Abstract

Recent computations of the backscattering cross section of randomly-oriented disk-like particles (refractive index, 1.20) with small-scale internal structure, using the discrete-dipole approximation (DDA), have been repeated using the Rayleigh-Gans approximation (RGA). As long as the thickness of the disks is approximately 20% of the wavelength (or less), the RGA agrees reasonably well quantitatively with the DDA. The comparisons show that the RGA is sufficiently accurate to be useful as a quantitative tool for exploring the backscattering features of disk-like particles with complex structure. It is used here to develop a zeroth-order correction for the neglect of birefringence on modeling the backscattering of detached coccoliths from E. huxleyi.

© 2007 Optical Society of America

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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-Verlag, 1983).
    [CrossRef]
  2. D. A. Siegel, A. C. Thomas, and J. Marra, "Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue," Deep Sea Res. II,  511-3 (2004).
    [CrossRef]
  3. D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, "The role of seawater constituents in light backscattering in the ocean," Prog. Oceanogr. 61, 27-56 (2004).
    [CrossRef]
  4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis (Cambridge, 2002).
  5. H. R. Gordon and T. Du, "Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi," Limnol. Oceanogr. 46, 1438-1454 (2001).
    [CrossRef]
  6. 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).
    [CrossRef]
  7. 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).
    [CrossRef]
  8. 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).
    [CrossRef]
  9. H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
    [CrossRef]
  10. 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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
    [CrossRef]
  11. H. R. Gordon, "Backscattering of light from disk-like particles: is fine-scale structure or gross morphology more important?" Appl. Opt. 45, 7166-7173 (2006).
    [CrossRef] [PubMed]
  12. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333,848-872 (1988).
    [CrossRef]
  13. B. T. Draine and P. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A ll,1491-1499 (1994).
    [CrossRef]
  14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  15. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  16. L. D. Cohen, R. D. Haracz, A. Cohen, and C. Acquista, "Scattering of light from arbitrarily oriented cylinders," Appl. Opt. 22, 742-748 (1983).
    [CrossRef] [PubMed]
  17. K. Shimizu, "Modification of the Rayleigh-Debye approximation," J. Opt. Soc. Am. 73, 504-507 (1983).
    [CrossRef]
  18. B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. 405, 685-697 (1993).
    [CrossRef]
  19. J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
    [CrossRef]
  20. E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223-2249 (1996).
    [CrossRef]
  21. J. M. Bennett and H. E. Bennett, "Polarization," in Handbook of Optics, W.G. Driscoll and W. Vaughan, eds., (McGraw-Hill, 1978).

2006

2005

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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
[CrossRef]

2004

D. A. Siegel, A. C. Thomas, and J. Marra, "Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue," Deep Sea Res. II,  511-3 (2004).
[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, 27-56 (2004).
[CrossRef]

2002

2001

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[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).
[CrossRef]

1996

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).
[CrossRef]

E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223-2249 (1996).
[CrossRef]

1994

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

1993

B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. 405, 685-697 (1993).
[CrossRef]

1992

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
[CrossRef]

1991

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).
[CrossRef]

1988

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

1983

Aas, E.

E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223-2249 (1996).
[CrossRef]

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).
[CrossRef]

Acquista, C.

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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
[CrossRef]

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[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).
[CrossRef]

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).
[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, 27-56 (2004).
[CrossRef]

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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
[CrossRef]

Boss, E.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, "The role of seawater constituents in light backscattering in the ocean," Prog. Oceanogr. 61, 27-56 (2004).
[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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
[CrossRef]

Boynton, G. C.

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[CrossRef]

Brown, P. R.

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
[CrossRef]

Cohen, A.

Cohen, L. D.

Didymus, J. M.

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
[CrossRef]

Draine, B. T.

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

B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. 405, 685-697 (1993).
[CrossRef]

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

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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
[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).
[CrossRef]

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).
[CrossRef]

Flatau, P.

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

Goodman, J.

B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. 405, 685-697 (1993).
[CrossRef]

Gordon, H. R.

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

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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
[CrossRef]

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[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).
[CrossRef]

Groom, S. B.

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).
[CrossRef]

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[CrossRef]

Haracz, R. D.

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).
[CrossRef]

Harbour, D. S.

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[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).
[CrossRef]

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).
[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).
[CrossRef]

Land, P. E.

Mann, S.

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
[CrossRef]

Marra, J.

D. A. Siegel, A. C. Thomas, and J. Marra, "Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue," Deep Sea Res. II,  511-3 (2004).
[CrossRef]

Moore, G. F.

Prins, B.

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
[CrossRef]

Shimizu, K.

Siegel, D. A.

D. A. Siegel, A. C. Thomas, and J. Marra, "Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue," Deep Sea Res. II,  511-3 (2004).
[CrossRef]

Smyth, T. J.

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).
[CrossRef]

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[CrossRef]

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, 27-56 (2004).
[CrossRef]

Thomas, A. C.

D. A. Siegel, A. C. Thomas, and J. Marra, "Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue," Deep Sea Res. II,  511-3 (2004).
[CrossRef]

Tyrrell, T.

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, 27-56 (2004).
[CrossRef]

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).
[CrossRef]

Young, J. R.

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
[CrossRef]

Appl. Opt.

Astrophys. J.

B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. 405, 685-697 (1993).
[CrossRef]

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

Deep Sea Res. II

D. A. Siegel, A. C. Thomas, and J. Marra, "Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue," Deep Sea Res. II,  511-3 (2004).
[CrossRef]

Geophys. Res. Lett.

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001).
[CrossRef]

J. Geophys. Res.

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 Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560.
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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

J. Plankton Res.

E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223-2249 (1996).
[CrossRef]

Limnol. Oceanogr.

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

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).
[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).
[CrossRef]

Nature

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992).
[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, 27-56 (2004).
[CrossRef]

Other

M. I. Mishchenko, L. D. Travis, and A. A. Lacis (Cambridge, 2002).

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

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

J. M. Bennett and H. E. Bennett, "Polarization," in Handbook of Optics, W.G. Driscoll and W. Vaughan, eds., (McGraw-Hill, 1978).

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

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Figures (12)

Fig. 1.
Fig. 1.

A volume element dVi , located at a point Di from the origin of coordinates (O). The incident radiation is propagating in the κ0 direction and the scattered radiation is propagating in the κ direction. The vector ri is from the volume element dVi , to a distant point at which the scattered field is measured. The vector r is from the origin to the same distant point, which is sufficiently far from the particle that the vectors ri and r are considered to be parallel.

Fig. 2.
Fig. 2.

The plane formed by the propagation vector κ⃗0 of the incident wave and the propagation vector κ⃗ of the scattered wave is the scattering plane. The incident and scattered fields are resolved into components parallel and perpendicular to the scattering plane, i.e., along (ê (0) l , ê (0) r ) and (êl , êr ), respectively. Θ is the scattering angle.

Fig. 3.
Fig. 3.

Comparison of the backscattering cross section computed with the RGA and the DDA for a homogeneous disk of diameter 2.75 μm and thicknesses 0.05, 0.10, and 0.15 μm.

Fig. 4.
Fig. 4.

Comparison of RGA and DDA computations for the Gordon and Du [5] “fishing reel” models of a detached coccolith.

Fig. 5.
Fig. 5.

Sectorized disks (“pinwheels”) for various values of n used in this study.

Fig. 6.
Fig. 6.

Comparison of DDA and RGA backscattering by sectorized disks in Fig. 5.

Fig. 7.
Fig. 7.

Comparison of DDA and RGA backscattering by sectorized disks in Fig. 5.

Fig. 8.
Fig. 8.

Comparison of DDA and RGA backscattering by parallel sectorized (n = 5 and 6) washers. The individual washers have an outside diameter of 1.50 μm, inside diameter 1.00 μm and thickness 0.05 μm. They are separated by a space of 0.30 μm.

Fig. 9.
Fig. 9.

Comparison of RGA computations of backscattering for a randomly-oriented birefringent disk and an isotropic disk with mi = (2mo + me )/3.

Fig. A1.
Fig. A1.

A schematic of scattering by a disk. The cylindrical coordinate system ( ρ , η , z′) is fixed with respect to the disk (z′ is in the direction of the normal, n).

Fig. A2.
Fig. A2.

Relationship between the laboratory-fixed coordinate system (x, y, z) and the body-fixed system (x′, y′, z′) or ( ρ , η , z′). θ, ϕ, and ψ are the Euler angles. Because of the symmetry of the disk the angle ψ can be set to zero.

Fig. A3.
Fig. A3.

The required integrals for the evaluation of the matrix A. The J’s are Bessel functions.

Tables (1)

Tables Icon

Table 1: Parameters of the Gordon and Du [5] “Fishing-reel” model of a detached coccolith.

Equations (62)

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

E D i t = E ( 0 ) D i t + j C D i D j E D j t ,
d p D i t = ρ n α E D i t d V i ,
d E ( s ) = 1 4 π ε 0 r i κ × [ κ × d p i ( D i , t r i c ) ] ,
E ( s ) r t = d E ( s ) r t ,
E ( 0 ) D i t = E ( 0 ) exp [ i ( κ 0 D i ωt ) ] ,
E ( 0 ) = E ( 0 ) e ̂ ( 0 ) + E r ( 0 ) e ̂ r ( 0 ) = ( E r ( 0 ) E ( 0 ) ) .
E ( s ) = 1 i κ r A E ( 0 ) exp [ i ( κ r ω t ) ] or ( E r ( s ) E ( s ) ) = 1 i κ r ( A r r A l r A r l A l l ) ( E r ( 0 ) E ( 0 ) ) exp [ i ( κ r ωt ) ] ,
E ( s ) = E ( s ) e ̂ + E r ( s ) e ̂ r = ( E r ( s ) E ( s ) ) .
S = κ ̂ 2 μ 0 c E E * + E r E r * = κ ̂ 2 μ 0 c E ˜ * E = κ ̂ d P d A ,
d σ d Ω d P ( s ) d Ω d P ( 0 ) d A = r 2 S ( s ) S ( 0 ) ,
S ( 0 ) = 1 2 μ 0 c E ˜ ( 0 ) * E ( 0 ) and S ( s ) = 1 2 μ 0 c 1 κ 2 r 2 E ˜ ( 0 ) * A ˜ * A E ( 0 ) ,
d σ d Ω = 1 κ 2 E ˜ ( 0 ) * A ˜ * A E ( 0 ) E ˜ ( 0 ) * E ( 0 ) .
E r ( 0 ) * E r ( 0 ) = E ( 0 ) * E ( 0 ) and E r ( 0 ) * E ( 0 ) = 0 = E ( 0 ) * E r ( 0 ) ,
d σ d Ω = 1 κ 2 E ˜ ( 0 ) * A ˜ * A E ( 0 ) E ˜ ( 0 ) * E ( 0 ) = 1 2 κ 2 ( A r r 2 + A r 2 + A r 2 + A 2 ) .
d p i ( D i , t r i c ) = ρ n α ( D i ) E ( 0 ) ( D i , t r i c ) d V i
= ρ n α ( D i ) E ( 0 ) exp [ i ( κ 0 D i ω t + r i ω c ) ] d V i .
d p i ( D i , t r i c ) = ρ n α ( D i ) E ( 0 ) exp [ i ( κ 0 κ ) D i ] exp [ i ( κ r ω t ) ] d V i .
d E ( s ) = 1 i κ r d A E ( 0 ) exp [ i ( κ r ω t ) ] ,
d A i = i ρ n κ 4 π ε 0 κ × [ κ × α ( D i ) ] exp [ i ( κ 0 κ ) D i ] d V i .
E ( 0 ) = E ( 0 ) e ̂ 0 + E r ( 0 ) e ̂ r 0 = E ( 0 ) e ̂ x E r ( 0 ) e ̂ y ,
E ( s ) = E ( s ) e ̂ + E r ( s ) e ̂ r = E ( s ) cos Θ e ̂ x E r ( s ) e ̂ y E ( s ) sin Θ e ̂ z ,
κ = κ ( e ̂ x sin Θ + e ̂ z cos Θ ) ,
κ 0 = κ e ̂ z .
d A i = i ρ n κ 3 4 π ε 0 ( α y y α x y α y z sin Θ + α x y cos Θ α x z sin Θ α x x cos Θ ) exp [ i ( κ 0 κ ) D i ] d V i ,
A = i ρ n κ 3 4 π ε 0 V ( α y y α x y α y z sin Θ + α x y cos Θ α x z sin Θ α x x cos Θ ) exp [ i ( κ 0 κ ) D i ] d V .
A = i ρ n α κ 3 4 π ε 0 ( 1 0 0 cos Θ ) V exp [ i ( κ 0 κ ) D ] d V ,
d σ d Ω = ( ρ n α κ 2 4 π ε 0 ) 2 ( 1 + cos 2 Θ ) 2 V exp [ i ( κ 0 κ ) D ] d V 2 .
ρ n α ε 0 = 3 ( m 2 1 m 2 + 2 ) ;
R V exp [ i ( κ 0 κ ) D ] d V ,
d σ d Ω = 9 16 π 2 κ 4 ( m 2 1 m 2 + 2 ) 2 ( 1 + cos 2 Θ 2 ) R 2
σ = 4 π d σ d Ω d Ω and σ b = Back 2 π d σ d Ω d Ω ,
P ( Θ ) = 4 π σ d σ ( Θ ) d Ω and β ( Θ ) = N d σ ( Θ ) d Ω ,
Δ α = 2 π 2 n
ρ n a ε 0 = 3 ( m e 2 1 m e 2 + 2 ) and ρ n b ε 0 = 3 ( m o 2 1 m o 2 + 2 )
ρ n α ε 0 = 3 ( m i 2 1 m i 2 + 2 )
α B = ( α ρ ρ 0 0 0 α η η 0 0 0 α z z ) ( a 0 0 0 b 0 0 0 b ) .
α = U ˜ B ˜ α B BU ,
U = ( cos ψ cos ϕ cos θ sin ϕ sin ψ cos ψ sin ϕ + cos θ cos ϕ sin ψ sin ψ sin θ sin ψ cos ϕ cos θ sin ϕ cos ψ sin ψ sin ϕ + cos θ cos ϕ cos ψ cos ψ sin θ sin θ sin ϕ sin θ cos ϕ cos θ )
B = ( cos η sin η 0 sin η cos η 0 0 0 1 ) .
( κ 0 κ ) D = 2 κ sin ( Θ 2 ) [ ρ cos ( η γ ) sin β + z cos β ]
cos β = cos θ sin ( Θ 2 ) sin θ sin ϕ cos ( Θ 2 ) ,
0 t 0 R 0 2 π α yy exp { i 2 κ sin ( Θ 2 ) [ ρ cos ( η γ ) sin β + z cos β ] } ρ d η d ρ d z .
α x x = 1 2 cos 2 η [ a + b + ( a b ) cos 2 ϕ ]
+ 2 ( a + b ) cos η sin η cos θ sin ϕ
+ sin 2 η [ b sin 2 θ + cos 2 θ ( b cos 2 ϕ + a sin 2 ϕ ) ] ,
α x y = ( a b ) ( cos θ cos ϕ sin η + cos η sin ϕ )
× ( cos η cos ϕ cos θ sin η sin ϕ ) ,
α x z = ( a b ) sin η sin θ ( cos η cos ϕ cos θ sin η sin ϕ ) ,
α y y = 1 2 cos 2 η [ a + b ( a b ) cos 2 ϕ ]
2 ( a + b ) cos η sin η cos θ cos ϕ sin ϕ ,
+ sin 2 η [ b sin 2 θ + cos 2 θ ( a cos 2 ϕ + b sin 2 ϕ ) ] ,
α y z = ( a b ) sin η sin θ ( cos η sin ϕ + cos θ sin η cos ϕ ) ,
α z z = b cos 2 η + sin 2 η ( a sin 2 θ + b cos 2 θ ) ,
α y x = α x y , α z x = α x z , α z y = α y z .
0 t 0 R 0 2 π ( cos 2 η cos η sin 2 η sin η ) exp { i 2 κ sin ( Θ 2 ) [ ρ cos ( η γ ) sin β + z cos β ] } ρ d η d ρ dz .
2 π κ 2 ( 1 J 0 ( κ R ) 0 κ R J 1 ( κ R ) + J 0 ( κ R ) 1 ) 2 k sin ( k t 2 ) ,
κ = 2 κ sin ( Θ 2 ) sin β ,
k = 2 κ sin ( Θ 2 ) cos β ,
cos β = cos θ sin ( Θ 2 ) sin θ sin ϕ cos ( Θ 2 ) .
A = i ρ n κ 3 a 4 π ε 0 ( 1 0 0 cos Θ ) V exp [ i ( κ 0 κ ) D ] d V
= i ρ n κ 3 a 4 π ε 0 ( 1 0 0 cos Θ ) 2 π κ 2 κ R J 1 ( κ R ) 2 k sin ( k t 2 ) .
d σ d Ω = ( ρ n κ 2 a 4 π ε 0 ) 2 ( 2 V J 1 ( κ R ) κ R sin ( k t 2 ) k t 2 ) 2 ( 1 + cos 2 Θ 2 ) .

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