Abstract

We study shape-induced variability in the scattered intensity from randomly oriented nonspherical particles. Up to 21 different Chebyshev shapes contribute to defining a shape-induced standard deviation about each of the mean nonspherical intensity vs angle curves shown in part 2 of this series. Bands of shape-induced variability (defined as plus and minus one standard deviation) for six size intervals within the size parameter range 1 ≤ x ≤ 20 are compared with corresponding spherical intensities. Averaging spherical intensities over narrow size ranges produces effects qualitatively similar to mildly distorting a single sphere. Nevertheless, among all shapes, the sphere is often the most anomalous scatterer; nonspherical scattered intensities tend to be closer to one another than to corresponding spherical intensities. For Chebyshev particles which are neither small nor large compared to the wavelength, shape-induced variability is often comparable to the mean. Furthermore, outside the forward-scattering region, this variability is large relative to the deformation from a sphere. The standard deviation is up to 50% of the mean scattered intensity for particles with an average deformation of only ~10%. This exaggerated sensitivity to shape will make it difficult to define representative angular scattering curves for many real-world nonspherical scattering problems which involve imperfect shape information.

© 1989 Optical Society of America

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References

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  1. W. J. Wiscombe, A. Mugnai, “Scattering from Nonspherical Chebyshev Particles. 2: Means of Angular Scattering Patterns,” Appl. Opt. 27, 2405–2421 (1988).
    [CrossRef] [PubMed]
  2. A. Mugnai, W. J. Wiscombe, “Scattering from Nonspherical Chebyshev Particles. 1: Cross Sections, Single-Scattering Albedo, Asymmetry Factor, and Backscattered Fraction,” Appl. Opt. 25, 1235–1244 (1986).
    [CrossRef] [PubMed]
  3. A. Mugnai, W. J. Wiscombe, “Scattering of Radiation by Moderately Nonspherical Particles,” J. Atmos. Sci. 37, 1291–1298 (1980).
    [CrossRef]
  4. W. J. Wiscombe, A. Mugnai, “Single Scattering from Nonspherical Chebyshev Particles: A Compendium of Calculations,” NASA Reference Publication 1157 (NASA/Goddard Space Flight Center, Greenbelt, MD, 1986).
  5. C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Ch. 12.
  6. T. Nevitt, C. Bohren, “Infrared Backscattering by Irregularly Shaped Particles: A Statistical Approach,” J. Clim. Appl. Meteor. 23, 1342–1349 (1984).
    [CrossRef]
  7. S. Hill, A. Hill, P. Barber, “Light Scattering by Size/Shape Distributions of Soil Particles and Spheroids,” Appl. Opt. 23, 1025–1031 (1984).
    [CrossRef] [PubMed]
  8. M. Iskander, A. Lakhtakia, C. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
    [CrossRef]
  9. D. W. Schuerman, R. Wang, B. Gustafson, R. Schaefer, “Systematic Studies of Light Scattering. 1: Particle Shape,” Appl. Opt. 20, 4039–4050 (1981).
    [CrossRef] [PubMed]
  10. V. Erma, “Perturbation Solution for the Scattering of Electromagnetic Waves from Conductors of Arbitrary Shape. II. General Case,” Phys. Rev. 176, 1544–1564 (1968).
    [CrossRef]

1988 (1)

1986 (1)

1984 (2)

S. Hill, A. Hill, P. Barber, “Light Scattering by Size/Shape Distributions of Soil Particles and Spheroids,” Appl. Opt. 23, 1025–1031 (1984).
[CrossRef] [PubMed]

T. Nevitt, C. Bohren, “Infrared Backscattering by Irregularly Shaped Particles: A Statistical Approach,” J. Clim. Appl. Meteor. 23, 1342–1349 (1984).
[CrossRef]

1983 (1)

M. Iskander, A. Lakhtakia, C. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

1981 (1)

1980 (1)

A. Mugnai, W. J. Wiscombe, “Scattering of Radiation by Moderately Nonspherical Particles,” J. Atmos. Sci. 37, 1291–1298 (1980).
[CrossRef]

1968 (1)

V. Erma, “Perturbation Solution for the Scattering of Electromagnetic Waves from Conductors of Arbitrary Shape. II. General Case,” Phys. Rev. 176, 1544–1564 (1968).
[CrossRef]

Barber, P.

Bohren, C.

T. Nevitt, C. Bohren, “Infrared Backscattering by Irregularly Shaped Particles: A Statistical Approach,” J. Clim. Appl. Meteor. 23, 1342–1349 (1984).
[CrossRef]

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Ch. 12.

Durney, C.

M. Iskander, A. Lakhtakia, C. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Erma, V.

V. Erma, “Perturbation Solution for the Scattering of Electromagnetic Waves from Conductors of Arbitrary Shape. II. General Case,” Phys. Rev. 176, 1544–1564 (1968).
[CrossRef]

Gustafson, B.

Hill, A.

Hill, S.

Huffman, D.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Ch. 12.

Iskander, M.

M. Iskander, A. Lakhtakia, C. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Lakhtakia, A.

M. Iskander, A. Lakhtakia, C. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Mugnai, A.

W. J. Wiscombe, A. Mugnai, “Scattering from Nonspherical Chebyshev Particles. 2: Means of Angular Scattering Patterns,” Appl. Opt. 27, 2405–2421 (1988).
[CrossRef] [PubMed]

A. Mugnai, W. J. Wiscombe, “Scattering from Nonspherical Chebyshev Particles. 1: Cross Sections, Single-Scattering Albedo, Asymmetry Factor, and Backscattered Fraction,” Appl. Opt. 25, 1235–1244 (1986).
[CrossRef] [PubMed]

A. Mugnai, W. J. Wiscombe, “Scattering of Radiation by Moderately Nonspherical Particles,” J. Atmos. Sci. 37, 1291–1298 (1980).
[CrossRef]

W. J. Wiscombe, A. Mugnai, “Single Scattering from Nonspherical Chebyshev Particles: A Compendium of Calculations,” NASA Reference Publication 1157 (NASA/Goddard Space Flight Center, Greenbelt, MD, 1986).

Nevitt, T.

T. Nevitt, C. Bohren, “Infrared Backscattering by Irregularly Shaped Particles: A Statistical Approach,” J. Clim. Appl. Meteor. 23, 1342–1349 (1984).
[CrossRef]

Schaefer, R.

Schuerman, D. W.

Wang, R.

Wiscombe, W. J.

W. J. Wiscombe, A. Mugnai, “Scattering from Nonspherical Chebyshev Particles. 2: Means of Angular Scattering Patterns,” Appl. Opt. 27, 2405–2421 (1988).
[CrossRef] [PubMed]

A. Mugnai, W. J. Wiscombe, “Scattering from Nonspherical Chebyshev Particles. 1: Cross Sections, Single-Scattering Albedo, Asymmetry Factor, and Backscattered Fraction,” Appl. Opt. 25, 1235–1244 (1986).
[CrossRef] [PubMed]

A. Mugnai, W. J. Wiscombe, “Scattering of Radiation by Moderately Nonspherical Particles,” J. Atmos. Sci. 37, 1291–1298 (1980).
[CrossRef]

W. J. Wiscombe, A. Mugnai, “Single Scattering from Nonspherical Chebyshev Particles: A Compendium of Calculations,” NASA Reference Publication 1157 (NASA/Goddard Space Flight Center, Greenbelt, MD, 1986).

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

M. Iskander, A. Lakhtakia, C. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

J. Atmos. Sci. (1)

A. Mugnai, W. J. Wiscombe, “Scattering of Radiation by Moderately Nonspherical Particles,” J. Atmos. Sci. 37, 1291–1298 (1980).
[CrossRef]

J. Clim. Appl. Meteor. (1)

T. Nevitt, C. Bohren, “Infrared Backscattering by Irregularly Shaped Particles: A Statistical Approach,” J. Clim. Appl. Meteor. 23, 1342–1349 (1984).
[CrossRef]

Phys. Rev. (1)

V. Erma, “Perturbation Solution for the Scattering of Electromagnetic Waves from Conductors of Arbitrary Shape. II. General Case,” Phys. Rev. 176, 1544–1564 (1968).
[CrossRef]

Other (2)

W. J. Wiscombe, A. Mugnai, “Single Scattering from Nonspherical Chebyshev Particles: A Compendium of Calculations,” NASA Reference Publication 1157 (NASA/Goddard Space Flight Center, Greenbelt, MD, 1986).

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Ch. 12.

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Figures (8)

Fig. 1
Fig. 1

Perpendicular (left column) and parallel (right column) scattered intensities for spheres with size parameter x = 2 (top row) and 20 (bottom row), and for micro-size-averaged spheres (over Δx = 0.1x). The hatched band indicates the minimum-to-maximum variation of the intensity over the micro-range Δx.

Fig. 2
Fig. 2

Perpendicular (left column) and parallel (right column) scattered intensities for spheres (top row), for a T4 (+0.1) particle (middle row), and for a mixture of all calculated randomly oriented 〈Tn〉 particles (bottom row), for an equal-volume-sphere size parameter x = 10. The hatched bands indicate the minimum-to-maximum variability over size (top row), orientation (middle row), and shape (bottom row).

Fig. 3
Fig. 3

Standard deviation over shape of the size-averaged perpendicular and parallel scattered intensities for the various Chebyshev shapes contributing to the size parameter ranges x = 5–8 and 8–12 [see Eq. 2].

Fig. 4
Fig. 4

Relative standard deviation over shape of the size-averaged perpendicular and parallel scattered intensities for the various Chebyshev shapes contributing to the six size parameter ranges from Table I.

Fig. 5
Fig. 5

Parallel scattered intensities in the 0–60° angular range for mixtures of all 〈Tn〉 particles, and for micro-size-averaged spheres, averaged over the largest four size parameter ranges from Table I (x = 5–8, 8–12, 12–16, and 16–20). The hatched band indicates the shape variability; it is plus and minus one standard deviation about the shape-averaged nonspherical intensity (solid line).

Fig. 6
Fig. 6

Parallel scattered intensities in the 60–180° angular range for mixtures of 〈Tn〉 particles, and for micro-size-averaged spheres, averaged over the six size parameter ranges from Table I. The hatched band indicates the shape variability; it is plus and minus one standard deviation about the shape-averaged nonspherical intensity (solid line).

Fig. 7
Fig. 7

As in Fig. 6, but for perpendicular intensities.

Fig. 8
Fig. 8

Parallel scattered intensities in the 60–180° angular range for mixtures of all 〈Tn〉 particles with relative deformation from a sphere |ɛ| = 0.05 (left column) and 0.10 (right column), and for micro-size-averaged spheres, averaged over three size parameter ranges x = 3–5, 5–8, and 8–12. The hatched band indicates the shape variability; it is plus and minus one standard deviation about the shape-averaged nonspherical intensity (solid line).

Tables (1)

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Table I Number of Chebyshev shapes Ns in each range of size parameter, as a function of the relative deformation from a sphere |ɛ|

Equations (4)

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r = r o [ 1 + ε T n ( cos θ ) ] ,
i q , s ( θ ) = 1 N x j = 1 N x i q , s ( x j , θ ) ,
i q ( θ ) = 1 N s s = 1 N s i q , s ( θ ) ,
σ q ( θ ) = 1 N s 1 s = 1 N s [ i q , s ( θ ) i q ( θ ) ] 2 .

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