## Rayleigh-Gans scattering approximation: surprisingly useful for understanding backscattering from disk-like particles

Optics Express, Vol. 15, Issue 9, pp. 5572-5588 (2007)

http://dx.doi.org/10.1364/OE.15.005572

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### 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

## 1. Introduction

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 , **51**1–3 (2004). [CrossRef]

*out*of the water. This radiance is proportional to the ratio of the backscattering coefficient

*b*(the differential scattering cross section per unit volume integrated over the backward

_{b}*hemisphere*) and the absorption coefficient

*a*of the medium (water plus constituents) [1

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]

*b*. Thus, understanding the backscattering coefficient of the suspended constituents of the natural waters is a central problem in marine optics. However, the backscattering coefficient of marine particles is arguably the poorest known of the inherent optical properties of natural waters [3

_{b}/a3. 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]

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]

*E. huxleyi*. This particular marine particle was chosen for study because (1) its shape is rather precisely known, resembling a disk or two roughly parallel disks; (2) its composition is known (Calcite), providing its refractive index relative to water (∼1.20); (3) its backscattering properties have been measured [6–8

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]

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 **II**, 1491–1499 (1994). [CrossRef]

*E. huxleyi*coccoliths. When the scale of the periodicity (the length of an open or closed sector measured along the circumference of the disk) was <

*λ*/4 (where

*λ*is the wavelength of the light

*in*the medium, i.e., water), I found the backscattering to be nearly identical to that of a homogeneous disk possessing a reduced refractive index. However, significant increases in backscattering were observed when the scale of the periodicity was greater than

*λ*/4.

*m*) is close to unity, and the “size” is ≪ the wavelength of light divided by |

*m*- 1| [14, 15]. Thus the size need not be ≪ the wavelength. It is computationally fast when compared to any other method because analytical formulas are available for many particle shapes. Moreover, extension to particles of any shape is straightforward.

*E. huxleyi*. The validity of the RGA for such particles allows investigation of the influence of their birefringence on backscattering.

## 2. The Electromagnetic scattering problem

_{E⃗(0)(D⃗i,t)}, then a volume element

*dV*at a position

_{i}*within the particle (Fig. 1) will experience an electric field*

_{D⃗t}_{E⃗(D⃗i,t)}given by

*excludes i=j*. The

*E*’s on both sides of this equation are unknown, while

*E*

^{(0)}and

*C*are known functions of position and time. This electric field induces a dipole moment (

*dp*) in

*dV*given by

_{i}**α**is the polarizability tensor and

*ρ*is the number density of atoms (molecules). At a great distance

_{n}*r⃗*from the particle the field due to the dipole moment induced in

_{i}*dV*is

_{i}_{|κ⃗| = 2π/λ}, where

*λ*is the wavelength of the incident field in the medium in which the particle is immersed, and |κ⃗| = |κ⃗

_{0}|) and

*c*is the speed of light. The vector

*r⃗*is assumed to be sufficiently far from the origin (

_{i}*O*) that it may be replaced by

*r⃗*except where it occurs in a phase. The total field at

*r⃗*, given by

*x,y,z*), the incident electric field propagating in the κ⃗

_{0}direction is given by

*E⃗*

^{(s)}into components parallel and perpendicular to the scattering plane as well (note that

*ê*

^{(0)}

_{l}and

*ê*, are not parallel), the scattered field at

_{l}*r̂*, which is in the form of a spherical wave, can then be written

**A**is the 2×2 scattering amplitude matrix, and

## 3. The differential scattering cross section

^{*}indicates the complex conjugate, the tilde indicates the transposed matrix and,

*dP*is the power crossing an area

*dA*oriented normal to the propagation direction κ⃗ (i.e., the

*irradiance*associated with the propagating field). The differential scattering cross section is defined to be the power scattered into a solid angle

*d*Ω. divided by the irradiance of the incident beam, i.e.,

*s*” stands for “scattered” and the superscript “0” stands for “incident.” The required Poynting vectors are given by

## 4. The Rayleigh-Gans approximation

*C*= 0), i.e., the only field experienced by

*dV*is the incident field. Thus, the RGA provides the “zeroth-order” approximation to the scattered field. In the laboratory reference frame (

_{i}*x,y,z*), the incident electric field is given by Eq. (5) so the induced dipole moment [Eq. (2)] is

_{κ = ω/c}, we have

_{riω/c = κri = κr-κ⃗●D⃗i},

*d*

**A**

_{i}is the contribution to the matrix

**A**from

*dV*and is given by

_{i}*x-z*plane (Fig. 2), so

*d*

**A**

_{i}yields

*α*’s are the components of

**α**in the laboratory reference frame (

*x,y,z*). The total scattered field is found by integration over the volume of the object:

*α*=

_{ij}*α δ*) then

_{ij}**A**reduces to

*V*. If the particle is immersed in a refracting medium, then

*m*is the refractive index of the particle

*relative*to the medium.

*σ*) and back (

*σ*) scattering cross sections are, respectively,

_{b}*N*is the number density of scatterers. The contribution that particles of a given size and shape make to the total scattering coefficient (

*b*) and the backscattering coefficient (

*b*) are

_{b}*b*=

*Nσ*and

*b*=

_{b}*Nσ*, respectively. In the RGA, the shape of the particle enters only through the computation of

_{b}*R*. Analytic formulas are available for simple shapes, e.g., spheres and cylinders; however, it is easy to carry out the integrations numerically for particles of any shape. For particles other than spheres,

*R*depends on the orientation of the particle. For particles with a given orientational distribution function,

*dσ/d*Ω. must be computed for a large number of orientations and the appropriate weighted average formed.

*dV*is subjected only to the incident field requires that two conditions must hold for the RGA to have validity: (1) there must be insignificant refraction or reflection at the surface of the particle, which implies |

_{i}*m*- 1| must be ≪ 1; and (2) the phase of the incident field must not shift significantly over distances of the order of the “size” (

*L*) of the particle, which requires

*κL*|

*m*- 1|≪1.

## 5. Comparison between RGA and DDA for disk-like particles

*σ*) of

_{b}*randomly-oriented*disk-like objects computed via the RGA and the DDA. The DDA results are taken as “exact” computations (the DDA-computed

*σ*’s are expected to be in error by no more than 5%). As an early motivation for such comparisons was interest in the backscattering of coccoliths detached from

_{b}*E. huxleyi*suspended in water, I consider disks with diameters 1.5 to 2.75 μm with

*m*= 1.2 (Calcite in water). Figure 3 provides such a comparison for a 2.75 μm homogeneous disk of various thicknesses (

*t*). The comparison shows that the RGA is close to the DDA for

*t/λ*

_{Water}less than, or approximately equal to, 0.20 to 0.25. Perhaps more importantly, the comparison shows that

*σ*can be expected to oscillate with increasing

_{b}*t*(or decreasing

*λ*

_{Water}), i.e., the RGA also provides the

*qualitative character*of the spectral variation of

*σ*. It should be pointed out that the (approximate) “physical optics” developed by model of Gordon and Du [5

_{b}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]

*t/λ*

_{Water}> 0.2, following the DDA reasonably well up to a

*t/λ*

_{Water}of 0.8.; however, it cannot be applied to the more complex particle shapes of interest here, e.g., disk-like particles with periodic angular fine structure.

*σ*for the Gordon and Du [5

_{b}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]

*D*

_{o}with material removed from a concentric circle of diameter

*D*(i.e., a washer-like object). The two disks are joined together by a hollow cylinder of inner diameter

_{i}*D*and outer diameter

_{i}*D*. The axis of the cylinder passes through the center of both disks. The individual disks have a thickness of 50 nm and the space between them (the height of the joining cylinder) is

_{r}*t*. Table 1 provides values of the parameters of the three fishing-reel models investigated. The three models all have the same volume (∼0.587 μm

^{3}). This is accomplished by decreasing the thickness of the wall of the connecting cylinder as shown in Table 1.

*t/λ*

_{Water}< 0.2 criterion from Fig. 3 in the visible. As in Fig. 3, Fig. 4 shows that the RGA and DDA produce

*qualitatively similar*spectral variations, and surprisingly good quantitative agreement even though the total thickness of the particle exceeds

*λ*

_{Water}in some cases, and the total diameter is several times

*λ*

_{Water}. Comparison of the two suggests that the RGA can be a valuable tool in exploring problems involving multiple disks as long as the

*individual disks*satisfy the

*t/λ*

_{Water}< 0.2 criterion.

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]

*α*and alternate sectors were removed. The angle Δ

*α*was given by

*n*is an integer. Figure 5 provides the positions of one layer of dipoles for the resulting structures for

*n*= 4 to 7. I will refer to these objects as “pinwheels.” If we let s be the arc length of the open (or closed) regions at the perimeter of the pinwheel, then

*s*=

*D*Δ

_{d}*α*/2, where

*D*is the diameter of the disk. The values of

_{d}*s*for the various cases that I examined (

*D*= 1.5 μm) were such that at a wavelength (

_{d}*λ*) of 400 nm in vacuum (300 nm in water), as

*n*progresses from 4 to 7,

*s*took on the values

*λ*,

*λ*/2,

*λ*/4, and

*λ*/8 in water. One of the main goals of my study was to determine if a relationship exists between

*s*and

*λ*where the periodic structure becomes important (or unimportant) to the backscattering.

*σ*, carried out for 1.5 μm pinwheels are provided in Fig. 6 (DDA on the left from Ref 11, and RGA on the right), which displays

_{b}*σ*, as function of the thickness (

_{b}*t*) of the disk divided by the wavelength of the light

*in water*(

*λ*

_{Water}). Three thicknesses of the disk are used: 0.05, 0.10, and 0.15 μm. The wavelength

*λ*

_{Water}covers the range from 200 nm to over 1000 nm. Note the qualitative similarity between the DDA and the RGA computations. Both show that the backscattering appears to follow a “universal curve” that is close to that for a homogeneous disk with a reduced index

*m*= 1.10 rather than 1.20 (labeled 1.10 in the key to the figure); however, as the wavelength decreases

*σ*, suddenly departs from the universal curve and increases dramatically. This was first observed through extensive computations using the DDA; however, in this case, the behavior could have been predicted based on the RGA computations. (The departure of

_{b}*σ*, from the universal curve occurs when the maximum arc length of the open or closed regions of the pinwheel exceeds

_{b}*λ*

_{Water}/4.). In Fig. 7 the comparisons in Fig. 6 are carried to larger values of

*t/λ*

_{Water}, and show that the RGA agrees well with the DDA for values of

*t/λ*

_{Water}up to, and somewhat beyond the first maximum that occurs in

*σ*

_{b}*after*the departure from the “universal curve.” This maximum is near

*s/λ*

_{Water}= 1/2. For larger values of

*t/λ*

_{Water}the RGA still provides the qualitative nature of the variation of

*σ*with

_{b}*t/λ*

_{Water}; however, it no longer quantitatively reproduces the DDA computations.

*Emiliania huxleyi*,” Limnol. Oceanogr. **46**, 1438–1454 (2001). [CrossRef]

*n*= 5 and 6 rather than homogeneous disks and the outside diameter is 1.50 μm rather than 2.75 μm. Again, the agreement between RGA and DDA is quite good throughout the visible, even in a

*quantitative*sense.

## 6. Application: estimate of the influence of *E. huxleyi* birefringence on backscattering

*E. huxleyi*coccolith is composed of calcite, one would expect it to be birefringent. This is indeed the case. The c-axis (optical axis) of the component parts of the

*E. huxleyi*coccolith is radial, i.e., along the “spoke-like” structures [19

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]

*t/λ*

_{Water}< 0.2, which is satisfied by the individual coccolith plates throughout the visible, we expect that it would apply equally well to a birefringent disk. Thus, we will investigate the possible influence of birefringence on

*E. huxleyi*backscattering by comparing the backscattering in the RGA of a birefringent and an isotropic disk. Computation of the scattering matrix

**A**for an anisotropic disk, for which the optical axis at any point is radial, is sketched out in the Appendix. A uniaxial crystal, Calcite has two refractive indices:

*m*for propagation with the electric vector parallel to the c-axis; and

_{e}*m*for propagation with the electric vector perpendicular to the c-axis. Letting

_{o}*m*represent the refractive index of the isotropic disk, we take

_{i}*a*and

*b*(see the Appendix) of the birefringent disk, and

*m*must depend on

_{i}*m*and

_{o}*m*in some manner, and one of the goals of this exercise is to find the combination that provides the best agreement for backscattering of the isotropic and the anisotropic cases. If the disk were composed of small grains of Calcite in random orientation, one would expect [20

_{e}20. E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. **18**, 2223–2249 (1996). [CrossRef]

*m*+

_{o}*m*)/3. Using tabulated values for the refractive indices of Calcite [21] near 500 nm and taking 1.338 for the index of water, we have

_{e}*m*= 1.241,

_{o}*m*= 1.113, and

_{e}*m*= 1.198 (close to the value 1.20 used in the earlier computations). Comparison of the RGA-computed

_{i}*σ*, for the radially-anisotropic disk with these values of

_{b}*m*and

_{o}*m*with the isotropic disk with index

_{e}*m*= (2

_{i}*m*+

_{o}*m*)/3 is provided in Fig. 9. The isotropic disk’s backscattering cross section is seen to be somewhat higher than the birefringent disk. The two can be brought into agreement by taking

_{e}*m*= 1.188 = 0.57

_{i}*m*+ 0.43

_{o}*m*. (For the total scattering cross section,

_{e}*m*= (2

_{i}*m*+

_{o}*m*)/3 provides better agreement between the two than

_{e}*m*= 1.188.) This suggests that for the computation of backscattering by model coccoliths with the DDA, a zeroth-order account of the birefringence can be effected by using

_{i}*m*= 1.188 rather than 1.198.

_{i}## 7. Concluding Remarks

*σ*, for disk-like particles. Rather, because it is computationally fast compared to the DDA, it can be used to

_{b}*explore*the backscattering of disk-like models of marine particles for the purpose of either excluding models with unacceptable qualitative behavior, or selecting promising models for further study using the more time-consuming DDA.

## 8. Appendix: scattering by a birefringent disk

*E. huxleyi*coccoliths we take the disk to be uniaxial with the optical axis at any point in the disk in the radial direction. We develop the anisotropic case first and then reduce these formulas to the isotropic case.

## A. Anisotropic case

*the angle*ρ ′

*and the coordinate*η ′

*z′*normal to the axis of the disk. In the integral for

**A**, Eq. (15), the required elements of the polarizability matrix (

*α*,

_{yy}*α*,

_{xy}*α*,

_{xz}*α*, and

_{xy}*α*) must be provided in laboratory-fixed reference system and depend on the particle’s orientation. However, in the body-fixed reference system the polarizability matrix assumes a particularly simple form:

_{xx}*ψ*is redundant and may be set to zero. The matrix elements of

**α**thus depend on

*θ*,

*ϕ*, and

*.*η ′

**A**, we need

_{D⃗ = ρ⃗′ + z⃗′}, and by resolving

_{(κ⃗0 - κ⃗)}into components parallel and normal to the disk’s surface, we find

*γ*is the angle between the component of

_{(κ⃗0 - κ⃗)}parallel to the plane of the disk and the

*x*’axis. A typical integral that must be evaluated to find

**A**is then

**α**are:

*θ*and

*ϕ*), all of the required integrals are of the form

*integrals, and this integrates to*η ′

*α*′s with

_{ij}*θ*and

*ϕ*as well as through

*k*′.

## B. Isotropic case

*a*=

*b*, then

*α*=

_{ij}*aδ*, and the scattering amplitude matrix becomes

_{ij}*k′*

## Acknowledgments

## References and links

1. | H. R. Gordon and A. Y. Morel, |

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 , |

3. | D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. |

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 |

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. |

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. |

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. |

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. |

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. |

11. | H. R. Gordon, “Backscattering of light from disk-like particles: is fine-scale structure or gross morphology more important?” Appl. Opt. |

12. | B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. |

13. | B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

14. | C. F. Bohren and D. R. Huffman, |

15. | H. C. van de Hulst, |

16. | L. D. Cohen, R. D. Haracz, A. Cohen, and C. Acquista, “Scattering of light from arbitrarily oriented cylinders,” Appl. Opt. |

17. | K. Shimizu, “Modification of the Rayleigh-Debye approximation,” J. Opt. Soc. Am. |

18. | B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. |

19. | J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, “Crystal assembly and phylogenetic evolution in heterococcoliths,” Nature |

20. | E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. |

21. | J. M. Bennett and H. E. Bennett, “Polarization,” in |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(290.1350) Scattering : Backscattering

**ToC Category:**

Atmospheric and ocean optics

**History**

Original Manuscript: March 20, 2007

Revised Manuscript: April 19, 2007

Manuscript Accepted: April 19, 2007

Published: April 23, 2007

**Citation**

Howard R. Gordon, "Rayleigh-Gans scattering approximation: surprisingly useful for understanding backscattering from disk-like particles," Opt. Express **15**, 5572-5588 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5572

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### References

- H. R. Gordon and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983). [CrossRef]
- 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]
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis (Cambridge, 2002).
- 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]
- 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]
- 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, "Backscattering of light from disk-like particles: is fine-scale structure or gross morphology more important?" Appl. Opt. 45, 7166-7173 (2006). [CrossRef] [PubMed]
- B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333,848-872 (1988). [CrossRef]
- B. T. Draine and P. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A ll,1491-1499 (1994). [CrossRef]
- 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).
- 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]
- K. Shimizu, "Modification of the Rayleigh-Debye approximation," J. Opt. Soc. Am. 73, 504-507 (1983). [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]
- 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]
- E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223-2249 (1996). [CrossRef]
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