Abstract

Typically, explanation/interpretation of observed light scattering and absorption properties of marine particles is based on assuming a spherical shape and homogeneous composition. We examine the influence of shape and homogeneity by comparing the optics of randomly-oriented cylindrically-shaped particles with those of equal-volume spheres, in particular the influence of aspect ratio (AR = length/diameter) on extinction and backscattering. Our principal finding is that the when AR > ~3–5 and the diameter is of the order of the wavelength, the extinction efficiency and the backscattering probability are close to those of an infinite cylinder. In addition, we show the spherical-based interpretation of extinction and absorption can lead to large error in predicted backscattering.

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References

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  1. S. G. Ackleson, “Optical determinations of suspended sediment dynamics in western Long Island Sound and the Connecticut River plume,” J. Geophys. Res. 111(C7), C07009 (2006), doi:.
    [CrossRef]
  2. N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993).
    [CrossRef]
  3. H. R. Gordon, and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983).
  4. D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
    [CrossRef]
  5. H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46(6), 1438–1454 (2001).
    [CrossRef]
  6. A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006).
    [CrossRef]
  7. W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007).
    [CrossRef]
  8. H. R. Gordon, T. J. Smyth, W. M. Balch, G. C. Boynton, and G. A. Tarran, “Light scattering by coccoliths detached from Emiliania huxleyi,” Appl. Opt. 48(31), 6059–6073 (2009).
    [CrossRef] [PubMed]
  9. H. R. Gordon, “Backscattering of light from disklike particles: is fine-scale structure or gross morphology more important?” Appl. Opt. 45(27), 7166–7173 (2006).
    [CrossRef] [PubMed]
  10. H. R. Gordon, “Backscattering of light from disk-like particles with aperiodic angular fine structure,” Opt. Express 15(25), 16424–16430 (2007).
    [CrossRef] [PubMed]
  11. L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005).
    [CrossRef]
  12. L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998).
    [CrossRef]
  13. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  14. H. C. Van de Hulst, Light Scattering by Small Particles, (Wiley, 1957).
  15. G. R. Fournier and B. T. Evans, “Approximations to extinction from randomly oriented circular and elliptical cylinders,” Appl. Opt. 35(21), 4271–4282 (1996).
    [CrossRef] [PubMed]
  16. L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19(1), 1–12 (1956).
    [CrossRef] [PubMed]
  17. A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. 28(11), 1375–1393 (1981).
    [CrossRef]
  18. A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986).
    [CrossRef] [PubMed]
  19. J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995).
    [CrossRef]
  20. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  21. B. T. Draine and P. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
    [CrossRef]
  22. R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985).
    [CrossRef]
  23. G. R. Fournier and B. T. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991).
    [CrossRef] [PubMed]
  24. M. Jonasz, and G. R. Fournier, Light Scattering by Particles in Water, Theoretical and Experimental Foundations (Academic Press, 2007).

2009

2007

H. R. Gordon, “Backscattering of light from disk-like particles with aperiodic angular fine structure,” Opt. Express 15(25), 16424–16430 (2007).
[CrossRef] [PubMed]

W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007).
[CrossRef]

2006

S. G. Ackleson, “Optical determinations of suspended sediment dynamics in western Long Island Sound and the Connecticut River plume,” J. Geophys. Res. 111(C7), C07009 (2006), doi:.
[CrossRef]

A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006).
[CrossRef]

H. R. Gordon, “Backscattering of light from disklike particles: is fine-scale structure or gross morphology more important?” Appl. Opt. 45(27), 7166–7173 (2006).
[CrossRef] [PubMed]

2005

L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005).
[CrossRef]

2004

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[CrossRef]

2001

H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46(6), 1438–1454 (2001).
[CrossRef]

1998

L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998).
[CrossRef]

1996

1995

J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995).
[CrossRef]

1994

1993

N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993).
[CrossRef]

1991

1988

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1986

1985

R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985).
[CrossRef]

1981

A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. 28(11), 1375–1393 (1981).
[CrossRef]

1956

L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19(1), 1–12 (1956).
[CrossRef] [PubMed]

Ackleson, S. G.

S. G. Ackleson, “Optical determinations of suspended sediment dynamics in western Long Island Sound and the Connecticut River plume,” J. Geophys. Res. 111(C7), C07009 (2006), doi:.
[CrossRef]

Balch, W. M.

Bernard, S.

A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006).
[CrossRef]

Bogucki, D.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[CrossRef]

Boss, E.

W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007).
[CrossRef]

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[CrossRef]

Boynton, G. C.

Bricaud, A.

A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986).
[CrossRef] [PubMed]

A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. 28(11), 1375–1393 (1981).
[CrossRef]

Chan, T. L.

L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005).
[CrossRef]

Clavano, W. R.

W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007).
[CrossRef]

Cohen, A.

R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985).
[CrossRef]

Cohen, L. D.

R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985).
[CrossRef]

Draine, B. T.

B. T. Draine and P. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Du, T.

H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46(6), 1438–1454 (2001).
[CrossRef]

Duysens, L. N. M.

L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19(1), 1–12 (1956).
[CrossRef] [PubMed]

Evans, B. T.

Flatau, P.

Fournier, G. R.

Gordon, H. R.

Haracz, R. D.

R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985).
[CrossRef]

Hoepffner, N.

N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993).
[CrossRef]

Jumars, P. A.

L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998).
[CrossRef]

Karp-Boss, L.

W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007).
[CrossRef]

L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998).
[CrossRef]

Kitchen, J. C.

J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995).
[CrossRef]

Lin, J.-Z.

L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005).
[CrossRef]

Morel, A.

A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986).
[CrossRef] [PubMed]

A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. 28(11), 1375–1393 (1981).
[CrossRef]

Quirantes, A.

A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006).
[CrossRef]

Sathyendranath, S.

N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993).
[CrossRef]

Smyth, T. J.

Stramski, D.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[CrossRef]

Tarran, G. A.

Voss, K. J.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[CrossRef]

Zaneveld, J. R. V.

J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995).
[CrossRef]

Zhang, L.-X.

L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005).
[CrossRef]

Appl. Opt.

Astrophys. J.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Biochim. Biophys. Acta

L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19(1), 1–12 (1956).
[CrossRef] [PubMed]

Deep-Sea Res.

A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. 28(11), 1375–1393 (1981).
[CrossRef]

J. Appl. Phys.

R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985).
[CrossRef]

J. Geophys. Res.

J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995).
[CrossRef]

S. G. Ackleson, “Optical determinations of suspended sediment dynamics in western Long Island Sound and the Connecticut River plume,” J. Geophys. Res. 111(C7), C07009 (2006), doi:.
[CrossRef]

N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006).
[CrossRef]

Limnol. Oceanogr.

L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998).
[CrossRef]

H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46(6), 1438–1454 (2001).
[CrossRef]

Oceanogr. Mar. Biol.

W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007).
[CrossRef]

Opt. Express

Phys. Fluids

L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005).
[CrossRef]

Prog. Oceanogr.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[CrossRef]

Other

H. R. Gordon, and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983).

C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

H. C. Van de Hulst, Light Scattering by Small Particles, (Wiley, 1957).

M. Jonasz, and G. R. Fournier, Light Scattering by Particles in Water, Theoretical and Experimental Foundations (Academic Press, 2007).

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Figures (13)

Fig. 1
Fig. 1

Specifications of the cylinders examined in this study.

Fig. 2
Fig. 2

Extinction and absorption efficiencies computed for randomly orientated, homogeneous or coated cylindrically shaped particles, given that their diameter and aspect ratio are known so that their orientationally-averaged projected area is πD(L + D/2)/4. Here, ρ = 2α (mr −1) and ρ′ = 4α mi , where mr imi , is the refractive index of the particle relative to water, and α = πD/λ, with D the cylinder’s (outer) diameter and λ the wavelength of light in the water. Solid lines are the exact computations for randomly oriented, infinite cylinders. The notation “m=1.05-040i_1.05-000i” indicates that the refractive index of the core is 1.05 – 0.040i, and the refractive index of the coating is 1.05 – 0.000i, etc. In the case of coated cylinders, ρ and ρ′ are computed using the mi of the associated homogeneous particle. Top: all aspect ratios (1/3 – 30). Bottom: all aspect ratios ≥ 3.

Fig. 3
Fig. 3

This figure provides the ratio of absorption efficiencies (coated to homogeneous) of strongly absorbing cylinders as a function of wavelength. The diameter of the cylinder (in μm) is specified in the legend. For each diameter, the symbols refer to cylinder lengths ranging from 0.5 to 15 μm. The figure shows that the effect of the absorbing pigment packaging is greatest in the blue region of the spectrum and for larger-diameter cylinders. Although the symbols do not differentiate between cylinder lengths, for a given diameter the packaging effect is smallest in the shortest cylinder and depends very little on the length once the aspect ratio (length/diameter) exceeds unity.

Fig. 4
Fig. 4

Qc as a function of ρ, computed for non-absorbing cylinders (m = 1.20 – 0.000i) with diameters (D) ranging from 0.5 μm to 2.0 μm. Left: 0.25 ≤ AR ≤ 30 (points colored in red are AR = 2). Right: 3 ≤ AR ≤ 30. The solid curve is the extinction efficiency (for a unit length) of randomly-oriented infinite cylinders. As in Fig. 2, ρ = 2α (m −1) with α = πD/λ.

Fig. 5
Fig. 5

Orientationally averaged scattering phase functions for long cylinders as a function of aspect ratio (AR). The values of the computed backscattering probabilities are 0.0212, 0.0212, 0.0209, 0.0218, and 0.0220, for AR = 5, 10, 100, 200, and 270, respectively. The values of the extinction efficiency (Qc ) are 0.999, 1.007, 1.032, 1.033, 1.033, and 1.034 for AR = 5, 10, 100, 200, 270, and ∞, respectively. The value of ρ for these efficiencies is 1.0472, so these computations fall very close to the continuous curve in Fig. 4. (Note, L = 1.25, 2.50, 25.0, 50.0, and 67.5 μm for AR = 5, 10, 100, 200, and 270, respectively.) D = 0.25 μm, m = 1.20, λ = 400 nm.

Fig. 6
Fig. 6

Orientationally-averaged scattering phase functions (left) and degree of linear polarization (right) at 600 nm (vacuum) for homogeneous cylinders with a diameter of 1 μm and length of 3 μm (D1xL3) and 20 μm (D1xL20). The refractive index is 1.05 – 0.002i. The oscillatory nature of the phase function is determined mostly by D/λ, but with some dependence on m.

Fig. 7
Fig. 7

Examples of the variation of the backscattering probability with aspect ratio for cylinder diameters between 0.5 and 1.5 μm and refractive indices ranging from 1.02 to 1.20. The black curves are for a vacuum wavelength of 400 nm and the red curves for 700 nm.

Fig. 13
Fig. 13

Backscattering probability as a function of aspect ratio and the imaginary part of the refractive index for non-absorbing to strongly absorbing, homogeneous and structured cylinders. D is in micrometers.

Fig. 8
Fig. 8

The extinction (left) and absorption (right) efficiencies computed by dividing the associated cross sections by the projected area of a volume-equivalent sphere as a function of ρ and ρ ′. Here, ρ = 2α (mr −1) and ρ′ = 4α mi , where mr imi , is the refractive index of the particle relative to water, and α = πd/λ, where d is now the diameter of the volume-equivalent sphere. For a given experimentally-determined Qa , the dashed vertical arrow provides the correct ρ ′ (and, hence mi ), while the solid vertical arrow provides the retrieved value of mi .

Fig. 9
Fig. 9

An example of retrievals of the real and imaginary parts of the refractive index for coated cylinders for all the combinations of diameter and length, using the van de Hulst approximation. Ideally one should derive a real part of 1.05 and an imaginary part of 0.010. The scatter shows that mi is retrieved to within ± 20% (somewhat better in the red) with an average (over all sizes) close to 0.010, and that the retrieved mr appears to be too low in almost all cases, but averages ~1.044.

Fig. 10
Fig. 10

The extinction efficiency computed by dividing the associated extinction cross section by the projected area of a volume-equivalent sphere as a function of ρ. Here, ρ = 2α (mr −1) and α = πd/λ, where d is the diameter of the volume-equivalent sphere. Points for some given diameters and lengths are connected by smooth curves (for which λ varies from 400 to 700 nm). The red curves are for diameters of 1.0 and 1.5 μm with AR = 10. The thick curve is the Van de Hulst approximation to Qc for spheres.

Fig. 11
Fig. 11

Backscattering by a cylinder divided by backscattering by an equal-volume sphere. Black curves: refractive index in the computation of σbb for spheres is that derived using the refractive index determined from the extinction and absorption cross sections using the equivalent-volume sphere assumption. Red curves: refractive index in the computation of σbb for spheres is the same value used for the cylinders, i.e., the correct value. Diameter (D) is in micrometers, and the true value of the refractive index is 1.05–0.010i.

Fig. 12
Fig. 12

Backscattering by a cylinder divided by backscattering by an equal-volume sphere. As the refractive index in this case cannot be derived from the extinction efficiency, in the computation of σbb for spheres m is the same value used for the cylinders. Diameter (D) is in micrometers, the wavelength is 400 nm, and the true value of the refractive index is 1.20–0.000i.

Tables (1)

Tables Icon

Table 1 The RMS the Backscattering Probability for Spheroids with Aspect Ratio AR = 10 to that for Spheroids with the Same Minor Axes but Aspect Ratio AR. D = 0.5, 1.0, and 1.5 μm, and λ = 400, 500, 600, and 700 nm

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

d σ b ( Θ , Φ ) = ( S b ( Θ , Φ ) ) A v g r ^ ( Θ , Φ ) d A | ( S I n c ) A v g | ,
d σ b ( Θ , Φ ) d Ω = r 2 ( S b ( Θ , Φ ) ) A v g r ^ ( Θ , Φ ) d A | ( S I n c ) A v g | .
σ b = Φ = 0 Φ = 2 π Θ = 0 Θ = π d σ b ( Θ , Φ ) d Ω sin Θ d Θ d Φ ,
σ b b = Φ = 0 Φ = 2 π Θ = π / 2 Θ = π d σ b ( Θ , Φ ) d Ω sin Θ d Θ d Φ ,
P ( Θ , Φ ) 1 σ b d σ b ( Θ , Φ ) d Ω .
d σ b ( Θ , Φ ) d Ω = r 2 ( S b ( Θ , Φ ) r ^ ( Θ , Φ ) ) A v g | ( S I n c ) A v g | ,
σ b = Φ = 0 Φ = 2 π Θ = 0 Θ = π d σ b ( Θ , Φ ) d Ω sin Θ d Θ d Φ ,
σ b b = Φ = 0 Φ = 2 π Θ = π / 2 Θ = π d σ b ( Θ , Φ ) d Ω sin Θ d Θ d Φ ,
P ( Θ , Φ ) 1 σ b d σ b ( Θ , Φ ) d Ω ,
β ( Θ ) = n d σ b ( Θ , Φ ) d Ω   and   b = n σ b ,
b b = 2 π Θ = π / 2 Θ = π β ( Θ ) sin Θ d Θ ,
σ a = A ( S t ( Θ , Φ ) ) A v g r ^ ( Θ , Φ ) d A | ( S I n c ) A v g | ,
σ c ( D , m , A R ) ( A R + 1 / 2 ) σ c ( D , m , A R ) ( A R + 1 / 2 ) .
σ b b ( D , m , A R ) ( A R + 1 / 2 ) σ b b ( D , m , A R ) ( A R + 1 / 2 ) ,
Q a ( ρ ) = 1 + 2 exp ( ρ ) ρ + 2 exp ( ρ ) 1 ρ 2 ,
Q c ( ρ ) = 2 4 exp ( ρ tan β ) [ cos β ρ sin ( ρ β ) + cos 2 β ρ 2 cos ( ρ 2 β ) ] + 4 cos 2 β ρ 2 cos ( 2 β ) ,
σ b ( D , m , A R ) A R 2 σ b ( D , m , A R ) A R 2 .

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