Abstract

We present an analytical model for light backscattering by coccoliths and coccolithophores of the marine calcifying phytoplankter Emiliania huxleyi. The model is based on the separation of the effects of diffraction, refraction, and reflection on scattering, a valid assumption for particle sizes typical of coccoliths and coccolithophores. Our model results match closely with results from an exact scattering code that uses complex particle geometry and our model also mimics well abrupt transitions in scattering magnitude. Finally, we apply our model to predict changes in the spectral backscattering coefficient during an Emiliania huxleyi bloom with results that closely match in situ measurements. Because our model captures the key features that control the light backscattering process, it can be generalized to coccoliths and coccolithophores of different morphologies which can be obtained from size-calibrated electron microphotographs. Matlab codes of this model are provided as supplementary material.

© 2017 Optical Society of America

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References

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    [Crossref]
  3. W. Balch, H. Gordon, B. Bowler, and E. Booth, “Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectroradiometer Data,” J. Geophys. Res. 110(C7), C07001 (2005), doi:.
    [Crossref]
  4. 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(8), 1587–1590 (2001).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2016 (1)

C. J. O’Brien, M. Vogt, and N. Gruber, “Global coccolithophore diversity: Drivers and future change,” Prog. Oceanogr. 140, 27–42 (2016).
[Crossref]

2015 (1)

2014 (1)

G. R. Fournier, V. Sanjuan-Calzado, and C. Trees, “Implications of a new phase function for autonomous underwater imaging,” Proc. SPIE VI, 911119 (2014).

2013 (1)

2009 (1)

2007 (1)

2006 (1)

2005 (2)

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

W. Balch, H. Gordon, B. Bowler, and E. Booth, “Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectroradiometer Data,” J. Geophys. Res. 110(C7), C07001 (2005), doi:.
[Crossref]

2002 (1)

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

2001 (4)

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(8), 1587–1590 (2001).
[Crossref]

D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. 40(18), 2929–2945 (2001).
[Crossref] [PubMed]

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]

E. Paasche, “A review of the coccolithophorid Emiliania huxleyi (Prymnesiophyceae), with particular reference to growth, coccolith formation, and calcification-photosynthesis interactions,” Phycologia 40(6), 503–529 (2001).
[Crossref]

1998 (1)

K. Voss, W. Balch, and K. Kilpatrick, “Scattering and attenuation properties of Emiliania huxleyi cells and their detached coccoliths,” Limnol. Oceanogr. 43(5), 870–876 (1998).
[Crossref]

1996 (1)

E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18(12), 2223–2249 (1996).
[Crossref]

1993 (1)

D. Risović, “Two-component model of sea particle size distribution,” Deep-Sea Res. 40(7), 1459–1473 (1993).
[Crossref]

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. 36(4), 629–643 (1991).
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

1929 (1)

E. Schoenberg, “Handb.,” Astrophysik 2, 255 (1929).

Aas, E.

E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18(12), 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. 36(4), 629–643 (1991).
[Crossref]

Balch, W.

W. Balch, H. Gordon, B. Bowler, and E. Booth, “Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectroradiometer Data,” J. Geophys. Res. 110(C7), C07001 (2005), doi:.
[Crossref]

K. Voss, W. Balch, and K. Kilpatrick, “Scattering and attenuation properties of Emiliania huxleyi cells and their detached coccoliths,” Limnol. Oceanogr. 43(5), 870–876 (1998).
[Crossref]

Balch, W. M.

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]

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(8), 1587–1590 (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. 36(4), 629–643 (1991).
[Crossref]

Bellerby, R. G.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Booth, E.

W. Balch, H. Gordon, B. Bowler, and E. Booth, “Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectroradiometer Data,” J. Geophys. Res. 110(C7), C07001 (2005), doi:.
[Crossref]

Borges, A. V.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Bowler, B.

W. Balch, H. Gordon, B. Bowler, and E. Booth, “Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectroradiometer Data,” J. Geophys. Res. 110(C7), C07001 (2005), doi:.
[Crossref]

Boynton, G. C.

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]

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(8), 1587–1590 (2001).
[Crossref]

Bricaud, A.

Brown, C. W.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

Chou, L.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Delille, B.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Doney, S. C.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[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]

Falkowski, P. G.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

Fournier, G. R.

G. R. Fournier, V. Sanjuan-Calzado, and C. Trees, “Implications of a new phase function for autonomous underwater imaging,” Proc. SPIE VI, 911119 (2014).

Frankignoulle, M.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Freeman, S. A.

Gattuso, J. P.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Gordon, H.

W. Balch, H. Gordon, B. Bowler, and E. Booth, “Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectroradiometer Data,” J. Geophys. Res. 110(C7), C07001 (2005), doi:.
[Crossref]

Gordon, H. R.

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]

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

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]

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]

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(8), 1587–1590 (2001).
[Crossref]

Groom, S. B.

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(8), 1587–1590 (2001).
[Crossref]

Gruber, N.

C. J. O’Brien, M. Vogt, and N. Gruber, “Global coccolithophore diversity: Drivers and future change,” Prog. Oceanogr. 140, 27–42 (2016).
[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(8), 1587–1590 (2001).
[Crossref]

Harlay, J.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Harris, L. A.

Hayes, P. K.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

Holligan, P. M.

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. 36(4), 629–643 (1991).
[Crossref]

Hu, Y.

Iglesias-Rodríguez, M. D.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

Jacquet, S.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Josset, D. B.

Kattawar, G. W.

Kilpatrick, K.

K. Voss, W. Balch, and K. Kilpatrick, “Scattering and attenuation properties of Emiliania huxleyi cells and their detached coccoliths,” Limnol. Oceanogr. 43(5), 870–876 (1998).
[Crossref]

Kleypas, J.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

Kolber, D.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

Kolber, Z.

M. D. Iglesias-Rodríguez, C. W. Brown, S. C. Doney, J. Kleypas, D. Kolber, Z. Kolber, P. K. Hayes, and P. G. Falkowski, “Representing key phytoplankton functional groups in ocean carbon cycle models: Coccolithophorids,” Global Biogeochem. Cycles 16(4), 47 (2002).
[Crossref]

Lucker, P. L.

Morel, A.

Neeley, A. R.

O’Brien, C. J.

C. J. O’Brien, M. Vogt, and N. Gruber, “Global coccolithophore diversity: Drivers and future change,” Prog. Oceanogr. 140, 27–42 (2016).
[Crossref]

Paasche, E.

E. Paasche, “A review of the coccolithophorid Emiliania huxleyi (Prymnesiophyceae), with particular reference to growth, coccolith formation, and calcification-photosynthesis interactions,” Phycologia 40(6), 503–529 (2001).
[Crossref]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Riebesell, U.

B. Delille, J. Harlay, I. Zondervan, S. Jacquet, L. Chou, R. Wollast, R. G. Bellerby, M. Frankignoulle, A. V. Borges, U. Riebesell, and J. P. Gattuso, “Response of primary production and calcification to changes of pCO2 during experimental blooms of the coccolithophorid Emiliania huxleyi,” Global Biogeochem. Cycles 19(2), GB2023 (2005), doi:.
[Crossref]

Risovic, D.

D. Risović, “Two-component model of sea particle size distribution,” Deep-Sea Res. 40(7), 1459–1473 (1993).
[Crossref]

Sanjuan-Calzado, V.

G. R. Fournier, V. Sanjuan-Calzado, and C. Trees, “Implications of a new phase function for autonomous underwater imaging,” Proc. SPIE VI, 911119 (2014).

Schoenberg, E.

E. Schoenberg, “Handb.,” Astrophysik 2, 255 (1929).

Smyth, T. J.

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]

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(8), 1587–1590 (2001).
[Crossref]

Stramski, D.

Tarran, G. A.

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[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Scanning Electron Micrograph of Emiliania huxleyi (Dr. Jeremy Young, University College London, London, with permission)
Fig. 2
Fig. 2 Realistic model of an Emiliania huxleyi coccolith used by Zhai et al. [9]
Fig. 3
Fig. 3 Rough to smooth backscatter ratio as a function of index of refraction. The red line is the index of calcite relative to water used by Zhai et al. [9] in their model and for comparison the blue line is the index of silica (1.094) which forms diatom frustules
Fig. 4
Fig. 4 Backscatter efficiency of coccoliths (a) and coccolithophores (b). Points are the results from the exact ADDA code of Zhai et al. [9] with no absorption k = 0.0 (red) and with and absorption of k = 0.01 (black). The red lines are the corresponding results from the analytical model.
Fig. 5
Fig. 5 (a) Backscatter efficiency of a 2.5 micron diameter coccolith as a function of wavelength. (b) Backscatter efficiency of a 6.25 micron diameter coccosphere as a function of wavelength. Points are from the code of Zhai et al. [9] for coccospheres with no absorption k = 0.0 (red) and with an absorption of k = 0.01 (black). The red lines are the corresponding results from the analytical model.
Fig. 6
Fig. 6 Backscatter cross-section for (a) a coccolith population with mean diameter of 2.4 microns and with a size distribution standard deviation of 0.23 microns and for (b) a coccolithophore population with mean naked core diameter of 5.2 microns and an overlap factor of 2.The solid lines are theory. The green dots are the laboratory data of Voss et al. [5] and the red dot is the field data of Gordon et al. [23].
Fig. 7
Fig. 7 Compound bloom spectral signatures (a) for a core covered with a single layer of liths at the start of shedding, and (b) for a core with two layers of liths at the start of shedding. The coccolith size distribution was modeled as a shifted Gamma function with a mean diameter of 2.4 microns and a standard deviation of 0.23 microns.
Fig. 8
Fig. 8 Simple disk model of an Emiliania huxleyi coccolith used in this paper. The radii of both the distal and proximal sheets are equal to rm in the full Zhai et al. [9] model shown in Fig. 2. The sum of their respective thicknesses, td and tp, is equal to the total thickness tt of a single disk of radius rm that would contain the same volume of solid material as the actual complex Zhai coccolith shape. nd , np and kd, kp denote the real and imaginary part of the refractive index of both distal sheets, respectively, while ng and kg denote the index of the gap between the sheets.

Equations (41)

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σ ¯ bb = σ ¯ g [ Q bdiff + Q brefr + Q brefl ]= σ ¯ g [ Q bdiff + Q brefr +ω F brefl ]
σ ¯ bb = σ ¯ g [ ω F brefl ]
Q brefl = ω bsm ( 1R F rg )+ ω brg R F rg
r 1 r 0 =0.18, r 2 r 0 =0.46, r 3 r 0 =0.54, r m r 0 =0.50, d c r 0 =0.07, d h r 0 =0.18
r tot = r c + d h + d c =1.215 r 0
Δ w = λ 4 = 2π r tr 2N
x tr 2N = 1 4 x tr = N 2
R= x mxs 2 x tr 2 x mxs 2 x mns 2 , x mns x tr x mxs
R= 1 ( x tr x mxs ) 2 1 ( x mns x mxs ) 2 = 1 ( N/2 x mxs ) 2 1 ( x mns x mxs ) 2
x tr x mns R=1 x tr x mxs R=0
F rg = x mxs 2 x mns 2 x lith 2
ω bsm =2π π/2 π ( 1 4π ) i | r i ( θ ) | 2 sinθdθ
ω bi ( n )= 3 n 4 16 n 3 +12 n 2 1+2 ( 2 n 2 1 ) 3/2 6 ( n 2 1 ) 2
ω bi|| ( n )=[ ( 3ln16 )+ 37 40 ( n1 n+1 ) ] ω b
ω bsm = i ω bi ( n ) = i ω bi + ω bi|| 2
ω t =2π 0 π ( 1 4π ) i | r i ( θ ) | 2 sinθdθ
ω i = ( 3n+1 )( n1 ) 3 ( n+1 ) 2
ω i|| = 1 ( n 2 +1 ) 3 ( n 2 1 ) 2 { ( n 4 1 )( n 6 4 n 5 7 n 4 +4 n 3 n 2 1 ) + 2 n 2 [ ( n 2 1 ) 4 ln( n1 n+1 )+8 n 2 ( n 4 +1 )ln( n ) ] }
ω t = i ω i ( n ) = i ω i ( n )+ ω i|| ( n ) 2
p rg ( θ )= ω t ( 2 3 π 2 )( sinθθcosθ )
ω brg =( 5 6 ) ω t
ω brg ω bsm =( 5 6 ) ω t 6 ( n 2 1 ) 2 3 n 4 16 n 3 +12 n 2 1+2 ( 2 n 2 1 ) 3/2 [ 2 1+( 3ln16 )+ 37 40 ( n1 n+1 ) ]
σ ¯ bb = σ ¯ g ω bsm
σ ¯ bb = σ ¯ g ω bsm [ ( 1 F rg )+ ω brg ω bsm F rg ]
b bb b = Q bscat Q scat Q brefl Q scat Q brefl Q scat ( b bb b )
t t = V c ( r 0 ) π r m 2 = t d + t p t t = 2π( t d + t p ) λ
r r min p( r )=0,r> r min p( r )= β 2 ( r r min ) e β( r r min )
β= 2 σ , r min =μ 2 σ
r s = r c +( O vl 1 )Δ r l
Δ r l = d h + d c =0.25 r o
σ bloom ( λ )=(1 F s ) O vl Q bs ( λ ) σ gs + N l F s Q bl ( λ ) σ ¯ gl + F s Q bc σ gc
σ gs =π r s 2 =π [ r c +( O vl 1 )Δ r l ] 2
σ ¯ gl =( 1 4 )( 2π r m 2 +2π r m t t )=( π r m 2 2 )( 1+ f t )
N l = O vl 2 ( 1 1 ( r m r s ) 2 )
n eff =1+ ( n1 ) 2 + k 2
ω t = ω 1 ( n d , k d )+ ω 2 ( n d , k d )( 1 ω 1 ( n d , k d ) ) ( 1 Q abs ( k d , t d ) ) 2 + ω 3 ( n d , k d )( 1 ω 1 ( n d , k d ) )( 1 ω 2 ( n d , k d ) ) ( 1 Q abs ( k d , t d ) ) 2 ( 1 Q abs ( k g , t g ) ) 2 + ω 4 ( n d , k d )( 1 ω 1 ( n d , k d ) )( 1 ω 2 ( n d , k d ) )( 1 ω 3 ( n d , k d ) ) ( 1 Q abs ( k d , t d ) ) 2 ( 1 Q abs ( k g , t g ) ) 2 ( 1 Q abs ( k p , t p ) ) 2
ω bsm = ω b1 ( n d , k d )+ ω b2 ( n d , k d )( 1 ω b1 ( n d , k d ) ) ( 1 Q abs ( k d , t d ) ) 2 + ω b3 ( n d , k d )( 1 ω b1 ( n d , k d ) )( 1 ω b2 ( n d , k d ) ) ( 1 Q abs ( k d , t d ) ) 2 ( 1 Q abs ( k g , t g ) ) 2 + ω b4 ( n d , k d )( 1 ω b1 ( n d , k d ) )( 1 ω b2 ( n d , k d ) )( 1 ω b3 ( n d , k d ) ) ( 1 Q abs ( k d , t d ) ) 2 ( 1 Q abs ( k g , t g ) ) 2 ( 1 Q abs ( k p , t p ) ) 2
Q abs =1+2 n 2 [ ( n 2 1 n 2 ) E 3 ( 2k 2πt λ n n 2 1 ) E 3 ( 2k 2πt λ ) ]
Q bs = Q bl + Q bl [ 1 ( r c r s ) 2 Q abssphere ( k c ,2 r c ) ]
Q abssphere ( k c , r c )=2[ 1 2 + e δ c ( λ ) r c δ c ( λ ) r c + ( e δ c ( λ ) r c 1 ) ( δ c ( λ ) r c ) 2 ]
δ c ( λ )=4k(λ) 2π λ

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