Abstract

The small-angle approximation to the radiative transport equation is used extensively in imaging models in which the transport medium is optically thick. The small-angle approximation is generally considered valid when the particles are very large compared with the wavelength, when the refractive-index ratio of the particle to the medium is close to 1, and when the optical thickness is not too large. We report results showing the limits of the validity of the small-angle approximation as a function of particle size and concentration for a particle-to-medium fixed refractive-index ratio of 1.196. This refractive-index ratio is comparable with that of minerals or diatoms suspended in water.

© 2001 Optical Society of America

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References

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  1. W. H. Wells, “Loss of resolution in water as a result of multiple small-angle scattering,” J. Opt. Soc. Am. 59, 686–691 (1969).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. 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]
  8. J. W. McLean, K. J. Voss, “Point spread function in ocean water: comparison between theory and experiment,” Appl. Opt. 30, 2027–2030 (1991).
    [CrossRef] [PubMed]
  9. W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1-3.3-19 (1973).
  10. H. Hodara, “Experimental results of small angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.4-1-3.4-17 (1973).
  11. J. W. McLean, J. D. Freeman, “Effects of ocean waves on airborne lidar imaging,” Appl. Opt. 35, 3261–3269 (1996).
    [CrossRef] [PubMed]
  12. G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
    [CrossRef]
  13. R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
    [CrossRef] [PubMed]
  14. R. W. Spinrad, J. R. V. Zaneveld, H. Pak, “Volume scattering function of suspended particulate matter at near-forward angles: a comparison of experimental and theoretical values,” Appl. Opt. 17, 1125–1130 (1978).
    [CrossRef] [PubMed]
  15. See, for example, K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), p. 6.
  16. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 12.

1996

1995

1991

1990

1987

1986

1985

1978

1977

1975

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

1973

W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1-3.3-19 (1973).

H. Hodara, “Experimental results of small angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.4-1-3.4-17 (1973).

1972

1969

Arnush, D.

Chang, H. W.

Chapin, A. L.

Ciervo, A. P.

Crawford, D. R.

Freeman, J. D.

Hall, G. J.

Hindman, C. L.

Hodara, H.

H. Hodara, “Experimental results of small angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.4-1-3.4-17 (1973).

Ishimaru, A.

Kattawar, G. W.

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

Kuga, Y.

Lutomirski, R. F.

McLean, J. W.

Mertens, L. E.

Pak, H.

Replogle, F. S.

Shifrin, K. S.

See, for example, K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), p. 6.

Spinrad, R. W.

Tsang, L.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 12.

Voss, K. J.

Wells, W. H.

W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1-3.3-19 (1973).

W. H. Wells, “Loss of resolution in water as a result of multiple small-angle scattering,” J. Opt. Soc. Am. 59, 686–691 (1969).
[CrossRef]

Zaneveld, J. R. V.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

Optics of the Sea

W. H. Wells, “Theory of small-angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.3-1-3.3-19 (1973).

H. Hodara, “Experimental results of small angle scattering,” in Optics of the Sea, AGARD Lect. Ser. 61, 3.4-1-3.4-17 (1973).

Other

See, for example, K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), p. 6.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 12.

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Figures (13)

Fig. 1
Fig. 1

Experimental setup for measuring the VSF: f1, fixed neutral-density filter; sf, spatial filter and collimator; AP, aperture; f2, removable neutral-density filters; s, sample cell; l, lens; pda, photodiode array detector; m, small mirror on a translation stage to block and unblock the main beam.

Fig. 2
Fig. 2

Measured versus theoretical VSF’s. The vertical scale has been arbitrarily offset for each particle size to present the data on a single plot. The indicated percentages are the percent difference in the normalized curves between the measured and the theoretical values at that point. The solid curves are the theoretical Mie curves.

Fig. 3
Fig. 3

Results of theoretical and measured PSF’s for 9.6-μm polystyrene spheres in water. Each set of curves corresponds to a value of τ given at the right of the graph.

Fig. 4
Fig. 4

Results of theoretical and measured PSF’s for 6.2-μm spheres in water.

Fig. 5
Fig. 5

Results of theoretical and measured PSF’s for 3.9-μm spheres in water.

Fig. 6
Fig. 6

Results of theoretical and measured PSF’s for 3.0-μm spheres in water.

Fig. 7
Fig. 7

Results of theoretical and measured PSF’s for 2.0-μm spheres in water.

Fig. 8
Fig. 8

Results of theoretical and measured PSF’s for 1.4-μm spheres in water.

Fig. 9
Fig. 9

Results of theoretical and measured PSF’s for 1.3-μm spheres in water.

Fig. 10
Fig. 10

Results of theoretical and measured PSF’s for 1.0 μm-spheres in water.

Fig. 11
Fig. 11

Percent error versus size parameter for the range of integration 0°θ14°.

Fig. 12
Fig. 12

Percent error versus size parameter for the range of integration 0°θ180°.

Fig. 13
Fig. 13

Percentage of the area under the normalized VSF curves for 0°θ14° as a function of particle size. The refractive-index ratio is 1.196. The solid curve was obtained by numerically integrating the normalized Mie curves from 0° to 14°. The particle sizes for which the PSF’s and the VSF’s were measured are indicated on the curve.

Tables (1)

Tables Icon

Table 1 Range of τ Where Breakdown of SAA Occurs for Each Particle Size a

Equations (19)

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f(θ, r)=2π0J0(2πθψ)F(ψ, r)ψdψ,
F(ψ, r)=2π0θmaxJ0(2πθψ)f(θ, r)θdθ,
σ(θ)=2π0J0(2πθψ)Σ(ψ)ψdψ,
Σ(ψ)=2π0θmaxJ0(2πθψ)σ(θ)θdθ.
4πσ(θ)dΩ=1.
F(ψ, r)=exp[-ξr+sfrΣ(ψ)],
η=2π0θmaxσ(θ)sin(θ)dθ.
F(ψ, r)=exp{-τ[1-ηΣ(ψ)]}.
σ(θ)=dI(θ)sΦndV,
dPσ(θ)=dI(θ)Ω exp[-ξ(L-x)].
Φn=(P0/A)exp(-ξx),
dPσ(θ)=σ(θ)sΩP0exp(-ξL)dx,
Pσ(θ)=σ(θ)sΩP0exp(-ξL)0Ldx=σ(θ)τP0Ω exp(-ξL),
Pr1(θ)=Pnexp(-ξ1L)+σ(θ)τ1P0Ω exp(-ξ1L),
Pr2(θ)=Pnexp(-ξ2L)+σ(θ)τ2P0Ω exp(-ξ2L),
σ(θ)={[Pr2(θ)]/[Pr2(0)]}-{[Pr1(θ)]/[Pr1(0)]}Ω(τ2-τ1),
σ(θ)={[E2(θ)]/[E2(0)]}-{[E1(θ)/E1(0)]}(τ2-τ1)(a1/R2)cos3[arctan(y/R)],
θmax=arcsin{(n1/n2)sin[arctan(d/R)]},
f(θ, L)=[P(θ, L)](P0Ω)=E(θ, L)E0(a/R2)cos3[arctan(y/R)].

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