Light scattering and absorption by randomly-oriented cylinders: dependence on aspect ratio for refractive indices applicable for marine particles

Howard R. Gordon

doi:10.1364/OE.19.004673

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]; marine photosynthesis [2]; and remote sensing of ocean color [3]). 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 (b_b). 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 ∝ b_b/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]. 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 [5–7]. Gordon and Du [5] and Gordon et al. [8] 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 [9,10] showed that small-scale structures (size ≤ λ/4) had little influence on the backscattering of disk-like particles. Clavano et al. [7] 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] 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]. 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, b_b/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.

I begin by reviewing scattering and absorption concepts for finite and infinite cylinders. Next I provide computed absorption and extinction efficiencies and backscattering probability of homogeneous and structured cylinders as a function of their diameter, refractive index, and aspect ratio. These are then compared to those of equal-volume spheres, showing that the spherical assumption is particularly poor at even moderate aspect ratios. Finally, I briefly compare scattering by cylinders and spheroids.

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, $d σ_{b} (Θ, Φ)$ is given by

d σ_{b} (Θ, Φ) = \frac{{({\vec{S}}_{b} (Θ, Φ))}_{A v g} • \hat{r} (Θ, Φ) d A}{| {({\vec{S}}_{I n c})}_{A v g} |},

where

{({\vec{S}}_{b} (Θ, Φ))}_{A v g}

is the time-averaged Poynting vector (irradiance) of the scattered field at the position

\vec{r}

in the direction (Θ,Φ),

\hat{r} (Θ, Φ)

is a unit vector from the scattering center to the detector of area dA with it’s normal parallel to

\hat{r} (Θ, Φ)

, and

{({\vec{S}}_{I n c})}_{A v g}

is the time averaged Poynting vector of the incident field. The angle Θ is the angle between the direction of the incident radiation and

\hat{r} (Θ, Φ)

, and Φ is the angle between the scattering plane (plane containing the incident direction and

\hat{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

d Ω = d A / r^{2}

, where r is the distance from the detector to the particle, this gives the conventional definition of the differential scattering cross section:

\frac{d σ_{b} (Θ, Φ)}{d Ω} = r^{2} \frac{{({\vec{S}}_{b} (Θ, Φ))}_{A v g} • \hat{r} (Θ, Φ) d A}{| {({\vec{S}}_{I n c})}_{A v g} |} .

Far from the scattering center the scattered fields are ∝ 1/r, so the scattered Poynting vector is

\propto \hat{r} / r^{2},

and the differential cross section is independent of r. The total scattering cross section is defined by

σ_{b} = \int_{Φ = 0}^{Φ = 2 π} \int_{Θ = 0}^{Θ = π} \frac{d σ_{b} (Θ, Φ)}{d Ω} \sin Θ d Θ d Φ,

the backscattering cross section by

σ_{b b} = \int_{Φ = 0}^{Φ = 2 π} \int_{Θ = π / 2}^{Θ = π} \frac{d σ_{b} (Θ, Φ)}{d Ω} \sin Θ d Θ d Φ,

and the scattering phase function by

P (Θ, Φ) \equiv \frac{1}{σ_{b}} \frac{d σ_{b} (Θ, Φ)}{d Ω} .

Now, if we average over all orientations of the particle (to represent a collection of identical, randomly-oriented, particles) these relationships are replaced by

〈 \frac{d σ_{b} (Θ, Φ)}{d Ω} 〉 = r^{2} \frac{〈 {({\vec{S}}_{b} (Θ, Φ) • \hat{r} (Θ, Φ))}_{A v g} 〉}{| {({\vec{S}}_{I n c})}_{A v g} |},

〈 σ_{b} 〉 = \int_{Φ = 0}^{Φ = 2 π} \int_{Θ = 0}^{Θ = π} 〈 \frac{d σ_{b} (Θ, Φ)}{d Ω} 〉 \sin Θ d Θ d Φ,

〈 σ_{b b} 〉 = \int_{Φ = 0}^{Φ = 2 π} \int_{Θ = π / 2}^{Θ = π} 〈 \frac{d σ_{b} (Θ, Φ)}{d Ω} 〉 \sin Θ d Θ d Φ,

〈 P (Θ, Φ) 〉 \equiv \frac{1}{〈 σ_{b} 〉} 〈 \frac{d σ_{b} (Θ, Φ)}{d Ω} 〉,

where

〈 X 〉 \equiv \sum_{i = 1}^{N} X_{i} / 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) = n 〈 \frac{d σ_{b} (Θ, Φ)}{d Ω} 〉 and b = n 〈 σ_{b} 〉,

where n is the number of such particles per unit volume. The backscattering coefficient is

b_{b} = 2 π \int_{Θ = π / 2}^{Θ = π} β (Θ) \sin Θ d Θ,

and the backscattering probability is

{\tilde{b}}_{b} \equiv b_{b} / b = 〈 σ_{b b} 〉 / 〈 σ_{b} 〉 .

The scattering efficiency Q_b is defined by

Q_{b} \equiv 〈 σ_{b} 〉 / 〈 A_{p} 〉,

where

〈 A_{p} 〉

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 σ_a [13]:

σ_{a} = - \frac{\underset{A}{∯} {({\vec{S}}_{t} (Θ, Φ))}_{A v g} • \hat{r} (Θ, Φ) d A}{| {({\vec{S}}_{I n c})}_{A v g} |},

where

{({\vec{S}}_{t} (Θ, Φ))}_{A v g}

is the time-averaged Poynting vector of the total field on the sphere: incident plus scattered. Similarly, performing the orientational averaging,

〈 σ_{a} 〉 \equiv \sum_{i = 1}^{N} {(σ_{a})}_{i} / N

. The absorption coefficient, a, is defined by

a = n 〈 σ_{a} 〉

, and the absorption efficiency by

Q_{a} \equiv 〈 σ_{a} 〉 / 〈 A_{p} 〉 .

Finally, by combining the absorption and scattering cross sections, we obtain, respectively, the extinction, absorption, and scattering cross sections given by

〈 σ_{c} 〉 \equiv 〈 σ_{a} 〉 + 〈 σ_{b} 〉,

c \equiv a + b,

and

Q_{c} \equiv Q_{a} + Q_{b}

. 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 [13,14]. 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], so one can define scattering, absorption, and extinction efficiencies such that the cross sections of a given length of particle are finite, e.g., $σ_{a} = Q_{a} L D \cos ς,$ where π/2 – ς is the angle between the cylinder axis and the direction of the incident beam ( $L D \cos ς$ is A_p 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], e.g., for a length L of cylinder, $〈 Q_{a} 〉 \equiv 〈 σ_{a} 〉 / 〈 A_{p} 〉 = 〈 Q_{a} A_{p} 〉 / 〈 A_{p} 〉 = 〈 Q_{a} \cos ς 〉 / 〈 \cos ς 〉$ , etc. The scattering phase function is also independent of r, and one can formally define the its orientational average through $〈 P (Θ, Φ) 〉 \equiv [1 / N] \sum_{i = 1}^{N} P (Θ, Φ)$ ; 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 . The upper cylinder is homogeneous with refractive index m_r − im_i, 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 m_r − 4im_i. The index of the outer layer is m_r − 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 [16–18]. 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 [6,19]. It is the second that we examine by considering the two cases in Fig. 1.

Fig. 1 Specifications of the cylinders examined in this study.

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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 m_r − im_i = 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 [20,21] 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]. 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]; 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 $〈 σ_{c} 〉$ and $〈 σ_{a} 〉$ through $Q_{c} \equiv 〈 σ_{c} 〉 / 〈 A_{p} 〉$ and $Q_{a} \equiv 〈 σ_{a} 〉 / 〈 A_{p} 〉$ where $〈 A_{p} 〉 = π D (L + D / 2) / 4$ 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 (AR ≡ L/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 provides the computed values of Q_c and Q_a for the all aspect ratios that were examined with absorbing cylinders (m_i > 0) and m_r = 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]. 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.

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 (m_r−1) and ρ′ = 4α m_i, where m_r − im_i, 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 m_i of the associated homogeneous particle. Top: all aspect ratios (1/3 – 30). Bottom: all aspect ratios ≥ 3.

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This packaging effect is displayed more clearly in Fig. 3 , for which I provide the ratio of the absorption efficiencies Q_a for a coated cylinder with indices m_inside = 1.05−0.040i and m_outside 1.05−0.000i (Q_a(Packaged)) to that of a homogeneous cylinder m_inside = m_outside = 1.05−0.010i (Q_a(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 Q_a 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., m_inside = 1.05−0.008i and m_outside = 1.05−0.000i compared to that of a homogeneous cylinder m_inside = m_outside = 1.05−0.002i, similar results are obtained; however, the maximum decrease in Q_a due to the packaging is only about 10% (Fig. 2.).

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.

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The extinction efficiencies of finite cylinders with larger values of ρ are compared with those of infinite cylinders in Fig. 4 . In this case the refractive index is 1.20 and since there is no absorption Q_b = Q_c. 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%).

Fig. 4 Q_c 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.

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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., Q_c(D,m,AR) ≈Q_c(D,m,∞). This implies that $〈 σ_{c} (D, m, A R) 〉 \approx Q_{c} (D, m, \infty) π D^{2} (A R + 1 / 2) / 4$ , or for two aspect ratios AR and AR′ (both > about 3),

\frac{〈 σ_{c} (D, m, A R) 〉}{(A R + 1 / 2)} \approx \frac{〈 σ_{c} (D, m, A R^{'}) 〉}{(A R^{'} + 1 / 2)} .

Similar expressions hold for the orientationally averaged absorption and scattering efficiencies. Although

Q_{c} (D, m, \infty)

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 ${\tilde{b}}_{b}$ ) 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 . 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.

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 (Q_c) 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.

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Figure 6 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 for a wide range of refractive indices and particle diameters.

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.

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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.

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These computations show that when AR > about 3–5, $〈 σ_{b b} (D, m, A R) 〉 \approx {\tilde{σ}}_{b b} (D, m, \infty) 〈 σ_{b} (D, m, A R) 〉,$ and in a manner similar to Eq. (13),

\frac{〈 σ_{b b} (D, m, A R) 〉}{(A R + 1 / 2)} \approx \frac{〈 σ_{b b} (D, m, A R^{'}) 〉}{(A R^{'} + 1 / 2)},

with both AR′ > AR ≈3-5 (Figs. 7 and 13 ). 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.

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.

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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). I have tested this approach using the extinction and absorption cross sections described above for cylinders. Rather than using the exact computation of Q_c and Q_a 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]

Q_{a} (ρ^{'}) = 1 + 2 \frac{\exp (- ρ^{'})}{ρ^{'}} + 2 \frac{\exp (- ρ^{'}) - 1}{{ρ^{'}}^{2}},

and

Q_{c} (ρ) = 2 - 4 \exp (- ρ \tan β) [\frac{\cos β}{ρ} \sin (ρ - β) + \frac{\cos^{2} β}{ρ^{2}} \cos (ρ - 2 β)] + 4 \frac{\cos^{2} β}{ρ^{2}} \cos (2 β),

where ρ = 2α (m_r−1), ρ′ = 4α m_i, tan β = m_i/(m_r−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 provides the extinction and absorption efficiencies computed assuming the spherical shape, e.g.,

Q_{a} \equiv 4 〈 σ_{a} 〉 / π d^{2}

, along with Q_c and Q_a 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 Q_c (left figure) even if full Mie theory is used; (2) the volume-equivalent sphere assumption is better in the case of Q_a (right figure), especially if full Mie theory is used; and (3) the package effect, while relatively unimportant for Q_c, is important in Q_a (right figure) for the case with stronger absorption, but not for the case with weaker absorption.

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 (m_r−1) and ρ′ = 4α m_i, where m_r − im_i, 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 Q_a, the dashed vertical arrow provides the correct ρ ′ (and, hence m_i), while the solid vertical arrow provides the retrieved value of m_i.

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How much error does the deviation of the derived Q_c and Q_a 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 m_r and m_i. Figure 9 provides the resulting computations for a coated cylinder with indices m_inside = 1.05−0.040i and m_outside = 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, m_i is retrieved to within ± 20% with an average (over all sizes) close to 0.010, and the retrieved m_r 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 m_i (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 Q_a [Eq. (15)]. However, the exact Mie results cannot actually be used because m_r 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 m_i even for particles with large aspect ratios; however, the retrieved values of m_r 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 m_i 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.

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 m_i is retrieved to within ± 20% (somewhat better in the red) with an average (over all sizes) close to 0.010, and that the retrieved m_r appears to be too low in almost all cases, but averages ~1.044.

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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 , which shows Q_c 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 Q_c. 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.

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 (m_r−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 Q_c for spheres.

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Given the values of m_r and m_i 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 $σ_{b b}^{(S p h)}$ for comparison with its cylindrical counterpart $σ_{b b}^{(C y l)}$ for each wavelength and particle size. The ratio $R = σ_{b b}^{(C y l)} / σ_{b b}^{(S p h)}$ 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 (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 m_r as the aspect ratio increases (the retrieved m_r becomes smaller as the aspect ratio increases, and $σ_{b b}^{(S p h)}$ is a strong function of m_r). 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.

Fig. 11 Backscattering by a cylinder divided by backscattering by an equal-volume sphere. Black curves: refractive index in the computation of σ_bb 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 σ_bb 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.

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The backscattering ratio R for the higher index (1.20 – 0.000i) is shown in Fig. 12 . Note that in this case $σ_{b b}^{(C y l)} < σ_{b b}^{(S p h)}$ . 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.

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 σ_bb 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.

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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 m_r = 1.05. Note that when the absorption is weak (m_i = 0.002 with a homogeneous distribution of absorbing molecules, ■, or m_i = 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] have provided a highly accurate anomalous diffraction approximation to Q_c 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 Q_c 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 Q_c (and Q_a) are virtually independent of AR and agree well with the Fournier and Evans [23] result for large AR (e.g., AR = 100).

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

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

View Table

It should be noted that when D and L are << λ, and the polarizabilities are determined in the electrostatic approximation (Rayleigh approximation [13,14]), 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, $〚 σ_{b} 〛 \propto {(D^{2} L)}^{2} = D^{6} A R^{2},$ and

\frac{〈 σ_{b} (D, m, A R) 〉}{A R^{2}} \approx \frac{〈 σ_{b} (D, m, A R^{'}) 〉}{A {R^{'}}^{2}} .

Equation (17) replaces Eq. (13) in this regime, and since

{\tilde{σ}}_{b b} = 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]. 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], we found that for cylinders the absorption index (m_i) can be determined with reasonable accuracy, i.e., ~ ± 20%, when m_r is low as it usually is for phytoplankton [24]. Remarkably, this approximation is actually better in retrieving m_i 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 m_i for low-m_r particles that deviate significantly from spheres. When m_r 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 σ_bb for the equal-volume sphere, can lead to an underestimation ( $σ_{b b}^{(C y l)} > σ_{b b}^{(S p h)}$ ) of cylinder backscattering by a significant factor (Fig. 11), largely because of the inaccuracy in estimation of m_r; 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., $σ_{b b}^{(C y l)} < σ_{b b}^{(S p h)} .$ Thus, prediction of $σ_{b b}$ by this method for low index particles could account for some of the “missing” backscattering suggested for marine particles [4]; 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

The author thanks Bruce Draine for providing the DDSCAT code used for most of the computations, and Georges Fournier for providing the IPHASE25 program for computing the scattering by infinite cylinders and the extinction of spheroids. This research was supported by the Office of Naval Research/ Environmental Optics Program, Grant Number N00014-07-1-0226.

References and links

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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]

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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]

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m	AR = 3	AR = 5
1.05 – 0.010i	10.9	3.4
1.20 – 0.000i	18.6	2.5

Light scattering and absorption by randomly-oriented cylinders: dependence on aspect ratio for refractive indices applicable for marine particles

Abstract

1. Introduction

2. Review of light scattering and absorption concepts

2.1. Finite cylinders

2.2 Infinite cylinders

3. Cylinders examined in the present work

4. Extinction and absorption efficiencies

5. Phase function and backscattering probability of cylinders

6. Cylinders compared to equal-volume spheres

7. Comparison with spheroids

8. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (13)

Tables (1)

Equations (17)

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