## Inherent optical properties of the coccolithophore: Emiliania huxleyi |

Optics Express, Vol. 21, Issue 15, pp. 17625-17638 (2013)

http://dx.doi.org/10.1364/OE.21.017625

Acrobat PDF (1865 KB)

### Abstract

A realistic nonspherical model for Emiliania huxleyi (EHUX) is built, based on electron micrographs of coccolithophore cells. The Inherent Optical Properties (IOP) of the EHUX are then calculated numerically by using the discrete dipole approximation. The coccolithophore model includes a near-spherical core with the refractive index of 1.04 + *m _{i}j*, and a carbonate shell formed by smaller coccoliths with refractive index of 1.2 +

*m*, where

_{i}j*m*= 0 or 0.01 and

_{i}*j*

^{2}= −1. The reported IOP are the Mueller scattering matrix, backscattering probability, and depolarization ratio. Our calculation shows that the Mueller matrices of coccolithophores show different angular dependence from those of coccoliths.

© 2013 OSA

## 1. Introduction

*CaCO*

_{3}production which responds to

*CO*

_{2}partial pressure change [1

1. J. D. Milliman, “Production and accumulation of calcium in the ocean,” Global Biogeochem. Cy. **7**, 927–957, (1993) [CrossRef] .

2. S. Ackleson, W. M. Balch, and P.M. Holligan, “Response of water leaving radiance to particulate calcite and chlorophyll a concentrations: A model for Gulf of Maine coccolithophore blooms,”J. Geophys. Res. **99**, 7483–7499, (1994) [CrossRef] .

3. A. Sadeghi, T. Dinter, M. Vountas, B. Taylor, M. Altenburg-Soppa, and A. Bracher, “Remote sensing of coccolithophore blooms in selected oceanic regions using the PhytoDOAS method applied to hyper-spectral satellite data,” Biogeoscience **9**, 2127–2143, (2012) [CrossRef] .

## 2. Numerical model for coccoliths and coccolithophores

### 2.1. Coccolith model in a local coordinate system

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

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

**r**= (

*x,y,z*). The coccolith model has a symmetry axis (the broken blue line in Fig. 1(a)), which passes through the origin

*O*of this local coordinate system. The symmetry axis is coincident with the

*z*-axis of the local coordinate system. A coccolith with its axis pointing in an arbitrary direction will be obtained by rotating the original coccolith using Euler’s rotation theorem in the following section. In this paper we also use a polar coordinate system within which a point is denoted as

**r**= (

*r*,

*θ*,

*ϕ*), where

*x*=

*r*sin

*θ*cos

*ϕ*,

*y*=

*r*sin

*θ*sin

*ϕ*,

*z*=

*r*cos

*θ*. The length of the projection of

**r**on the

*x*−

*y*plane is

*V*so that any point

**r**belongs to

*V*, i.e.,

**r**∈

*V*.

*V*, which consists of three parts (regions): an inner spherical cap

_{c}*VI*, an outer spherical cap

*VII*, and a hollow cylinder connecting the two caps

*VIII*. The spherical caps are parts of concentric spherical shells, with the center of the spherical shells defined as the coordinate origin

*O*. Figure 1(b) shows a sketch of a cross section of the coccolith in a half plane containing the origin and axis. The radius and thickness of the inner cap

*VI*are

*r*

_{0}and

*d*, respectively. The hollow cylinder has an inner radius of

_{c}*r*

_{1}. The thickness of the cylinder is also

*d*. The height of the cylinder

_{c}*VIII*is

*d*, which is defined as the distance between the two centers where two caps intersect with the cylinder in Fig. 1(b). The thickness of the outer spherical cap is assumed to be the same as the inner cap. The projection of the inner or outer spherical cap in the x–y plane is shown in Fig. 1(c). It is fairly obvious that this projection is very similar to the pin-wheel model by Gordon (see Fig. 3 in [21

_{h}21. H. R. Gordon, “Backscattering of light from disklike particles: is fine-scale structure or gross morphology more important?” Appl. Opt. **45**, 7166–7173 (2006) [CrossRef] [PubMed] .

*N′*= 40, and

*N*= 2 *

*N′*, we have the following definitions:

*VI*that the inner spherical cap occupies can be defined in the following way: where ∩ and ∪ are used to show the intersection and union of two space volumes, respectively.

*VIII*, is defined as:

### 2.2. Coccolith model in a global coordinate system

**M**, we assume that

**r̂**

*= (sin*

_{c}*θ*cos

_{c}*ϕ*, sin

_{c}*θ*cos

_{c}*ϕ*, cos

_{c}*θ*) is the unit vector in the global coordinate system pointing in the direction of the coccolith axis. The local x-axis could be defined as

_{c}**ẑ**

*= (0, 0, 1)*

_{g}*is the unit vector of the z-axis of the global coordinate system; and × denotes the vector cross product. The local y axis is then*

^{T}**r̂**

*is the local z-axis expressed in the global coordinate system. The transformation matrix is written as: because this form of*

_{c}**M**can transform (1, 0, 0)

*, (0, 1, 0)*

^{T}*, and (0, 0, 1)*

^{T}*into*

^{T}**M**

^{−1}is the inverse matrix of

**M**.

### 2.3. Coccolithophore model

*m*of the spherical core of the symmetrical coccolithophore is given by: where

*m*is 0 and 0.01 in our study. These values are relative to pure water and consistent with [18

_{i}18. H. Volten, J.F. de Haan, J.W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. **43**, 1180–1197 (1998) [CrossRef] .

*m*= 0.01 is used to study the impacts of absorption on the IOP of coccoliths and coccolithophores.

_{i}*θ*and

_{c}*ϕ*. They are evenly distributed on and cover the surface of the spherical core with the radius of

_{c}*r*

_{0}. The number of

*θ*values is: as we intend to cover the zenith angle range from 0 to

_{c}*π*with coccoliths whose half angle span is

*θ*

_{3}. The

*NINT*() operator takes the integer part of the enclosed real number. The

*θ*values are: For each value of

_{c}*θ*, the number of

_{c}*ϕ*values are given by: The

_{c}*ϕ*is: For each pair of

_{c}*θ*and

_{c}*ϕ*values, the coccolith domain is defined by the two step procedure described in Sec 2.2. If a point is in the coccolith domains, the refractive index is specified as: which is defined relative to water [21

_{c}21. H. R. Gordon, “Backscattering of light from disklike particles: is fine-scale structure or gross morphology more important?” Appl. Opt. **45**, 7166–7173 (2006) [CrossRef] [PubMed] .

23. H. R. Gordon, “Rayleigh-Gans scattering approximation: surprisingly useful for understanding backscattering from disk-like particles,” Opt. Express **15**, 5572–5588 (2007) [CrossRef] [PubMed]

24. 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**, 6059–6073 (2009) [CrossRef] [PubMed]

*r′*

_{0}=

*r*

_{0}+

*d*+ 2

_{h}*d*is used to replace

_{c}*r*

_{0}from Eq. (1) to (5). In addition, the condition of is used to determine whether a coccolith in the extra outer layer should be kept before we put it in the model, where

*η*is a random number uniformly distributed in the interval of [0, 1]. This will effectively reduce the number of coccoliths by half in the additional layer of the carbonate shell. After the extra layer of coccoliths is added, we further compress the z coordinate values of all the points in the global space by 80%:

*z′*= 0.8

*z*. The coefficient of 0.8 is determined by the ratio of the smallest and largest dimensions of the SEM shown in Fig. 1 of [21

**45**, 7166–7173 (2006) [CrossRef] [PubMed] .

*r*

_{1}/

*r*

_{0}= 0.18,

*r*

_{2}/

*r*

_{0}= 0.46,

*d*

_{c}/r_{0}= 0.07, and

*d*

_{h}/r_{0}= 0.18. The size of the particle is then primarily determined by

*r*

_{0}. We have used

*r*

_{0}= 1.5, 1.8, 2.1, 2.4, and 2.7

*μm*in the simulation of light scattering by coccolithophores to study the effect of different sizes on the IOP. The maximum diameters of coccolithophores corresponding to these

*r*

_{0}values are roughly:

*D*= 2(

*r*

_{0}+

*d*+

_{h}*d*) = 3.75, 4.50, 5.25, 6.00 and 6.75

_{c}*μm*. The values of

*r*

_{1},

*r*

_{2},

*d*, and

_{c}*d*could be set to vary independently if future research finds better statistical representations for them. For the case of coccoliths, we have studied the cases of

_{h}*r*

_{0}= 3.0, 3.3, 3.6, 3.9, 4.2, and 4.5

*μm*, in additional to those used for coccolithophore cases. The diameters of the corresponding coccolith projections (see Fig. 1(c)) range from roughly 1.63

*μm*(for

*r*

_{0}= 1.5

*μm*) to 4.73

*μm*for (

*r*

_{0}= 4.5

*μm*). Note that the values of

*r*

_{0}> 3.0

*μm*are probably higher than a typical value suggested by SEM images, which are used here for theoretical considerations. For light scattering by coccolithophores,

*r*

_{0}> 3.0

*μm*requires too much computational resource which are ruled out in this study.

*x*= 2

*πr*to denote the particle size, where

_{v}/λ*λ*is the wavelength;

*r*= [3

_{v}*V*/(4

_{e}*π*)]

^{1/3}is the volume equivalent radius; and

*V*is the total geometric volume occupied by the particles. The usage of size parameters is motivated and justified by the scale invariance rule, i.e., the scattering properties of a particle depend on its relative size to the wavelength, not on its absolute size (see Sec. 5.8.2 of [27]), if the morphology is the same. It is evident that for particles with structures as diverse as detached coccoliths, symmetrical and asymmetrical coccolithophores, the volume equivalent size parameter is only one of many parameters which influence the particles interaction with the electromagnetic field. However, as its definition is valid for all the structures, it can be used to obtain a simplified but unified representation of the different particles.

_{e}## 3. Numerical methods and IOP

26. M.A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer **112**, 2234–2247 (2011) [CrossRef] .

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

32. P. Zhai, G. W. Kattawar, P. Yang, and C. Li, “Application of the symplectic finite-difference time-domain method to light scattering by small particles,” Appl. Opt. **44**, 1650–1656 (2005) [CrossRef] [PubMed]

33. M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer **111**, 650–658. (2010) [CrossRef]

26. M.A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer **112**, 2234–2247 (2011) [CrossRef] .

34. M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express **15**, 17902–17911 (2007) [CrossRef] [PubMed] .

*λ*/12 along each dimension, where

*λ*= 0.532

*μm*. Each grid is equivalent to a dipole and the coordinates of its center are used to pass the test described above to judge if it is occupied by the core, the carbonate shell, or empty. The refractive index is given to the dipole according to the region where it belongs. The coordinates and refractive indices of the dipoles are written into the input files of the ADDA code. Random orientation averaging is applied to the IOP calculated by the ADDA. The ADDA uses three Euler angles to define orientations of a scattering particle. Orientation averaging over one Euler angle is equivalent to rotating the scattering plane, which can be done with a single internal field calculation. Averaging over the other two Euler angles is performed by Romberg integration in an adaptive iterative mode, i.e., only a necessary number of orientation angles have to be evaluated. The termination of orientation averaging is based on estimations of relative errors of scattering cross sections. We have used the criteria of EPS=0.001 to perform the random orientation averaging, where EPS is the maximum relative error of the scattering cross sections. In general, the numbers of orientations are around 160, 120, and 220, for each size of detached coccolith, symmetrical and asymmetrical coccolithophores, respectively. Readers are referred to [26

26. M.A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer **112**, 2234–2247 (2011) [CrossRef] .

35. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles*(Wiley, 1998) [CrossRef] .

*I*,

*Q*,

*U*,

*V*are the Stokes parameters; the subscripts

*i*and

*s*denote the incident and scattered light, respectively;

*k*= 2

*π/λ*. For a mirror symmetric particle, the scattering matrix is block diagonal after random orientation averaging, i.e.,

*S*

_{13}=

*S*

_{14}=

*S*

_{23}=

*S*

_{24}=

*S*

_{31}=

*S*

_{32}=

*S*

_{41}=

*S*

_{42}= 0 [27]. Strictly speaking the coccolith and coccolithophore model described in this paper do not have mirror symmetry. However, it is somewhat close to a mirror symmetric particle. The calculation result confirms that the off block diagonal elements are virtually zero (

*S*

_{ij}/S_{11}< 10

^{−7}) for the detached coccoliths, and symmetrical coccolithophores. The off block diagonal elements

*S*

_{ij}/S_{11}of asymmetrical coccolithophores are smaller than 0.1 for all the scattering angles. They will not be presented in the following. In addition, we have confirmed that the simulated scattering matrix elements satisfy the following inequality for all scattering angles and all particles [36

36. E. S. Fry and G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. **20**, 2811–2814, (1981) [CrossRef] [PubMed] .

*C*from

_{scat}*S*

_{11}, which is defined as: so that Note that in the atmospheric radiative transfer community, the term phase function is used to denote

*p′*=

*p** 4

*π*. We have adopted the convention in Eq. (19) to be consistent with the ocean optics community [6]. It is also convenient to define and study the scattering efficiency in the light scattering theory: where

*r*is the volume equivalent radius in this application. Moreover, the term “reduced scattering matrix” is often used to denote the scattering matrix divided by

_{v}*S*

_{11}.

*B*and the depolarization ratio

_{bp}*δ*: where Θ is the scattering angle. The backscattering probability

_{L}*B*has important applications in ocean optics [6, 7], while

_{bp}*δ*is important for particle categorization in lidar applications [37

_{L}37. Y. Hu, S. Rodier, K. Xu, W. Sun, J. Huang, B. Lin, P. Zhai, and D. Josset, “Occurrence, liquid water content, and fraction of supercooled water clouds from combined CALIOP/IIR/MODIS measurements,” J. Geophys. Res. **115**, D00H34, (2010) [CrossRef] .

*B*and

_{bp}*δ*are studied in the next section.

_{L}## 4. Results

*m*= 0 and

_{i}*m*= 0.01 are used. In addition to simulation results, the reduced matrix elements of the average ocean water from Voss and Fry [14

_{i}14. K. J. Voss and E. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. **23**, 4427–4439 (1984) [CrossRef] [PubMed] .

*m*= 0 and

*m*= 0.01 are small, especially for small sizes of coccoliths. The reduced scattering matrix elements do show similar features to those of the average ocean water. However, they are different in some ways. The two elements

*S*

_{12}/

*S*

_{11}and

*S*

_{33}/

*S*

_{11}are closer to those of Voss. The differences in

*S*

_{22}/

*S*

_{11}are larger. Note that

*S*

_{22}/

*S*

_{11}is always 1 for spherical particles so the deviation of

*S*

_{22}/

*S*

_{11}from 1 is an indicator of nonsphericity. Our coccolith model is highly nonspherical which is the reason that

*S*

_{22}/

*S*

_{11}is smaller than 1 except for Θ = 0. The

*S*

_{22}/

*S*

_{11}values for different sizes of coccoliths all shows smallest values around 90° to 120°, consistent with the measurements by Voss and Fry. However, there is no exact match, which is not surprising because the measurement in [14

14. K. J. Voss and E. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. **23**, 4427–4439 (1984) [CrossRef] [PubMed] .

*π*/2. Different from coccoliths, the reduced scattering matrix elements for coccolithophores show sharp peaks around backscatter. This suggests that it is difficult to extrapolate measurements towards 180°. A similar feature has been observed in the scattering matrix elements of large aerosol particles (see Fig. 3 in [38

38. M. Kahnert, T. Nousiainen, M. A. Thomas, and J. Tyynelä, “Light scattering by particles with small-scale surface roughness: Comparison off our classes of model geometries,” J. Quant. Spectrosc. Radiat. Transfer **113**, 2356–2367, (2012) [CrossRef] .

*S*

_{22}/

*S*

_{11}shift to larger scattering angles (Θ around 160°) for coccolithophores. This may mean that the average ocean water scattering matrix contains contributions largely from smaller particles. The contribution of large particles could be more significant for coccolithophore blooms, however. Svensen et al. have reported the measurement of the scattering matrix for EHUX (see Fig. 6 in [17

17. Ø. Svensen, J. J. Stamnes, M. Kildemo, L. Martin, S. Aas, S. R. Erga, and Ø. Frette, “Mueller matrix measurements of algae with different shape and size distributions,” Appl. Opt. **50**, 5149–5157 (2011) [CrossRef] [PubMed] .

*S*

_{12}and

*S*

_{33}is similar to what they have shown. However, it is hard to do a more quantitative comparison because the numerical values of the measurement are not available to us. In addition, they have assumed that EHUX is a spherical particle so (

*S*

_{33}+

*S*

_{44})/2 is used to represent

*S*

_{33}, which does not apply to our nonspherical model. Figure 6 shows the difference between

*S*

_{33}/

*S*

_{11}and

*S*

_{44}/

*S*

_{11}for symmetrical coccolithophores. In the forward scattering direction, the difference is small. However, they show large differences in the backscattering region, which is also an indicator of nonsphericity, in addition to the

*S*

_{22}element.

*Q*,

_{scat}*B*, and

_{bp}*δ*all decrease as the imaginary refractive index increases. Spherical particles also have similar trend for

_{L}*Q*[39

_{scat}39. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. **16**, 527–610, (1974) [CrossRef] .

*x*increases if particles are small (

*x*< 10). Then it increases as

*x*increases. After

*x*> 35, the rate of increase decreases a bit. Overall the

*B*for detached coccoliths is ranging from 0.005 to 0.025. Gordon et al. have shown that the backscattering probabilities are around 0.04 for their modified ”fishing reel” model [24

_{bp}24. 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**, 6059–6073 (2009) [CrossRef] [PubMed]

*μm*

^{2}per coccolith at 500 nm [24

24. 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**, 6059–6073 (2009) [CrossRef] [PubMed]

*μm*

^{2}to 0.07

*μm*

^{2}for coccoliths with diameters smaller than 2.83

*μm*, which is consistent with [24

**48**, 6059–6073 (2009) [CrossRef] [PubMed]

*x*. Asymmetrical coccolithophores do show larger

*δ*than symmetrical coccolithophores in general.

_{L}## 5. Conclusion

*S*

_{22}, which indicates the nonsphericity of the particles. The smallest value of

*S*

_{22}/

*S*

_{11}of the detached coccolith is at around 90° to 120°, which is similar to Voss and Fry [14

14. K. J. Voss and E. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. **23**, 4427–4439 (1984) [CrossRef] [PubMed] .

*S*

_{33}is not equal to

*S*

_{44}for coccolithophores, which is another indicator of nonsphericity. A particle with larger imaginary refractive index shows smaller scattering efficiency, backscattering probability, and depolarization ratio.

## Acknowledgments

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29. | A. Taflove and S. C. Hagness, |

30. | P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A |

31. | W. Sun, Q. Fu, and Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. |

32. | P. Zhai, G. W. Kattawar, P. Yang, and C. Li, “Application of the symplectic finite-difference time-domain method to light scattering by small particles,” Appl. Opt. |

33. | M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Transfer |

34. | M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express |

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

36. | E. S. Fry and G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. |

37. | Y. Hu, S. Rodier, K. Xu, W. Sun, J. Huang, B. Lin, P. Zhai, and D. Josset, “Occurrence, liquid water content, and fraction of supercooled water clouds from combined CALIOP/IIR/MODIS measurements,” J. Geophys. Res. |

38. | M. Kahnert, T. Nousiainen, M. A. Thomas, and J. Tyynelä, “Light scattering by particles with small-scale surface roughness: Comparison off our classes of model geometries,” J. Quant. Spectrosc. Radiat. Transfer |

39. | J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(290.1350) Scattering : Backscattering

(290.5850) Scattering : Scattering, particles

(290.7050) Scattering : Turbid media

(010.4458) Atmospheric and oceanic optics : Oceanic scattering

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 17, 2013

Revised Manuscript: June 20, 2013

Manuscript Accepted: July 6, 2013

Published: July 16, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Peng-Wang Zhai, Yongxiang Hu, Charles R. Trepte, David M. Winker, Damien B. Josset, Patricia L. Lucker, and George W. Kattawar, "Inherent optical properties of the coccolithophore: Emiliania huxleyi," Opt. Express **21**, 17625-17638 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-15-17625

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