Abstract

The backscattering of light from disklike objects possessing periodic structures (e.g., resembling a wheel with spokes, hereafter called a pinwheel) or an object with a wavelength-sized deviation from a flat disk (e.g., a spherical cap) has been computed by using the discrete dipole approximation. The disks ranged in diameter from 1.5to2.7μm with thicknesses from 0.04to0.15μm. The goal of the study was to obtain some understanding of the differences between the backscattering of a collection of such objects in random orientation and a collection of randomly oriented homogeneous disks of the same size, i.e., the conditions under which the gross morphology (e.g., disklikeness) of these objects determines their backscattering. The computations for pinwheels showed that their backscattering cross sections were nearly identical to those of homogeneous disks of similar size (but with reduced effective refractive indices that are easily estimated) as long as the maximum separations between the spokes was less than one quarter of the wavelength. In this regime the backscattering is totally governed by the particle's gross morphology and effective index. For larger spoke separation, departures from a homogeneous disk are observed and manifest as significant increases (many times) in backscattering. In the case of spherical caps with the same projected area as the associated disk, the computations again show a complete similarity in their backscattering, and when the disks are sufficiently thin (with thickness divided by wavelength <0.15to0.25) there is very little difference between the backscattering of a cap and the associated disk.

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References

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  1. H. R. Gordon and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer, 1983).
  2. P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
    [CrossRef]
  3. W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991).
  4. W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).
  5. T. J. Smyth, G. F. Moore, S. B. Groom, P. E. Land, and T. Tyrrell, "Optical modeling and measurements of a coccolithophore bloom," Appl. Opt. 41, 7679-7688 (2002).
  6. W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
    [CrossRef]
  7. H. R. Gordon and T. Du, "Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi," Limnol. Oceanogr. 46, 1438-1454. (2001).
  8. P. Chylek, G. Videen, J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, "Effective medium approximations for heterogeneous particles," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 273-308.
  9. H. A. Lorentz, "Über die Beziehung zwischen del' Fortpflanzungsgeschwindigkeit des Lichtes und del' Lorperdichte," Ann. Phys. Chem. 9, 641-665 (1880).
  10. L. Lorenz, "Über die Refractionconstante," Ann. Phys. Chem. 11, 70-103 (1880).
  11. J. C. Maxwell-Garnett, "Colours in metal glasses and metallic films," Philos. Trans. R. Soc. London , Ser. A 203, 385-420 (1904).
  12. D. A. G. Bruggeman, "Berechnung vershiedener physikalischer Konstanten von heterogenen Substanzen. 1. Dielektrizitatskonstanten und Leitfahigkeiten del' Mischkoper aus isotropen Substanzen," Ann. Phys. 24, 636-664 (1935).
  13. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
    [CrossRef]
  14. B. T. Draine and P. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
  15. B. T. Draine, "The discrete-dipole approximation for light scattering by irregular targets," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 131-145.
  16. M. I. Mishchenko and L. D. Travis, "Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers," J. Quant. Spectrosc. Radiat. Transf. 60, 7206-7225 (1998).
    [CrossRef]

2005 (1)

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
[CrossRef]

2002 (1)

2001 (1)

H. R. Gordon and T. Du, "Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi," Limnol. Oceanogr. 46, 1438-1454. (2001).

1998 (1)

M. I. Mishchenko and L. D. Travis, "Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers," J. Quant. Spectrosc. Radiat. Transf. 60, 7206-7225 (1998).
[CrossRef]

1996 (1)

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).

1994 (1)

1991 (1)

W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991).

1988 (1)

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

1983 (1)

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, "Berechnung vershiedener physikalischer Konstanten von heterogenen Substanzen. 1. Dielektrizitatskonstanten und Leitfahigkeiten del' Mischkoper aus isotropen Substanzen," Ann. Phys. 24, 636-664 (1935).

1904 (1)

J. C. Maxwell-Garnett, "Colours in metal glasses and metallic films," Philos. Trans. R. Soc. London , Ser. A 203, 385-420 (1904).

1880 (2)

H. A. Lorentz, "Über die Beziehung zwischen del' Fortpflanzungsgeschwindigkeit des Lichtes und del' Lorperdichte," Ann. Phys. Chem. 9, 641-665 (1880).

L. Lorenz, "Über die Refractionconstante," Ann. Phys. Chem. 11, 70-103 (1880).

Ackleson, S. G.

W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991).

Balch, W. M.

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
[CrossRef]

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).

W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991).

Booth, E. S.

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
[CrossRef]

Bowler, B. C.

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, "Berechnung vershiedener physikalischer Konstanten von heterogenen Substanzen. 1. Dielektrizitatskonstanten und Leitfahigkeiten del' Mischkoper aus isotropen Substanzen," Ann. Phys. 24, 636-664 (1935).

Camus, P.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
[CrossRef]

Champagne-Philippe, M.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
[CrossRef]

Chylek, P.

P. Chylek, G. Videen, J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, "Effective medium approximations for heterogeneous particles," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 273-308.

Dobbie, J. S.

P. Chylek, G. Videen, J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, "Effective medium approximations for heterogeneous particles," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 273-308.

Draine, B. T.

B. T. Draine and P. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).

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

B. T. Draine, "The discrete-dipole approximation for light scattering by irregular targets," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 131-145.

Drapeau, D. T.

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
[CrossRef]

Du, T.

H. R. Gordon and T. Du, "Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi," Limnol. Oceanogr. 46, 1438-1454. (2001).

Fernandez, E.

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).

Flatau, P.

Geldart, J. W.

P. Chylek, G. Videen, J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, "Effective medium approximations for heterogeneous particles," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 273-308.

Gordon, H. R.

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
[CrossRef]

H. R. Gordon and T. Du, "Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi," Limnol. Oceanogr. 46, 1438-1454. (2001).

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

Groom, S. B.

Harbour, D.

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).

Harbour, D. S.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
[CrossRef]

Holligan, P. M.

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).

W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991).

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
[CrossRef]

Kilpatrick, K.

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).

Land, P. E.

Lorentz, H. A.

H. A. Lorentz, "Über die Beziehung zwischen del' Fortpflanzungsgeschwindigkeit des Lichtes und del' Lorperdichte," Ann. Phys. Chem. 9, 641-665 (1880).

Lorenz, L.

L. Lorenz, "Über die Refractionconstante," Ann. Phys. Chem. 11, 70-103 (1880).

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, "Colours in metal glasses and metallic films," Philos. Trans. R. Soc. London , Ser. A 203, 385-420 (1904).

Mishchenko, M. I.

M. I. Mishchenko and L. D. Travis, "Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers," J. Quant. Spectrosc. Radiat. Transf. 60, 7206-7225 (1998).
[CrossRef]

Moore, G. F.

Morel, A. Y.

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

Smyth, T. J.

Travis, L. D.

M. I. Mishchenko and L. D. Travis, "Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers," J. Quant. Spectrosc. Radiat. Transf. 60, 7206-7225 (1998).
[CrossRef]

Tso, H. C. W.

P. Chylek, G. Videen, J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, "Effective medium approximations for heterogeneous particles," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 273-308.

Tyrrell, T.

Videen, G.

P. Chylek, G. Videen, J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, "Effective medium approximations for heterogeneous particles," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 273-308.

Viollier, M.

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
[CrossRef]

Voss, K. J.

W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991).

Ann. Phys. (1)

D. A. G. Bruggeman, "Berechnung vershiedener physikalischer Konstanten von heterogenen Substanzen. 1. Dielektrizitatskonstanten und Leitfahigkeiten del' Mischkoper aus isotropen Substanzen," Ann. Phys. 24, 636-664 (1935).

Ann. Phys. Chem. (2)

H. A. Lorentz, "Über die Beziehung zwischen del' Fortpflanzungsgeschwindigkeit des Lichtes und del' Lorperdichte," Ann. Phys. Chem. 9, 641-665 (1880).

L. Lorenz, "Über die Refractionconstante," Ann. Phys. Chem. 11, 70-103 (1880).

Appl. Opt. (1)

Astrophys. J. (1)

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

J. Geophys. Res. (1)

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on moderate-resolution imaging spectraradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:.
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

M. I. Mishchenko and L. D. Travis, "Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers," J. Quant. Spectrosc. Radiat. Transf. 60, 7206-7225 (1998).
[CrossRef]

Limnol. Oceanogr. (3)

H. R. Gordon and T. Du, "Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi," Limnol. Oceanogr. 46, 1438-1454. (2001).

W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991).

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996).

Nature (1)

P. M. Holligan, M. Viollier, D. S. Harbour, P. Camus, and M. Champagne-Philippe, "Satellite and ship studies of coccolithophore production along the continental shelf edge," Nature 304, 339-342 (1983).
[CrossRef]

Philos. Trans. R. Soc. London (1)

J. C. Maxwell-Garnett, "Colours in metal glasses and metallic films," Philos. Trans. R. Soc. London , Ser. A 203, 385-420 (1904).

Other (3)

B. T. Draine, "The discrete-dipole approximation for light scattering by irregular targets," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 131-145.

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

P. Chylek, G. Videen, J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, "Effective medium approximations for heterogeneous particles," in Light Scattering by Nonspherical Particles, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000) pp. 273-308.

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Figures (10)

Fig. 1
Fig. 1

Scanning electron micrograph image of a single cell of E. Huxleyi (the spherically shaped object) resting on a filter pad. The individual disklike structures covering the cell are the coccoliths (Fig. 2.). The pore size of the filter (small holes in the background) is 0.2 μm . The horizontal white bar in the lower left has a length of 2 μm . (Photo courtesy of Jeremy Young, The Natural History Museum, London.)

Fig. 2
Fig. 2

Two scanning electron micrograph views of an isolated coccolith along with the associated disklike fishing reel model of Gordon and Du (Ref. 7). (Photos courtesy of Jeremy Young, The Natural History Museum, London).

Fig. 3
Fig. 3

Positions of a single layer of dipoles in the pinwheel models: (a) n = 4 , (b) n = 5 , (c) n = 6 , and (d) n = 7 .

Fig. 4
Fig. 4

Schematic showing the relationship between the disk and the associated spherical cap.

Fig. 5
Fig. 5

(Color online) Computations of the backscattering cross section of randomly oriented disklike particles with D d = 1.5 mm , and various thicknesses t as described in the text. Also shown is the physical optics model of Gordon and Du (Ref. 7) for m = 1.10 .

Fig. 6
Fig. 6

Comparison of the backscattering coefficients of a disk and the associated spherical cap as a function of the thickness of the disk. Also shown are the results of the physical optics model of Gordon and Du (Ref. 7) for m = 1.20 .

Fig. 7
Fig. 7

Comparison of the backscattering cross section of a pinwheel with D d = 2.7 mm and n = 6 with that of a full disk with m = 1.10 . Also shown is the physical optics model of Gordon and Du (Ref. 7) for m = 1.10 .

Fig. 8
Fig. 8

Comparison of the backscattering coefficients of a disk ( D d = 2.7 μm ) and the associated spherical cap as a function of the thickness of the disk. Also shown are the results of the physical optics model of Gordon and Du (Ref. 7) for m = 1.20 .

Fig. 9
Fig. 9

(Color online) Total scattering cross section as a function of wavelength for the disklike objects with D d = 1.5 μm studied in the text. Note that for a given thickness the scattering is nearly independent of the detailed morphology and depends only on the gross morphology.

Fig. 10
Fig. 10

Comparison of the total scattering cross section of disks and the associated spherical caps as a function of wavelength.

Tables (2)

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Table 1 s as a Function of n

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Table 2 Value of s∕λWater just Before σb Departs From the Universal Curve a

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Δα = 2 π 2 n ,

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