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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 5 — Feb. 28, 2011
  • pp: 4673–4691
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Light scattering and absorption by randomly-oriented cylinders: dependence on aspect ratio for refractive indices applicable for marine particles

Howard R. Gordon  »View Author Affiliations


Optics Express, Vol. 19, Issue 5, pp. 4673-4691 (2011)
http://dx.doi.org/10.1364/OE.19.004673


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Abstract

Typically, explanation/interpretation of observed light scattering and absorption properties of marine particles is based on assuming a spherical shape and homogeneous composition. We examine the influence of shape and homogeneity by comparing the optics of randomly-oriented cylindrically-shaped particles with those of equal-volume spheres, in particular the influence of aspect ratio (AR = length/diameter) on extinction and backscattering. Our principal finding is that the when AR > ~3–5 and the diameter is of the order of the wavelength, the extinction efficiency and the backscattering probability are close to those of an infinite cylinder. In addition, we show the spherical-based interpretation of extinction and absorption can lead to large error in predicted backscattering.

© 2011 OSA

1. Introduction

The inherent optical properties (IOPs) of particles suspended in natural waters are of interest in several areas of marine science: sediment properties and transport [1

1. S. G. Ackleson, “Optical determinations of suspended sediment dynamics in western Long Island Sound and the Connecticut River plume,” J. Geophys. Res. 111(C7), C07009 (2006), doi:. [CrossRef]

]; marine photosynthesis [2

2. N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993). [CrossRef]

]; and remote sensing of ocean color [3

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

]). The IOPs include the absorption coefficient (a), the scattering coefficient (b), the extinction coefficient c = a + b, the volume scattering function β(Θ) (Θ is the scattering angle), and the backscattering coefficient (bb). The extinction coefficient is particularly important in sediment studies, the spectral absorption coefficient in photosynthesis, and the backscattering coefficient in remote sensing (water-leaving radiance ∝ bb/a).

For many years, the interpretation of measurements of the IOPs of particles suspended in natural waters (in particular the estimation of their refractive indices) has usually employed the assumption that the particles are spherically symmetric [4

4. D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004). [CrossRef]

]. The development of electromagnetic scattering codes for computing the scattering from particles with more complex shapes has stimulated interest in the influence of particle shape on the IOPs [5

5. 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]

7

7. W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007). [CrossRef]

]. Gordon and Du [5

5. 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]

] and Gordon et al. [8

8. 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]

] showed that a complex shape was required to reproduce the spectral variation and absolute magnitude of the backscattering cross section of coccoliths detached from the coccolithophore E. huxleyi. Gordon [9

9. 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]

,10

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

] showed that small-scale structures (size ≤ λ/4) had little influence on the backscattering of disk-like particles. Clavano et al. [7

7. W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007). [CrossRef]

] carried out a comprehensive study of scattering by spheroid-shaped particles (ellipses of revolution) in random orientation with refractive indices characteristic of particles suspended in water. They showed that the computed IOPs for such particles deviated significantly from those computed for spheres having the same volume and refractive index. The deviations increased as the aspect ratio of the spheroids increased. They also reported data suggesting that the most frequent aspect ratio of living marine particles was ~5.

In this work, I consider in detail the dependence of IOPs on aspect ratio. Rather than spheroids, I use homogeneous and structured cylinders as the study particle, but make comparisons with similarly-sized spheroids. I consider refractive indices that are within the range expected for marine particles. I also assume throughout that the particles are in random orientation; however, if the particles have a preferred orientation the influence on the IOPs will be large, particularly on backscattering. Generally, particles embedded in an isotropic turbulent flow, will have a random orientation. In contrast, if they are embedded in non-isotropic turbulence [11

11. L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005). [CrossRef]

] or in a shear flow, there will be a tendency for the particles to align with the flow. This tendency will be particularly important in aligning particles with large aspect ratios, e.g., diatom chains [12

12. L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998). [CrossRef]

]. In the absence of data concerning particle orientation in the natural waters, we have no basis on which to assume any orientation other than random.

As the principal focus is to study the influence of aspect ratio on the IOPs, the particle diameters are of necessity limited by the storage requirements of the computer code used in the computations. The largest particle that could be examined has a volume-equivalent spherical diameter of ~3.6 μm, and an aspect ratio of 10.

The principal result of the study is that the extinction, scattering, and absorption efficiencies, as well as the scattering phase function, β(Θ)/b, and the backscattering probability, bb/b, become nearly independent of the aspect ratio (length/diameter) when it becomes greater than ~3−5. This implies that the IOPs of longer cylindrical particles can be inferred from those of particles with aspect ratios in this range.

2. Review of light scattering and absorption concepts

2.1. Finite cylinders

For a cylinder (or any particle) of finite extent, the differential scattering cross section, dσb(Θ,Φ)is given by
dσb(Θ,Φ)=(Sb(Θ,Φ))Avgr^(Θ,Φ)dA|(SInc)Avg|,
(1)
where (Sb(Θ,Φ))Avgis the time-averaged Poynting vector (irradiance) of the scattered field at the position rin the direction (Θ,Φ), r^(Θ,Φ) is a unit vector from the scattering center to the detector of area dA with it’s normal parallel to r^(Θ,Φ), and (SInc)Avg is the time averaged Poynting vector of the incident field. The angle Θ is the angle between the direction of the incident radiation and r^(Θ,Φ), and Φ is the angle between the scattering plane (plane containing the incident direction and r^) and a laboratory-fixed reference plane containing the incident direction. Using the fact that the solid angle subtended by the detector at the particle is dΩ=dA/r2, where r is the distance from the detector to the particle, this gives the conventional definition of the differential scattering cross section:
dσb(Θ,Φ)dΩ=r2(Sb(Θ,Φ))Avgr^(Θ,Φ)dA|(SInc)Avg|.
(2)
Far from the scattering center the scattered fields are ∝ 1/r, so the scattered Poynting vector is r^/r2, and the differential cross section is independent of r. The total scattering cross section is defined by
σb=Φ=0Φ=2πΘ=0Θ=πdσb(Θ,Φ)dΩsinΘdΘdΦ,
(3)
the backscattering cross section by
σbb=Φ=0Φ=2πΘ=π/2Θ=πdσb(Θ,Φ)dΩsinΘdΘdΦ,
(4)
and the scattering phase function by
P(Θ,Φ)1σbdσb(Θ,Φ)dΩ.
(5)
Now, if we average over all orientations of the particle (to represent a collection of identical, randomly-oriented, particles) these relationships are replaced by
dσb(Θ,Φ)dΩ=r2(Sb(Θ,Φ)r^(Θ,Φ))Avg|(SInc)Avg|,
(6)
σb=Φ=0Φ=2πΘ=0Θ=πdσb(Θ,Φ)dΩsinΘdΘdΦ,
(7)
σbb=Φ=0Φ=2πΘ=π/2Θ=πdσb(Θ,Φ)dΩsinΘdΘdΦ,
(8)
P(Θ,Φ)1σbdσb(Θ,Φ)dΩ,
(9)
where Xi=1NXi/N, with the index i referencing one of N appropriately chosen orientations of the particle. Note that the resulting P(Θ,Φ) is actually independent of Φ. The volume scattering function, β(Θ), and the scattering coefficient, b, of a collection of such (randomly-oriented) particles are given by
β(Θ)=ndσb(Θ,Φ)dΩ  and  b=nσb,
(10)
where n is the number of such particles per unit volume. The backscattering coefficient is
bb=2πΘ=π/2Θ=πβ(Θ)sinΘdΘ,
(11)
and the backscattering probability is b˜bbb/b=σbb/σb. The scattering efficiency Qb is defined by Qbσb/Ap,where Ap is the orientationally-averaged projected area (shadow) of the particle.

By computing the net flow of energy into a large sphere surrounding the particle, one can determine the power absorbed by the particle and thus the absorption cross section σa [13

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

]:
σa=A(St(Θ,Φ))Avgr^(Θ,Φ)dA|(SInc)Avg|,
(12)
where (St(Θ,Φ))Avgis the time-averaged Poynting vector of the total field on the sphere: incident plus scattered. Similarly, performing the orientational averaging, σai=1N(σa)i/N. The absorption coefficient, a, is defined by a=nσa, and the absorption efficiency by Qaσa/Ap. Finally, by combining the absorption and scattering cross sections, we obtain, respectively, the extinction, absorption, and scattering cross sections given by σcσa+σb, ca+b,and QcQa+Qb. Equations (1)(12) provide operational definitions for most of the inherent optical properties of interest in marine optics (c is usually referred to as the beam attenuation coefficient).

2.2 Infinite cylinders

The problem of scattering of a plane electromagnetic wave from an infinite cylinder can be solved by separation of variables in cylindrical coordinates [13

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

,14

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

]. If a detector is placed a distance r from the origin of coordinates (located somewhere on the axis of the cylinder), for a given orientation of the cylinder the quantities defined in Eqs. (1)(4) can be formally computed in a straightforward manner. If r is sufficiently large, the scattered fields are ∝ √r, so both the differential and the total cross sections are ∝ r; however, the scattering phase function is independent of r. Thus, the cross sections defined in this way lose their meaning as characteristics of the particle (they depend on how they are measured, i.e., r). Nevertheless, by constructing a large-diameter coaxial cylindrical surface of some length around the cylindrical particle, one can show that the scattering, absorption, and extinction cross sections per unit length are finite [13

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

], so one can define scattering, absorption, and extinction efficiencies such that the cross sections of a given length of particle are finite, e.g., σa=QaLDcosς,where π/2 – ς is the angle between the cylinder axis and the direction of the incident beam (LDcosςis Ap for the length L of the cylindrical particle).

For infinite cylinders the orientational averaging of various quantities cannot be carried out in the normal way. The difficulty is that if one were to try to use Eq. (6), regardless of how large r is made, there will be some orientations for which the detector is not in the far-field zone (or scattered field zone). Thus, the notion of the scattered field (fields ∝ √r) is not applicable for all orientations and a fixed r. In contrast, quantities that are independent of r, i.e., the Q’s, can be averaged over orientation in a straightforward manner [15

15. G. R. Fournier and B. T. Evans, “Approximations to extinction from randomly oriented circular and elliptical cylinders,” Appl. Opt. 35(21), 4271–4282 (1996). [CrossRef] [PubMed]

], e.g., for a length L of cylinder, Qaσa/Ap=QaAp/Ap=Qacosς/cosς, etc. The scattering phase function is also independent of r, and one can formally define the its orientational average through P(Θ,Φ)[1/N]i=1NP(Θ,Φ); however, this has no real physical meaning, as it cannot be used to retrieve an orientationally-averaged differential cross section.

3. Cylinders examined in the present work

The cylinders studied in this work are shown schematically in Fig. 1
Fig. 1 Specifications of the cylinders examined in this study.
. The upper cylinder is homogeneous with refractive index mrimi, relative to water. The lower cylinder (coated) has the same outer diameter (D) as the homogeneous cylinder, however the core diameter is D/2 and the core index is mr − 4imi. The index of the outer layer is mr − 0i, so this simulates a situation in which the mass of absorbing material (or the number of absorbing molecules) is the same in the upper and lower cylinder, and allows simulation of the influence of the distribution of absorbing pigments in cylindrically shaped particles. In marine optics, there are two phenomena that are referred to as the “package effect.” The first is the difference in absorption between equal quantities of absorbing material in solution or in particles suspended in the same volume [16

16. L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19(1), 1–12 (1956). [CrossRef] [PubMed]

18

18. A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986). [CrossRef] [PubMed]

]. The second is the difference in absorption between cells in which the absorbing molecules are uniformly distributed within the cell walls, and cells in which the absorbing molecules are packaged in smaller structures, e.g., chloroplasts [6

6. A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006). [CrossRef]

,19

19. J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995). [CrossRef]

]. It is the second that we examine by considering the two cases in Fig. 1.

The computations were carried out for D = 0.5, 1.0, 1.5, 2.0 μm, L = 0.5, 1.0, 3.0, 5.0, 7.0, 10.0, 15.0, and 20 μm, λ = 400, 500, 600, and 700 nm (in vacuum), and mrimi = 1.02, 1.05, 1.05 – 0.002i, 1.05 – 0.008i, 1.05 – 0.010i, 1.05 – 0.040i, 1.10, 1.15, and 1.20, relative to water. The discrete-dipole approximation code [20

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

,21

21. B. T. Draine and P. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

] DDSCAT 7.0 was used for most of the computations presented here. The orientational averaging was carried out in the manner prescribed in the DDSCAT code. This averaging is not optimum for scattering by a very long cylinder because for a given orientation the scattered light is in the form of a thin cone containing the incident beam and the axis of which cone is coincident with the axis of the cylinder [13

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

]. The cone broadens as the cylinder’s length decreases. A more robust formal of the averaging for long cylinders has been described by Haracz, et al. [22

22. R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985). [CrossRef]

]; however, as we show later, we believe that averaging process in DDSCAT is sufficiently accurate for the cylinders studied here.

The computations were carried out on an 80 CPU cluster with a total memory of 160 GB. The dipole density was such that their lattice spacing was ~λ /18 at 400 nm and ~λ/31 at 700 nm. This insured that the backscattering cross section could be computed with an error < about 5%. Even with the multi-processor cluster, the computations were very time consuming for the larger particles: ~10 days were required to compute the scattering (at 4 wavelengths) by a cylinder with D = 1.5 μm, L = 15 μm and m = 1.20.

4. Extinction and absorption efficiencies

The extinction and absorption efficiencies were computed using from the orientationally-averaged extinction and absorption cross sections σcand σa through Qcσc/Ap and Qaσa/Apwhere Ap=πD(L+D/2)/4 is the orientationally-averaged projected area (shadow) of the particle. Using the computed efficiencies for finite-length cylinders, I asked several questions: (1) how do the efficiencies depend on the aspect ratio (ARL/D) and diameter of the cylinders?; (2) how do the efficiencies compare with those for an infinite cylinder with the same diameter?; and (3) how do the efficiencies depend on the distribution of absorbing material within the cylinders (Fig. 1)?

Figure 2
Fig. 2 Extinction and absorption efficiencies computed for randomly orientated, homogeneous or coated cylindrically shaped particles, given that their diameter and aspect ratio are known so that their orientationally-averaged projected area is πD(L + D/2)/4. Here, ρ = 2α (mr−1) and ρ′ = 4α mi, where mrimi, is the refractive index of the particle relative to water, and α = πD/λ, with D the cylinder’s (outer) diameter and λ the wavelength of light in the water. Solid lines are the exact computations for randomly oriented, infinite cylinders. The notation “m=1.05-040i_1.05-000i” indicates that the refractive index of the core is 1.05 – 0.040i, and the refractive index of the coating is 1.05 – 0.000i, etc. In the case of coated cylinders, ρ and ρ′ are computed using the mi of the associated homogeneous particle. Top: all aspect ratios (1/3 – 30). Bottom: all aspect ratios ≥ 3.
provides the computed values of Qc and Qa for the all aspect ratios that were examined with absorbing cylinders (mi > 0) and mr = 1.05. The notation in the legend is explained in the figure caption. The upper two panels are for all aspect ratios (1/3 – 30) and the lower panels for AR ≥ 3. The lines correspond to exact computations for infinite cylinders with the same diameter using the IPHASE code [15

15. G. R. Fournier and B. T. Evans, “Approximations to extinction from randomly oriented circular and elliptical cylinders,” Appl. Opt. 35(21), 4271–4282 (1996). [CrossRef] [PubMed]

]. For the case with weaker absorption, the results clearly show that the efficiencies closely follow those for infinite cylinders as long as AR ≥ 3, for both the coated and the homogeneous cylinders. For the case with stronger absorption, the “package effect,” the decrease in absorption when the absorbing molecules are not uniformly distributed with in the particle, results in lower absorption efficiency.

This packaging effect is displayed more clearly in Fig. 3
Fig. 3 This figure provides the ratio of absorption efficiencies (coated to homogeneous) of strongly absorbing cylinders as a function of wavelength. The diameter of the cylinder (in μm) is specified in the legend. For each diameter, the symbols refer to cylinder lengths ranging from 0.5 to 15 μm. The figure shows that the effect of the absorbing pigment packaging is greatest in the blue region of the spectrum and for larger-diameter cylinders. Although the symbols do not differentiate between cylinder lengths, for a given diameter the packaging effect is smallest in the shortest cylinder and depends very little on the length once the aspect ratio (length/diameter) exceeds unity.
, for which I provide the ratio of the absorption efficiencies Qa for a coated cylinder with indices minside = 1.05−0.040i and moutside 1.05−0.000i (Qa(Packaged)) to that of a homogeneous cylinder minside = moutside = 1.05−0.010i (Qa(Homo)). Note that both cylinders contain the same number of absorbing molecules. The figure shows that the effect of the absorbing pigment packaging is greatest in the blue region of the spectrum and for larger-diameter cylinders. The maximum decrease in Qa due to the packaging is about 25%. Although the symbols do not differentiate between cylinder lengths, for a given diameter the packaging effect is smallest in the shortest cylinder and depends very little on the length once AR exceeds unity. In the case with less overall absorption, i.e., minside = 1.05−0.008i and moutside = 1.05−0.000i compared to that of a homogeneous cylinder minside = moutside = 1.05−0.002i, similar results are obtained; however, the maximum decrease in Qa due to the packaging is only about 10% (Fig. 2.).

The extinction efficiencies of finite cylinders with larger values of ρ are compared with those of infinite cylinders in Fig. 4
Fig. 4 Qc as a function of ρ, computed for non-absorbing cylinders (m = 1.20 – 0.000i) with diameters (D) ranging from 0.5 μm to 2.0 μm. Left: 0.25 ≤ AR ≤ 30 (points colored in red are AR = 2). Right: 3 ≤ AR ≤ 30. The solid curve is the extinction efficiency (for a unit length) of randomly-oriented infinite cylinders. As in Fig. 2, ρ = 2α (m −1) with α = πD/λ.
. In this case the refractive index is 1.20 and since there is no absorption Qb = Qc. The figure clearly shows that for AR ≥ 3, the extinction efficiency is again close to that of an infinite cylinder, with all cases except two differing by less than ± 10% (RMS difference ~5%).

Thus, it is clear that for homogeneous cylindrically-shaped particles with aspect ratios > about 3, the extinction efficiency becomes close to that of an infinite cylinder, i.e., Qc(D,m,AR) ≈Qc(D,m,∞). This implies that σc(D,m,AR)Qc(D,m,)πD2(AR+1/2)/4, or for two aspect ratios AR and AR′ (both > about 3),
σc(D,m,AR)(AR+1/2)σc(D,m,AR)(AR+1/2).
(13)
Similar expressions hold for the orientationally averaged absorption and scattering efficiencies. Although Qc(D,m,)has not been computed for coated cylinders, presumably Eq. (15) should hold for such particles as well.

5. Phase function and backscattering probability of cylinders

We have seen that the extinction and absorption efficiencies of micrometer-sized cylindrical particles depend little on the aspect ratios as long as AR ≥ 3. Is this the case for the phase function and backscattering probability? I mentioned earlier that the default orientational-averaging scheme used in DDSCAT 7.0 was employed in the present computations. Is this default sufficient to provide orientational averaging for long cylinders? To examine this question, I computed the orientationally-averaged phase function (and b˜b) for cylinders with small and large AR. To achieve the very large variation in AR, I used D = 0.25 μm. The results of this computation are shown in Fig. 5
Fig. 5 Orientationally averaged scattering phase functions for long cylinders as a function of aspect ratio (AR). The values of the computed backscattering probabilities are 0.0212, 0.0212, 0.0209, 0.0218, and 0.0220, for AR = 5, 10, 100, 200, and 270, respectively. The values of the extinction efficiency (Qc) are 0.999, 1.007, 1.032, 1.033, 1.033, and 1.034 for AR = 5, 10, 100, 200, 270, and ∞, respectively. The value of ρ for these efficiencies is 1.0472, so these computations fall very close to the continuous curve in Fig. 4. (Note, L = 1.25, 2.50, 25.0, 50.0, and 67.5 μm for AR = 5, 10, 100, 200, and 270, respectively.) D = 0.25 μm, m = 1.20, λ = 400 nm.
. One sees that with the exception of small scattering angles (Θ ≤ 8°), as AR increases the computed phase function simply becomes “noisier.” This is what would be expected, as the scattering pattern (for a given orientation) degenerates into an infinitely thin cone as L → ∞. Careful examination shows that there is a high correlation between the “noise” at AR = 100 and 200, etc., as would be expected as the scattering cone thins. Since the thickness of the scattering cone depends mostly on λ / L (the thickness decreases as L increases), I conclude that for the values of L examined in this work (≤ 25 μm) the averaging procedure in DDSCAT is sufficiently accurate to yield reliable phase functions and backscattering probabilities.

Figure 6
Fig. 6 Orientationally-averaged scattering phase functions (left) and degree of linear polarization (right) at 600 nm (vacuum) for homogeneous cylinders with a diameter of 1 μm and length of 3 μm (D1xL3) and 20 μm (D1xL20). The refractive index is 1.05 – 0.002i. The oscillatory nature of the phase function is determined mostly by D/λ, but with some dependence on m.
provides another example of the weak dependence of the scattering phase function (and degree of linear polarization) on aspect ratio. Virtually the only differences in the phase function between AR = 20 and AR = 3 are the enhanced scattering near zero degrees and the deeper minima near 30°, 50°, 80°, and 130° for AR = 20. This weak dependence on aspect ratios as long as AR ≥ 5 is also displayed by the backscattering probability as shown in Fig. 7
Fig. 7 Examples of the variation of the backscattering probability with aspect ratio for cylinder diameters between 0.5 and 1.5 μm and refractive indices ranging from 1.02 to 1.20. The black curves are for a vacuum wavelength of 400 nm and the red curves for 700 nm.
for a wide range of refractive indices and particle diameters.

These computations show that when AR > about 3–5, σbb(D,m,AR)σ˜bb(D,m,)σb(D,m,AR), and in a manner similar to Eq. (13),
σbb(D,m,AR)(AR+1/2)σbb(D,m,AR)(AR+1/2),
(14)
with both AR′ > AR ≈3-5 (Figs. 7 and 13
Fig. 13 Backscattering probability as a function of aspect ratio and the imaginary part of the refractive index for non-absorbing to strongly absorbing, homogeneous and structured cylinders. D is in micrometers.
). I used Eq. (14) to compute the orientationally averaged backscattering cross section for cylinders with D = 1 μm and AR′ ≥ 5 from AR = 3 and for AR′ ≥ 7 from AR = 5 with m = 1.05 – 0.010i. The rms error was 2.5% for AR = 3 and 1.2% for AR = 5. Similar computations with m = 1.20 – 0.000i resulted in rms errors of 8.1% and 2.6% for AR = 3, and 5, respectively.

6. Cylinders compared to equal-volume spheres

A well-known approach to estimating the complex refractive index of marine particles, e.g., phytoplankton, is to measure their extinction and absorption coefficients and the particle volume (e.g., with a Coulter Counter). The particles are then assumed to be homogeneous spheres and the extinction and absorption efficiencies are computed. Real and imaginary parts of the refractive index are then found which, for a spherical particle, would yield the same extinction and absorption efficiencies (see, for example, Ref. 18

18. A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986). [CrossRef] [PubMed]

). I have tested this approach using the extinction and absorption cross sections described above for cylinders. Rather than using the exact computation of Qc and Qa for spheres, it is much simpler to use the analytical formulas of the van de Hulst anomalous diffraction approximation for the scattering and absorption efficiencies of a homogeneous sphere. These are [14

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

]
Qa(ρ)=1+2exp(ρ)ρ+2exp(ρ)1ρ2,
(15)
and
Qc(ρ)=24exp(ρtanβ)[cosβρsin(ρβ)+cos2βρ2cos(ρ2β)]+4cos2βρ2cos(2β),
(16)
where ρ = 2α (mr−1), ρ′ = 4α mi, tan β = mi/(mr−1), and α = πd/λ, with d the sphere’s diameter and λ the wavelength of light in the water. In this analysis, we take d to be the diameter of a sphere with the same volume as the cylinder, i.e., the equal-volume sphere. Fig. 8
Fig. 8 The extinction (left) and absorption (right) efficiencies computed by dividing the associated cross sections by the projected area of a volume-equivalent sphere as a function of ρ and ρ ′. Here, ρ = 2α (mr−1) and ρ′ = 4α mi, where mrimi, is the refractive index of the particle relative to water, and α = πd/λ, where d is now the diameter of the volume-equivalent sphere. For a given experimentally-determined Qa, the dashed vertical arrow provides the correct ρ ′ (and, hence mi), while the solid vertical arrow provides the retrieved value of mi.
provides the extinction and absorption efficiencies computed assuming the spherical shape, e.g., Qa4σa/πd2, along with Qc and Qa computed with the Van de Hulst anomalous diffraction theory (VdH), and with exact the Mie theory (MIE). There are three important observations to be made from Fig. 8: (1) the volume-equivalent sphere assumption is not very good in the case of Qc (left figure) even if full Mie theory is used; (2) the volume-equivalent sphere assumption is better in the case of Qa (right figure), especially if full Mie theory is used; and (3) the package effect, while relatively unimportant for Qc, is important in Qa (right figure) for the case with stronger absorption, but not for the case with weaker absorption.

How much error does the deviation of the derived Qc and Qa from Eqs. (18) and (19) make in estimating the refractive index? Given the volume of the particle, and assuming a spherical shape, d and α are determined. Calculations of the beam attenuation coefficient and absorption coefficients for coated cylinders were inserted into Eqs. (15) and (16) to find mr and mi. Figure 9
Fig. 9 An example of retrievals of the real and imaginary parts of the refractive index for coated cylinders for all the combinations of diameter and length, using the van de Hulst approximation. Ideally one should derive a real part of 1.05 and an imaginary part of 0.010. The scatter shows that mi is retrieved to within ± 20% (somewhat better in the red) with an average (over all sizes) close to 0.010, and that the retrieved mr appears to be too low in almost all cases, but averages ~1.044.
provides the resulting computations for a coated cylinder with indices minside = 1.05−0.040i and moutside = 1.05−0.000i, and all combinations of diameter and length. Recall that the package effect is larger for this case. Ideally one should derive an index of 1.05−0.010i, based on the concentration of absorbing material. Clearly, mi is retrieved to within ± 20% with an average (over all sizes) close to 0.010, and the retrieved mr appears to be too low in almost all cases, but averages ~1.044. One notes that if the exact Mie theory were used in the retrieval of mi (Fig. 8) the retrieved values would be about 10% larger than shown in Fig. 9 due to the inaccuracy of the Van de Hulst approximation to Qa [Eq. (15)]. However, the exact Mie results cannot actually be used because mr is required, and as we see in the figure, it is strongly dependent on AR. Thus, it is clear that measuring particle volume and the extinction and absorption cross sections, assuming the particles are spherical, then applying Eqs. (15) and (16) does yield meaningful results for mi even for particles with large aspect ratios; however, the retrieved values of mr depend strongly on the aspect ratio. It is interesting to note that if one employed the homogeneous infinite-cylinder assumption in the analysis of the cross sections (Fig. 2) for this example, the error in the retrieved mi would actually be larger than for the equivalent-volume sphere approach: accurate retrieval would require consideration of the package effect, probably by using a coated infinite-cylinder retrieval model.

For larger values of the refractive index of the cylinder, this method fails completely when the cylinder becomes too large. A dramatic example of this failure is provided in Fig. 10
Fig. 10 The extinction efficiency computed by dividing the associated extinction cross section by the projected area of a volume-equivalent sphere as a function of ρ. Here, ρ = 2α (mr−1) and α = πd/λ, where d is the diameter of the volume-equivalent sphere. Points for some given diameters and lengths are connected by smooth curves (for which λ varies from 400 to 700 nm). The red curves are for diameters of 1.0 and 1.5 μm with AR = 10. The thick curve is the Van de Hulst approximation to Qc for spheres.
, which shows Qc for cylinders determined from the cross-sectional area of the equal-volume sphere (in a manner identical to that in Fig. 8), and ρ evaluated using the diameter of the equal-volume sphere. The thick solid line in the figure is the Van de Hulst approximation to Qc. Note that for ρ > about 3, for most cases shown, there is no refractive index value for equal-volume spheres that can produce the associated extinction efficiency. Thus, this method often fails to provide any value for m.

Given the values of mr and mi derived from measurements of the absorption and extinction efficiencies, the particle volume, and the assumption of sphericity (using the methodology described above, when it works, i.e., for low-index cylinders), how well does the predicted backscattering cross section reproduce the actual backscattering cross section of cylindrical particles? To shed light on this question, I used the retrieved refractive indices shown in Fig. 9 and Mie theory to compute σbb(Sph) for comparison with its cylindrical counterpart σbb(Cyl)for each wavelength and particle size. The ratio R=σbb(Cyl)/σbb(Sph) shows some spectral variation, but to gain a better perspective on the influence of particle shape, I averaged R over the visible spectrum for each size. Figure 11
Fig. 11 Backscattering by a cylinder divided by backscattering by an equal-volume sphere. Black curves: refractive index in the computation of σbb for spheres is that derived using the refractive index determined from the extinction and absorption cross sections using the equivalent-volume sphere assumption. Red curves: refractive index in the computation of σbb for spheres is the same value used for the cylinders, i.e., the correct value. Diameter (D) is in micrometers, and the true value of the refractive index is 1.05–0.010i.
(black points/lines) shows the resulting R for particles with diameters ~1-3 times the wavelength as a function of the aspect ratio. The red points/lines on the figure provide R values when the true value of the cylinder’s refractive index is used to compute the backscattering the sphere, rather than that determined from particle extinction, absorption, and volume. We note that for cylinders in this size range, R is greater than 1 and increases approximately linearly with aspect ratio. The large values of R when the index is retrieved from extinction, etc., is due to the increased error in the derived values of mr as the aspect ratio increases (the retrieved mr becomes smaller as the aspect ratio increases, and σbb(Sph) is a strong function of mr). Thus, the extinction, absorption, volume, and sphericity-assumption methodology cannot yield reliable values of the backscattering cross section of cylindrically shaped particles with even moderate (>5) aspect ratios; however, if the correct refractive index is known and used in the computation, the backscattering of the equal-volume sphere is much closer to that of the cylinder.

The backscattering ratio R for the higher index (1.20 – 0.000i) is shown in Fig. 12
Fig. 12 Backscattering by a cylinder divided by backscattering by an equal-volume sphere. As the refractive index in this case cannot be derived from the extinction efficiency, in the computation of σbb for spheres m is the same value used for the cylinders. Diameter (D) is in micrometers, the wavelength is 400 nm, and the true value of the refractive index is 1.20–0.000i.
. Note that in this case σbb(Cyl)<σbb(Sph). Thus, in the size range examined here, backscattering by cylinders appears to be larger than equal-volume spheres at low refractive indices and smaller at high refractive indices.

Absorption appears to have only a small effect on the backscattering by cylinders; however, the packaging of the absorbing substance within the cylinder can have a significant effect, especially when the absorption is large. Figure 13 provides the backscattering probability for cylinders (homogeneous and packaged) for all cases studied with mr = 1.05. Note that when the absorption is weak (mi = 0.002 with a homogeneous distribution of absorbing molecules, ■, or mi = 0.008 with the absorbing molecules confined to the inner cylinder of Fig. 1, ▲) there is little difference in the backscattering probability and that of a non-absorbing cylinder (♦). In contrast, for the more strongly absorbing, homogeneous (◊) or packaged (□), cylinders, absorption clearly influences the backscattering probability with larger values for the packaged case.

7. Comparison with spheroids

A limited number of computations have been carried out for prolate spheroids to see if the conclusions regarding the influence of aspect ratio on light scattering by cylinders applies to spheroids. Fournier and Evans [23

23. G. R. Fournier and B. T. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

] have provided a highly accurate anomalous diffraction approximation to Qc for spheroids. If one uses their relationships for a spheroid with minor axes D and major axis L (AR = L/D), it is seen that Qc becomes almost independent of AR for AR > about 3. I have carried out computations (using DDSCAT) for spheroids with D = 0.5, 1.0, and 1.5 μm, and AR = 3, 5, and 10, for m = 1.05 – 0.010i and m = 1.20 – 0.000i. The resulting values of Qc (and Qa) are virtually independent of AR and agree well with the Fournier and Evans [23

23. G. R. Fournier and B. T. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

] result for large AR (e.g., AR = 100).

The independence of the backscattering probability on AR for AR >˜ 3-5 for cylinders is also seen for spheroids of similar size. The results presented in Table 1

Table 1. The RMS the Backscattering Probability for Spheroids with Aspect Ratio AR = 10 to that for Spheroids with the Same Minor Axes but Aspect Ratio AR. D = 0.5, 1.0, and 1.5 μm, and λ = 400, 500, 600, and 700 nm

table-icon
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suggest that the backscattering probability for spheroids becomes essentially constant for AR > about 5. Thus, our conclusions regarding the dependenceof light scattering properties on AR for cylinders appear to apply equally well to spheroids.

It should be noted that when D and L are << λ, and the polarizabilities are determined in the electrostatic approximation (Rayleigh approximation [13

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

,14

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

]), the total scattering for spheroids is proportional to the square of the volume times a factor that is dependent on AR. This latter factor becomes nearly independent of AR for AR > ~ 3 for the refractive indices of interest here. In this case, σb(D2L)2=D6AR2, and
σb(D,m,AR)AR2σb(D,m,AR)AR2.
(17)
Equation (17) replaces Eq. (13) in this regime, and since σ˜bb=1/2, independent of AR, a similar expression replaces Eq. (14). Again, similar expressions also apply to cylinders.

8. Concluding remarks

As stated in the abstract, I have shown the extinction, absorption, and scattering efficiencies, and the backscattering probability of randomly oriented, homogeneous and structured, cylinders become nearly independent of the aspect ratio when AR > ~3-5, for refractive indices characteristic of marine particles (organic and inorganic). This applies to cylinders with diameters in the range 0.25 to 1.5 μm when illuminated with visible light (wavelength, 400-700 nm). Some long-chain phytoplankton, e.g., Prochlorotrix hollandica, fall in this size range [24

24. M. Jonasz, and G. R. Fournier, Light Scattering by Particles in Water, Theoretical and Experimental Foundations (Academic Press, 2007).

]. It should also apply to much larger cylindrically-shaped particles, i.e., in sizes for which geometrical optics is applicable. Computational schemes for intermediate sized cylinders with high aspect ratios are not available; however, as the validity of the observation does not appear to depend on the actual diameter of the cylinders (Figs. 7 and 13) in the size ranges examined, one would expect that it would apply to intermediate sized particles as well. A limited number of computations for prolate spheroids suggest that the observations apply equally well to particles with this shape. This should simplify the inclusion of AR-distributions in the characterization of scattering by marine particles.

In order to interpret measured particle extinction and absorption cross sections to obtain the refractive index for single-species of phytoplankton, it is of course best to use a close approximation to the particle’s shape in the necessary model calculations, i.e., there is no canonical shape that can be used for all particles. For cylindrical particles with aspect ratios greater than 3, it appears that infinitely-long cylinders (homogeneous or coated) with the same diameter (shorter dimension) are adequate for estimation of the refractive index. For other particles, other shapes will be appropriate; however, I expect that for particles that can be represented by “simple” shapes, e.g., spheroids, linear chains of spheres or spheroids (homogeneous or coated), etc., the cross sections will be proportional to the length, and the efficiencies will depend mostly on diameter for aspect ratios 3 or greater, as they do for cylinders. For the commonly used equivalent-volume-sphere approximation to obtain refractive index [18

18. A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986). [CrossRef] [PubMed]

], we found that for cylinders the absorption index (mi) can be determined with reasonable accuracy, i.e., ~ ± 20%, when mr is low as it usually is for phytoplankton [24

24. M. Jonasz, and G. R. Fournier, Light Scattering by Particles in Water, Theoretical and Experimental Foundations (Academic Press, 2007).

]. Remarkably, this approximation is actually better in retrieving mi than using a homogeneous cylinder to model (incorrectly) a coated cylinder (Fig. 2). This suggests that in the absence of shape information, the equivalent-volume-sphere approximation is capable of yielding realistic estimates of mi for low-mr particles that deviate significantly from spheres. When mr is high, the method fails completely (Fig. 10), and a more appropriate shape is required to interpret the observed cross sections.

In the case of backscattering, for the low index particles we examined, using the index retrieved through the equal-volume-sphere assumption, and computing σbb for the equal-volume sphere, can lead to an underestimation (σbb(Cyl)>σbb(Sph)) of cylinder backscattering by a significant factor (Fig. 11), largely because of the inaccuracy in estimation of mr; however, if the correct value of the refractive index is known, the error is significantly decreased. For the high-index case (for which the equal-volume-sphere analysis fails), given the correct value of the refractive index, the equal-volume sphere backscatters more than the cylinder, i.e., σbb(Cyl)<σbb(Sph). Thus, prediction of σbbby this method for low index particles could account for some of the “missing” backscattering suggested for marine particles [4

4. D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004). [CrossRef]

]; however, when the correct index is used in the computations, the underestimation is greatly reduced or eliminated completely.

Although the computations presented here represent a grossly inadequate span of cylinder sizes due to inadequate computer resources, they do suggest the manner in which particles with high aspect ratios can be included in scattering computations carried out at lower aspect ratios, particularly when the diameter of the particle is of the order of λ.

Acknowledgments

References and links

1.

S. G. Ackleson, “Optical determinations of suspended sediment dynamics in western Long Island Sound and the Connecticut River plume,” J. Geophys. Res. 111(C7), C07009 (2006), doi:. [CrossRef]

2.

N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993). [CrossRef]

3.

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

4.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 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(6), 1438–1454 (2001). [CrossRef]

6.

A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006). [CrossRef]

7.

W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007). [CrossRef]

8.

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]

9.

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]

10.

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

11.

L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005). [CrossRef]

12.

L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998). [CrossRef]

13.

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

14.

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

15.

G. R. Fournier and B. T. Evans, “Approximations to extinction from randomly oriented circular and elliptical cylinders,” Appl. Opt. 35(21), 4271–4282 (1996). [CrossRef] [PubMed]

16.

L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19(1), 1–12 (1956). [CrossRef] [PubMed]

17.

A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. 28(11), 1375–1393 (1981). [CrossRef]

18.

A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986). [CrossRef] [PubMed]

19.

J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995). [CrossRef]

20.

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

21.

B. T. Draine and P. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

22.

R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985). [CrossRef]

23.

G. R. Fournier and B. T. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

24.

M. Jonasz, and G. R. Fournier, Light Scattering by Particles in Water, Theoretical and Experimental Foundations (Academic Press, 2007).

OCIS Codes
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(010.1030) Atmospheric and oceanic optics : Absorption

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: December 21, 2010
Revised Manuscript: February 13, 2011
Manuscript Accepted: February 18, 2011
Published: February 24, 2011

Virtual Issues
Vol. 6, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Howard R. Gordon, "Light scattering and absorption by randomly-oriented cylinders: dependence on aspect ratio for refractive indices applicable for marine particles," Opt. Express 19, 4673-4691 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4673


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References

  1. S. G. Ackleson, “Optical determinations of suspended sediment dynamics in western Long Island Sound and the Connecticut River plume,” J. Geophys. Res. 111(C7), C07009 (2006), doi:. [CrossRef]
  2. N. Hoepffner and S. Sathyendranath, “Determination of major groups of phytoplankton pigments from absorption spectra of total particulate matter,” J. Geophys. Res. 98(C12), 22789–22803 (1993). [CrossRef]
  3. H. R. Gordon, and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983).
  4. D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 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(6), 1438–1454 (2001). [CrossRef]
  6. A. Quirantes and S. Bernard, “Light scattering methods for modeling algal particles as a collection of coated and/or nonspherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 315–324 (2006). [CrossRef]
  7. W. R. Clavano, E. Boss, and L. Karp-Boss, “Inherent Optical Properties of Non-Spherical Marine-Like Particles - From Theory to Observations,” Oceanogr. Mar. Biol. 45, 1–38 (2007). [CrossRef]
  8. 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]
  9. 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]
  10. H. R. Gordon, “Backscattering of light from disk-like particles with aperiodic angular fine structure,” Opt. Express 15(25), 16424–16430 (2007). [CrossRef] [PubMed]
  11. L.-X. Zhang, J.-Z. Lin, and T. L. Chan, “Orientation distribution of cylindrical particles suspended in turbulent pipe flow,” Phys. Fluids 17(9), 093105 (2005). [CrossRef]
  12. L. Karp-Boss and P. A. Jumars, “Motion of diatom chains in steady shear flow,” Limnol. Oceanogr. 43(8), 1767–1773 (1998). [CrossRef]
  13. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  14. H. C. Van de Hulst, Light Scattering by Small Particles, (Wiley, 1957).
  15. G. R. Fournier and B. T. Evans, “Approximations to extinction from randomly oriented circular and elliptical cylinders,” Appl. Opt. 35(21), 4271–4282 (1996). [CrossRef] [PubMed]
  16. L. N. M. Duysens, “The flattening of the absorption spectrum of suspensions, as compared to that of solutions,” Biochim. Biophys. Acta 19(1), 1–12 (1956). [CrossRef] [PubMed]
  17. A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. 28(11), 1375–1393 (1981). [CrossRef]
  18. A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25(4), 571–580 (1986). [CrossRef] [PubMed]
  19. J. R. V. Zaneveld and J. C. Kitchen, “The variation of inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure,” J. Geophys. Res. 100(C7), 13,309–13,320 (1995). [CrossRef]
  20. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]
  21. B. T. Draine and P. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]
  22. R. D. Haracz, L. D. Cohen, and A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58(9), 3322–3327 (1985). [CrossRef]
  23. G. R. Fournier and B. T. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]
  24. M. Jonasz, and G. R. Fournier, Light Scattering by Particles in Water, Theoretical and Experimental Foundations (Academic Press, 2007).

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