g 2 ( z ) Δ J 0 ( λ r ) for z < z , g 1 ( z ) = ( h γ 1 ) exp ( γ d + γ z ) + ( h γ + 1 ) exp ( γ d γ z ) , g 2 ( z ) = ( h γ + 1 ) exp ( γ z ) + ( h γ 1 ) exp ( γ z ) , Δ = 2 γ [ ( h γ + 1 ) 2 exp ( γ d ) ( h γ 1 ) 2 exp ( γ d ) ] , γ 2 = λ 2 + q 2 .
U d ( r , z ) = 0 λ d λ γ J 0 ( λ r ) W 2 2 × exp ( W 2 λ 2 4 ) [ C 0 2 A ( z ) + C 1 4 π h B ( z ) ] ,
A ( z ) = 1 γ α t [ exp ( α t z ) exp ( γ z ) ] 1 γ + α t [ exp ( γ z γ d α t d ) exp ( α t d ) ] g 11 ( γ + α t ) [ exp ( γ z γ d α t d ) exp ( γ z ) ] + g 12 γ α t [ exp ( γ z + γ d α t d ) exp ( γ z ) ] + g 21 γ α t [ exp ( γ z + γ d α t d ) exp ( γ z ) ] g 22 γ + α t [ exp ( γ z γ d α t d ) exp ( γ z ) ] ,
B ( z ) = exp ( γ z γ d α t d ) exp ( γ z ) + g 11 [ exp ( γ z γ d α t d ) exp ( γ z ) ] + g 12 [ exp ( γ z + γ d α t d ) exp ( γ z ) ] + g 21 [ exp ( γ z + γ d α t d ) exp ( γ z ) ] + g 22 [ exp ( γ z γ d α t d ) exp ( γ z ) ] ,
g 11 = ( h 2 γ 2 1 ) exp ( γ d ) / Δ , g 12 = ( h γ 1 ) 2 exp ( γ d ) / Δ , g 21 = ( h 2 γ 2 1 ) exp ( γ d ) / Δ , g 22 = ( h γ 1 ) 2 exp ( γ d ) / Δ , Δ = ( h γ + 1 ) 2 exp ( γ d ) ( h γ 1 ) 2 exp ( γ d ) .
F d ( r ) = σ s μ ¯ F 0 ( r ) σ t r exp ( ρ σ t z ) z ̂ 4 π 3 ρ σ t r grad U d ( r ) ,
σ t r = σ s ( 1 μ ¯ ) + σ a .
½ U d ( r ) + F d n ( r ) 4 π = 0 ,
F d z ( r , d ) = 2 π U d ( r ) .
F z ( r , d ) = F r i ( r , d ) + F d z ( r , d ) = I 0 exp ( r 2 / W 2 ) exp ( α t d ) + 2 π U d ( r , d ) .
F z ( r , d ) = F r i ( r , d ) + [ U d ( r ) + 3 4 π F d ( r ) z ̂ ] Δ ω = I 0 exp ( r 2 / W 2 ) exp ( α t d ) + 1.25 θ 0 2 F d z ( r , d ) ,
P R = A r ( ω ) [ I r i ( ω ) + I d ( ω ) ] d ω , I r i = F 0 exp ( ρ σ t d ) , I d ( θ ) = ρ σ t 4 π cos θ p ( θ ) 0 d exp [ ρ σ t d ( 1 cos θ 1 ) ρ σ t z 2 W 0 2 ( z ) 2 tan 2 θ ] d z ,
μ ¯