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

Howard R. Gordon

doi:10.1364/OE.15.005572

1. Introduction

Satellite remote sensing of ocean color [1] is now a well-developed tool for studying phytoplankton dynamics on regional to global scales ([2], and papers therein). The term “ocean color” refers to the water-leaving spectral radiance, i.e., the radiance resulting from the backscattering of sunlight out of the water. This radiance is proportional to the ratio of the backscattering coefficient b_b (the differential scattering cross section per unit volume integrated over the backward hemisphere) and the absorption coefficient a of the medium (water plus constituents) [1], i.e., b_b/a. Thus, understanding the backscattering coefficient of the suspended constituents of the natural waters is a central problem in marine optics. However, the backscattering coefficient of marine particles is arguably the poorest known of the inherent optical properties of natural waters [3], and much effort is being focused on remedying this situation. The inherent optical properties of marine particles are most-often modeled as homogeneous spheres using Mie Theory. Although this approach has been fruitful, the next logical step in modeling marine particles is to abandon the normally-employed spherical approximation and use more realistic approximations to their shape. The advent of computer codes capable of handling more complex shapes [4], and the increased computational speeds now available, suggest that particle modeling employing simple non-spherical shapes, e.g., disks, rods, etc., could become routine. Gordon and Du [5] used a two-disk model to try to reproduce the backscattering by coccoliths detached from E. huxleyi. This particular marine particle was chosen for study because (1) its shape is rather precisely known, resembling a disk or two roughly parallel disks; (2) its composition is known (Calcite), providing its refractive index relative to water (∼1.20); (3) its backscattering properties have been measured [6–8]; and (4) it is amenable to remote sensing [9,10]. They found that, while the resulting spectral variation of the backscattering cross section agreed with experiment, its magnitude was low by a factor of 2-3. Such simple shapes are still at best poor approximations to real particles. Thus, to try to understand this discrepancy, I asked [11] the following question: how far can the actual shape of a particle deviate from these simple shapes and still be realistically modeled by them? To shed some light on this issue, I used discrete dipole approximation (DDA) solutions [12, 13] to the scattering problem to investigate the backscattering by disk-like particles possessing periodic angular fine structure (disks divided into equal-angle sectors with alternate sectors removed), i.e., more closely resembling E. huxleyi coccoliths. When the scale of the periodicity (the length of an open or closed sector measured along the circumference of the disk) was < λ/4 (where λ is the wavelength of the light in the medium, i.e., water), I found the backscattering to be nearly identical to that of a homogeneous disk possessing a reduced refractive index. However, significant increases in backscattering were observed when the scale of the periodicity was greater than λ/4.

For many of the cases examined, I have also computed the backscattering cross section using the Rayleigh-Gans approximation (RGA) to scattering. The RGA is applicable when the relative refractive index of the particle (m) is close to unity, and the “size” is ≪ the wavelength of light divided by |m - 1| [14, 15]. Thus the size need not be ≪ the wavelength. It is computationally fast when compared to any other method because analytical formulas are available for many particle shapes. Moreover, extension to particles of any shape is straightforward.

In this paper I compare backscattering by disk-like particles (with refractive index 1.20 relative to water) computed using the RGA with exact (DDA) computations. The comparisons show that the RGA is sufficiently accurate to be useful as a quantitative tool for exploring the backscattering features of disk-like particles with complex structure, e.g., disks with angular periodicities or detached coccoliths from the coccolithophored E. huxleyi. The validity of the RGA for such particles allows investigation of the influence of their birefringence on backscattering.

2. The Electromagnetic scattering problem

Conceptually, the electromagnetic scattering problem can be developed in a simple manner. If a particle is subjected to an incident electromagnetic field _{E⃗⁽⁰⁾(D⃗_i,t)}, then a volume element dV_i at a position _{D⃗_t} within the particle (Fig. 1) will experience an electric field _{E⃗(D⃗_i,t)} given by

\vec{E} ({\vec{D}}_{i}, t) = {\vec{E}}^{(0)} ({\vec{D}}_{i}, t) + \sum_{j} C ({\vec{D}}_{i}, {\vec{D}}_{j}) \vec{E} ({\vec{D}}_{j}, t),

where the sum excludes i=j. The E’s on both sides of this equation are unknown, while E ⁽⁰⁾ and C are known functions of position and time. This electric field induces a dipole moment (dp) in dV_i given by

d \vec{p} ({\vec{D}}_{i}, t) = ρ_{n} α \vec{E} ({\vec{D}}_{i}, t) d V_{i},

where α is the polarizability tensor and ρ_n is the number density of atoms (molecules). At a great distance r⃗_i from the particle the field due to the dipole moment induced in dV_i is

d {\vec{E}}^{(s)} = \frac{- 1}{4 π ε_{0} r_{i}} \vec{κ} \times [\vec{κ} \times d {\vec{p}}_{i} (\frac{{\vec{D}}_{i}, t - r_{i}}{c})],

where κ⃗ is the vector shown in Fig. 1 (_{|κ⃗| = 2π/λ}, where λ is the wavelength of the incident field in the medium in which the particle is immersed, and |κ⃗| = |κ⃗₀|) and c is the speed of light. The vector r⃗_i is assumed to be sufficiently far from the origin (O) that it may be replaced by r⃗ except where it occurs in a phase. The total field at r⃗ , given by

{\vec{E}}^{(s)} (\vec{r}, t) = \int d {\vec{E}}^{(s)} (\vec{r}, t),

is the “scattered” field.

In the laboratory reference frame (x,y,z), the incident electric field propagating in the κ⃗₀ direction is given by

{\vec{E}}^{(0)} ({\vec{D}}_{i}, t) = {\vec{E}}^{(0)} \exp [i ({\vec{κ}}_{0} ● {\vec{D}}_{i} - ωt)],

where the field amplitude is resolved into components parallel and perpendicular to the scattering plane (See Fig. 2):

{\vec{E}}^{(0)} = E_{ℓ}^{(0)} {\hat{e}}_{ℓ}^{(0)} + E_{r}^{(0)} {\hat{e}}_{r}^{(0)} = (\begin{matrix} E_{r}^{(0)} \\ E_{ℓ}^{(0)} \end{matrix}) .

Fig. 1. A volume element dV_i, located at a point D_i from the origin of coordinates (O). The incident radiation is propagating in the κ₀ direction and the scattered radiation is propagating in the κ direction. The vector r_i is from the volume element dV_i, to a distant point at which the scattered field is measured. The vector r is from the origin to the same distant point, which is sufficiently far from the particle that the vectors r_i and r are considered to be parallel.

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Fig. 2. The plane formed by the propagation vector κ⃗₀ of the incident wave and the propagation vector κ⃗ of the scattered wave is the scattering plane. The incident and scattered fields are resolved into components parallel and perpendicular to the scattering plane, i.e., along (ê ⁽⁰⁾ _l , ê ⁽⁰⁾ _r) and (ê_l, ê_r), respectively. Θ is the scattering angle.

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Resolving the scattered field E⃗ ^(s) into components parallel and perpendicular to the scattering plane as well (note that ê ⁽⁰⁾ _l and ê_l, are not parallel), the scattered field at r̂, which is in the form of a spherical wave, can then be written

{\vec{E}}^{(s)} = \frac{1}{i κ r} A {\vec{E}}^{(0)} \exp [i (κ r - ω t)] or (\begin{matrix} E_{r}^{(s)} \\ E_{ℓ}^{(s)} \end{matrix}) = \frac{1}{i κ r} (\begin{matrix} A_{r r} & A_{l r} \\ A_{r l} & A_{l l} \end{matrix}) (\begin{matrix} E_{r}^{(0)} \\ E_{ℓ}^{(0)} \end{matrix}) \exp [i (κ r - ωt)],

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

{\vec{E}}^{(s)} = E_{ℓ}^{(s)} {\hat{e}}_{ℓ} + E_{r}^{(s)} {\hat{e}}_{r} = (\begin{matrix} E_{r}^{(s)} \\ E_{ℓ}^{(s)} \end{matrix}) .

3. The differential scattering cross section

To relate the scattered field to scattering cross sections, etc., we recall that the time averaged Poynting vector of the scattered field is

〈 \vec{S} 〉 = \frac{\hat{κ}}{2 μ_{0} c} 〈 E_{ℓ} E_{ℓ}^{*} + E_{r} E_{r}^{*} 〉 = \frac{\hat{κ}}{2 μ_{0} c} 〈 {\tilde{\vec{E}}}^{*} \vec{E} 〉 = \hat{κ} \frac{d P}{d A},

Where the superscript ^* indicates the complex conjugate, the tilde indicates the transposed matrix and, dP is the power crossing an area dA oriented normal to the propagation direction κ⃗ (i.e., the irradiance associated with the propagating field). The differential scattering cross section is defined to be the power scattered into a solid angle dΩ. divided by the irradiance of the incident beam, i.e.,

\frac{d σ}{d Ω} \equiv \frac{\frac{d P^{(s)}}{d Ω}}{\frac{d P^{(0)}}{d A}} = r^{2} \frac{∣ 〈 {\vec{S}}^{(s)} 〉 ∣}{∣ 〈 {\vec{S}}^{(0)} 〉 ∣},

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

∣ 〈 {\vec{S}}^{(0)} 〉 ∣ = \frac{1}{2 μ_{0} c} 〈 {\tilde{\vec{E}}}^{(0) *} {\vec{E}}^{(0)} 〉 and ∣ 〈 {\vec{S}}^{(s)} 〉 ∣ = \frac{1}{2 μ_{0} c} \frac{1}{κ^{2} r^{2}} 〈 {\tilde{\vec{E}}}^{(0) *} {\tilde{A}}^{*} A {\vec{E}}^{(0)} 〉,

so

\frac{d σ}{d Ω} = \frac{1}{κ^{2}} \frac{〈 {\tilde{\vec{E}}}^{(0) *} {\tilde{A}}^{*} A {\vec{E}}^{(0)} 〉}{〈 {\tilde{\vec{E}}}^{(0) *} {\vec{E}}^{(0)} 〉} .

If the incident field is unpolarized, then

〈 E_{r}^{{(0)}^{*}} E_{r}^{(0)} 〉 = 〈 E_{ℓ}^{{(0)}^{*}} E_{ℓ}^{(0)} 〉 and 〈 E_{r}^{{(0)}^{*}} E_{ℓ}^{(0)} 〉 = 0 = 〈 E_{ℓ}^{{(0)}^{*}} E_{r}^{(0)} 〉,

and the differential cross section becomes

\frac{d σ}{d Ω} = \frac{1}{κ^{2}} \frac{〈 {\tilde{\vec{E}}}^{(0) *} {\tilde{A}}^{*} A {\vec{E}}^{(0)} 〉}{〈 {\tilde{\vec{E}}}^{(0) *} {\vec{E}}^{(0)} 〉} = \frac{1}{2 κ^{2}} ({∣ A_{r r} ∣}^{2} + {∣ A_{ℓ r} ∣}^{2} + {∣ A_{r ℓ} ∣}^{2} + {∣ A_{ℓ ℓ} ∣}^{2}) .

(Note: the quantity S ₁₁ defined by Bohren and Huffman [14] is κ²dσ/dΩ.)

4. The Rayleigh-Gans approximation

In its simplest form, in the Rayleigh-Gans Approximation [4, 14, 15] (RGA) to scattering, the sum in the Eq. (1) is ignored (C = 0), i.e., the only field experienced by dV_i is the incident field. Thus, the RGA provides the “zeroth-order” approximation to the scattered field. In the laboratory reference frame (x,y,z), the incident electric field is given by Eq. (5) so the induced dipole moment [Eq. (2)] is

d {\vec{p}}_{i} (\frac{{\vec{D}}_{i}, t - r_{i}}{c}) = ρ_{n} α ({\vec{D}}_{i}) {\vec{E}}^{(0)} (\frac{{\vec{D}}_{i}, t - r_{i}}{c}) d V_{i}

Then noting that _{κ = ω/c}, we have _{r_iω/c = κr_i = κr-κ⃗●D⃗_i},

d {\vec{p}}_{i} (\frac{{\vec{D}}_{i}, t - r_{i}}{c}) = ρ_{n} α ({\vec{D}}_{i}) {\vec{E}}^{(0)} \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● {\vec{D}}_{i}] \exp [i (κ r - ω t)] d V_{i} .

Resolving the fields into components parallel and perpendicular to the scattering plane, as in Eq. (6) the scattered field is

d {\vec{E}}^{(s)} = \frac{1}{i κ r} d A {\vec{E}}^{(0)} \exp [i (κ r - ω t)],

where d A _i is the contribution to the matrix A from dV_i and is given by

d A_{i} = \frac{- i ρ_{n} κ}{4 π ε_{0}} \vec{κ} \times [\vec{κ} \times α ({\vec{D}}_{i})] \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● {\vec{D}}_{i}] d V_{i} .

The scattering plane is the x-z plane (Fig. 2), so

{\vec{E}}^{(0)} = E_{ℓ}^{(0)} {\hat{e}}_{ℓ}^{0} + E_{r}^{(0)} {\hat{e}}_{r}^{0} = E_{ℓ}^{(0)} {\hat{e}}_{x} - E_{r}^{(0)} {\hat{e}}_{y},

and

{\vec{κ}}_{0} = κ {\hat{e}}_{z} .

Inserting these into the equation for d A _i yields

d A_{i} = \frac{i ρ_{n} κ^{3}}{4 π ε_{0}} (\begin{matrix} - α_{y y} & α_{x y} \\ - α_{y z} \sin Θ + α_{x y} \cos Θ & α_{x z} \sin Θ - α_{x x} \cos Θ \end{matrix}) \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● {\vec{D}}_{i}] d V_{i},

where the α’s are the components of α in the laboratory reference frame (x,y,z). The total scattered field is found by integration over the volume of the object:

A_{} = \frac{i ρ_{n} κ^{3}}{4 π ε_{0}} \int \int_{V} \int (\begin{matrix} - α_{y y} & α_{x y} \\ - α_{y z} \sin Θ + α_{x y} \cos Θ & α_{x z} \sin Θ - α_{x x} \cos Θ \end{matrix}) \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● {\vec{D}}_{i}] d V .

If the α’s are independent of position within the particle, the matrix can be removed from the integration. If the particle’s polarizability tensor is isotropic (i.e., α_ij = α δ_ij) then A reduces to

A = \frac{- i ρ_{n} α κ^{3}}{4 π ε_{0}} (\begin{matrix} 1 & 0 \\ 0 & \cos Θ \end{matrix}) \int \int_{V} \int \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● \vec{D}] d V,

and the differential cross section becomes

\frac{d σ}{d Ω} = {(\frac{ρ_{n} α κ^{2}}{4 π ε_{0}})}^{2} \frac{(1 + \cos^{2} Θ)}{2} {∣ \int \int_{V} \int \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● \vec{D}] d V ∣}^{2} .

The polarizability of the particle can be related to the refractive index, m, through the Clausius-Mossotti equation:

\frac{ρ_{n} α}{ε_{0}} = 3 (\frac{m^{2} - 1}{m^{2} + 2});

and defining

R \equiv \int \int_{V} \int \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● \vec{D}] d V,

we have

\frac{d σ}{d Ω} = \frac{9}{16 π^{2}} κ^{4} {(\frac{m^{2} - 1}{m^{2} + 2})}^{2} (\frac{1 + \cos^{2} Θ}{2}) {∣ R ∣}^{2}

for the differential scattering cross section of a single particle of volume V. If the particle is immersed in a refracting medium, then m is the refractive index of the particle relative to the medium.

The total (σ) and back (σ_b) scattering cross sections are, respectively,

σ = \iint_{4 π} \frac{d σ}{d Ω} d Ω and σ_{b} = \iint_{Back 2 π} \frac{d σ}{d Ω} d Ω,

while the scattering phase and volume scattering functions are

P (Θ) = \frac{4 π}{σ} \frac{d σ (Θ)}{d Ω} and β (Θ) = N \frac{d σ (Θ)}{d Ω},

where N is the number density of scatterers. The contribution that particles of a given size and shape make to the total scattering coefficient (b) and the backscattering coefficient (b_b) are b = Nσ and b_b = Nσ_b, respectively. In the RGA, the shape of the particle enters only through the computation of R. Analytic formulas are available for simple shapes, e.g., spheres and cylinders; however, it is easy to carry out the integrations numerically for particles of any shape. For particles other than spheres, R depends on the orientation of the particle. For particles with a given orientational distribution function, dσ/dΩ. must be computed for a large number of orientations and the appropriate weighted average formed.

The fact that dV_i is subjected only to the incident field requires that two conditions must hold for the RGA to have validity: (1) there must be insignificant refraction or reflection at the surface of the particle, which implies |m - 1| must be ≪ 1; and (2) the phase of the incident field must not shift significantly over distances of the order of the “size” (L) of the particle, which requires κL|m - 1|≪1.

This is one of many forms of the RGA. The resulting cross section becomes coincident with the equations in van de Hulst¹⁵ by using the requirement that |m - 1| ≪ 1 so that ρ_nα/ε ₀ ≈ 2(m - 1). Refinements to the RGA have been proposed by several authors, such as using the electrostatic approximation to relate the polarizabilities to the refractive index [16], or replacing the magnitude of the propagation vector in the medium by that in the particle [17]. I employ the form presented here because it is more closely related to the DDA, i.e., the DDA [18] uses the Clausius-Mossotti relationship modified to include radiation reaction and the “lattice dispersion relation.”

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

In what follows we compare the backscattering cross sections (σ_b) of randomly-oriented disk-like objects computed via the RGA and the DDA. The DDA results are taken as “exact” computations (the DDA-computed σ_b’s are expected to be in error by no more than 5%). As an early motivation for such comparisons was interest in the backscattering of coccoliths detached from E. huxleyi suspended in water, I consider disks with diameters 1.5 to 2.75 μm with m = 1.2 (Calcite in water). Figure 3 provides such a comparison for a 2.75 μm homogeneous disk of various thicknesses (t). The comparison shows that the RGA is close to the DDA for t/λ _Water less than, or approximately equal to, 0.20 to 0.25. Perhaps more importantly, the comparison shows that σ_b can be expected to oscillate with increasing t (or decreasing λ _Water), i.e., the RGA also provides the qualitative character of the spectral variation of σ_b. It should be pointed out that the (approximate) “physical optics” developed by model of Gordon and Du [5] out performs the RGA when t/λ _Water > 0.2, following the DDA reasonably well up to a t/λ _Water of 0.8.; however, it cannot be applied to the more complex particle shapes of interest here, e.g., disk-like particles with periodic angular fine structure.

Fig. 3. Comparison of the backscattering cross section computed with the RGA and the DDA for a homogeneous disk of diameter 2.75 μm and thicknesses 0.05, 0.10, and 0.15 μm.

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For a more complex example, Fig. 4 compares RGA and DDA computations of σ_b for the Gordon and Du [5] “fishing-reel” model of a detached coccolith. The fishing-reel model consists of two parallel disks of diameter D _o with material removed from a concentric circle of diameter D_i (i.e., a washer-like object). The two disks are joined together by a hollow cylinder of inner diameter D_i and outer diameter D_r. The axis of the cylinder passes through the center of both disks. The individual disks have a thickness of 50 nm and the space between them (the height of the joining cylinder) is t. Table 1 provides values of the parameters of the three fishing-reel models investigated. The three models all have the same volume (∼0.587 μm³). This is accomplished by decreasing the thickness of the wall of the connecting cylinder as shown in Table 1.

For these models, the individual disks have a thickness of 0.05 μm, and therefore are within the t/λ _Water < 0.2 criterion from Fig. 3 in the visible. As in Fig. 3, Fig. 4 shows that the RGA and DDA produce qualitatively similar spectral variations, and surprisingly good quantitative agreement even though the total thickness of the particle exceeds λ _Water in some cases, and the total diameter is several times λ _Water. Comparison of the two suggests that the RGA can be a valuable tool in exploring problems involving multiple disks as long as the individual disks satisfy the t/λ _Water < 0.2 criterion.

Fig. 4. Comparison of RGA and DDA computations for the Gordon and Du [5] “fishing reel” models of a detached coccolith.

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Table 1:. Parameters of the Gordon and Du [5] “Fishing-reel” model of a detached coccolith.

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In an effort to understand the influence of small-scale periodic structure in disk-like objects on backscattering, I examined [11] backscattering by a “sectorized” disk formed by starting with a homogeneous disk and removing sectors. Specifically, the disk was divided into equal angle sectors of angle Δα and alternate sectors were removed. The angle Δα was given by

Δ α = \frac{2 π}{2^{n}}

where n is an integer. Figure 5 provides the positions of one layer of dipoles for the resulting structures for n = 4 to 7. I will refer to these objects as “pinwheels.” If we let s be the arc length of the open (or closed) regions at the perimeter of the pinwheel, then s = D_dΔα/2, where D_d is the diameter of the disk. The values of s for the various cases that I examined (D_d = 1.5 μm) were such that at a wavelength (λ) of 400 nm in vacuum (300 nm in water), as n progresses from 4 to 7, s took on the values λ, λ/2, λ/4, and λ/8 in water. One of the main goals of my study was to determine if a relationship exists between s and λ where the periodic structure becomes important (or unimportant) to the backscattering.

Fig. 5. Sectorized disks (“pinwheels”) for various values of n used in this study.

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Fig. 6. Comparison of DDA and RGA backscattering by sectorized disks in Fig. 5.

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The results of the computations of the backscattering cross section, σ_b, carried out for 1.5 μm pinwheels are provided in Fig. 6 (DDA on the left from Ref 11, and RGA on the right), which displays σ_b, as function of the thickness (t) of the disk divided by the wavelength of the light in water (λ _Water). Three thicknesses of the disk are used: 0.05, 0.10, and 0.15 μm. The wavelength λ _Water covers the range from 200 nm to over 1000 nm. Note the qualitative similarity between the DDA and the RGA computations. Both show that the backscattering appears to follow a “universal curve” that is close to that for a homogeneous disk with a reduced index m = 1.10 rather than 1.20 (labeled 1.10 in the key to the figure); however, as the wavelength decreases σ_b, suddenly departs from the universal curve and increases dramatically. This was first observed through extensive computations using the DDA; however, in this case, the behavior could have been predicted based on the RGA computations. (The departure of σ_b, from the universal curve occurs when the maximum arc length of the open or closed regions of the pinwheel exceeds λ _Water/4.). In Fig. 7 the comparisons in Fig. 6 are carried to larger values of t/λ _Water, and show that the RGA agrees well with the DDA for values of t/λ _Water up to, and somewhat beyond the first maximum that occurs in σ_b after the departure from the “universal curve.” This maximum is near s/λ _Water = 1/2. For larger values of t/λ _Water the RGA still provides the qualitative nature of the variation of σ_b with t/λ _Water; however, it no longer quantitatively reproduces the DDA computations.

In Fig. 8 a more complex geometry – parallel pinwheels – is examined. This is somewhat similar to the “fishing reel” model [5] for detached coccoliths, but uses sectorized disks (washers) with n = 5 and 6 rather than homogeneous disks and the outside diameter is 1.50 μm rather than 2.75 μm. Again, the agreement between RGA and DDA is quite good throughout the visible, even in a quantitative sense.

Fig. 7. Comparison of DDA and RGA backscattering by sectorized disks in Fig. 5.

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Fig. 8. Comparison of DDA and RGA backscattering by parallel sectorized (n = 5 and 6) washers. The individual washers have an outside diameter of 1.50 μm, inside diameter 1.00 μm and thickness 0.05 μm. They are separated by a space of 0.30 μm.

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

As the E. huxleyi coccolith is composed of calcite, one would expect it to be birefringent. This is indeed the case. The c-axis (optical axis) of the component parts of the E. huxleyi coccolith is radial, i.e., along the “spoke-like” structures [19]. How does this birefringence influence the backscattering? Since RGA provides an adequate description of the backscattering of homogeneous or structured disk-like objects as long as t/λ _Water < 0.2, which is satisfied by the individual coccolith plates throughout the visible, we expect that it would apply equally well to a birefringent disk. Thus, we will investigate the possible influence of birefringence on E. huxleyi backscattering by comparing the backscattering in the RGA of a birefringent and an isotropic disk. Computation of the scattering matrix A for an anisotropic disk, for which the optical axis at any point is radial, is sketched out in the Appendix. A uniaxial crystal, Calcite has two refractive indices: m_e for propagation with the electric vector parallel to the c-axis; and m_o for propagation with the electric vector perpendicular to the c-axis. Letting m_i represent the refractive index of the isotropic disk, we take

\frac{ρ_{n} a}{ε_{0}} = 3 (\frac{m_{e}^{2} - 1}{m_{e}^{2} + 2}) and \frac{ρ_{n} b}{ε_{0}} = 3 (\frac{m_{o}^{2} - 1}{m_{o}^{2} + 2})

for the polarizabilities a and b (see the Appendix) of the birefringent disk, and

\frac{ρ_{n} α}{ε_{0}} = 3 (\frac{m_{i}^{2} - 1}{m_{i}^{2} + 2})

for the isotropic disk. Clearly, m_i must depend on m_o and m_e in some manner, and one of the goals of this exercise is to find the combination that provides the best agreement for backscattering of the isotropic and the anisotropic cases. If the disk were composed of small grains of Calcite in random orientation, one would expect [20] the average refractive index for unpolarized light to be approximately (2m_o + m_e)/3. Using tabulated values for the refractive indices of Calcite [21] near 500 nm and taking 1.338 for the index of water, we have m_o = 1.241, m_e = 1.113, and m_i = 1.198 (close to the value 1.20 used in the earlier computations). Comparison of the RGA-computed σ_b, for the radially-anisotropic disk with these values of m_o and m_e with the isotropic disk with index m_i = (2m_o + m_e)/3 is provided in Fig. 9. The isotropic disk’s backscattering cross section is seen to be somewhat higher than the birefringent disk. The two can be brought into agreement by taking m_i = 1.188 = 0.57 m_o + 0.43 m_e. (For the total scattering cross section, m_i = (2m_o + m_e)/3 provides better agreement between the two than m_i = 1.188.) This suggests that for the computation of backscattering by model coccoliths with the DDA, a zeroth-order account of the birefringence can be effected by using m_i = 1.188 rather than 1.198.

Fig. 9. Comparison of RGA computations of backscattering for a randomly-oriented birefringent disk and an isotropic disk with m_i = (2m_o + m_e)/3.

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7. Concluding Remarks

The value of the RGA in obtaining qualitative information regarding backscattering by disk-like particles has been demonstrated. The success of the RGA in this case derives from the fact that, unlike a spherical particle with the same mass, most of the volume elements of a thin disk are far enough away from any given element that their interaction is small, i.e., a relatively small amount of the total mass of the particle is close to any one of the volume elements.

I am not advocating the use of the RGA for quantitative computations of σ_b, for disk-like particles. Rather, because it is computationally fast compared to the DDA, it can be used to explore the backscattering of disk-like models of marine particles for the purpose of either excluding models with unacceptable qualitative behavior, or selecting promising models for further study using the more time-consuming DDA.

8. Appendix: scattering by a birefringent disk

Here we develop the formulas for scattering from a birefringent disk. As our application is to the E. huxleyi coccoliths we take the disk to be uniaxial with the optical axis at any point in the disk in the radial direction. We develop the anisotropic case first and then reduce these formulas to the isotropic case.

A. Anisotropic case

Figure A1 provides the geometry of the scattering problem. The body-fixed coordinate system is cylindrical with radial coordinate $ρ'$ the angle $η'$ and the coordinate z′ normal to the axis of the disk. In the integral for A, Eq. (15), the required elements of the polarizability matrix (α_yy, α_xy, α_xz, α_xy, and α_xx) must be provided in laboratory-fixed reference system and depend on the particle’s orientation. However, in the body-fixed reference system the polarizability matrix assumes a particularly simple form:

α_{B} = (\begin{matrix} α_{ρ' ρ'} & 0 & 0 \\ 0 & α_{η' η'} & 0 \\ 0 & 0 & α_{z' z'} \end{matrix}) \equiv (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & b \end{matrix}) .

The transformation of this matrix to the laboratory-fixed system is straightforward:

α = \tilde{U} \tilde{B} α_{B} BU,

where the matrices U and B are related to the Euler angles (θ, ϕ, ψ) and $η'$ (Fig. A1 and Fig. A2) through

Fig. A1. A schematic of scattering by a disk. The cylindrical coordinate system ( $ρ'$ , $η'$ , z′) is fixed with respect to the disk (z′ is in the direction of the normal, n).

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Fig. A2. Relationship between the laboratory-fixed coordinate system (x, y, z) and the body-fixed system (x′, y′, z′) or ( $ρ'$ , $η'$ , z′). θ, ϕ, and ψ are the Euler angles. Because of the symmetry of the disk the angle ψ can be set to zero.

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U = (\begin{matrix} \cos ψ \cos ϕ - \cos θ \sin ϕ \sin ψ & \cos ψ \sin ϕ + \cos θ \cos ϕ \sin ψ & \sin ψ \sin θ \\ - \sin ψ \cos ϕ - \cos θ \sin ϕ \cos ψ & - \sin ψ \sin ϕ + \cos θ \cos ϕ \cos ψ & \cos ψ \sin θ \\ \sin θ \sin ϕ & - \sin θ \cos ϕ & \cos θ \end{matrix})

and

B = (\begin{matrix} \cos η' & \sin η' & 0 \\ - \sin η' & \cos η' & 0 \\ 0 & 0 & 1 \end{matrix}) .

Because of the symmetry of the disk, the Euler angle ψ is redundant and may be set to zero. The matrix elements of α thus depend on θ, ϕ, and $η'$ .

To carry out the required integrations to find A, we need _{D⃗ = ρ $⃗'$ + z $⃗'$}, and by resolving _{(κ⃗₀ - κ⃗)} into components parallel and normal to the disk’s surface, we find

({\vec{κ}}_{0} - \vec{κ}) ● \vec{D} = 2 κ \sin (\frac{Θ}{2}) [ρ' \cos (η' - γ) \sin β + z' \cos β]

and

\cos β = \cos θ \sin (\frac{Θ}{2}) - \sin θ \sin ϕ \cos (\frac{Θ}{2}),

where γ is the angle between the component of _{(κ⃗₀ - κ⃗)} parallel to the plane of the disk and the x’axis. A typical integral that must be evaluated to find A is then

\int_{0}^{t} \int_{0}^{R} \int_{0}^{2 π} α_{yy} \exp {i 2 κ \sin (\frac{Θ}{2}) [ρ' \cos (η' - γ) \sin β + z' \cos β]} ρ' d η' d ρ' d z' .

Explicit relationships for the components of α are:

α_{x x} = \frac{1}{2} \cos^{2} η' [a + b + (a - b) \cos 2 ϕ]

Thus, for a given orientation of the disk (given θ and ϕ), all of the required integrals are of the form

\int_{0}^{t} \int_{0}^{R} \int_{0}^{2 π} (\begin{matrix} \cos^{2} η' \\ \cos η' \\ \sin^{2} η' \end{matrix} \sin η') \exp {i 2 κ \sin (\frac{Θ}{2}) [ρ' \cos (η' - γ) \sin β + z' \cos β]} ρ' d η' d ρ' dz' .

Figure A3 provides the required $η'$ integrals, and this integrates to

\frac{2 π}{{κ'}^{2}} (\begin{matrix} 1 - J_{0} (κ' R) \\ 0 \\ κ' R J_{1} (κ' R) + J_{0} (κ' R) - 1 \end{matrix}) \frac{2}{k'} \sin (\frac{k' t}{2}),

where

κ' = 2 κ \sin (\frac{Θ}{2}) \sin β,

and

\cos β = \cos θ \sin (\frac{Θ}{2}) - \sin θ \sin ϕ \cos (\frac{Θ}{2}) .

The disk orientation enters through the variation of the α_ij′s with θ and ϕ as well as through $κ'$ and k′.

The complete formulas for the scattering amplitude matrix and cross sections are straightforward to write down, but are too complicated to be informative. Rather, we shall only provide results of numerical computations in the text. However, in the case of an isotropic disk, the formulas are simple and are provided in the next subsection.

B. Isotropic case

When the polarizability of the disk material is isotropic, i.e., a = b, then α_ij = aδ_ij, and the scattering amplitude matrix becomes

A = \frac{- i ρ_{n} κ^{3} a}{4 π ε_{0}} (\begin{matrix} 1 & 0 \\ 0 & \cos Θ \end{matrix}) \int \int_{V} \int \exp [i ({\vec{κ}}_{0} - \vec{κ}) ● \vec{D}] d V

The differential cross section is then

\frac{d σ}{d Ω} = {(\frac{ρ_{n} κ^{2} a}{4 π ε_{0}})}^{2} {(2 V \frac{J_{1} (κ' R)}{κ' R} \frac{\sin (\frac{k' t}{2})}{\frac{k' t}{2}})}^{2} (\frac{1 + \cos^{2} Θ}{2}) .

Note that the orientation of the disk enters only through the parameters $κ'$ and k′

Fig. A3. The required integrals for the evaluation of the matrix A. The J’s are Bessel functions.

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Acknowledgments

The author is indebted to K. Voss for many useful discussions, B. Draine and P. Flatau for providing their DDA code, and the Office of Naval Research for support under Grant Numbers N000140510004 and N000140710226.

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Model	D _o	D_i	D_r	t
	(μm)	(μm)	(μm)	(μm)
SR	2.75	1.38	1.93	0.1
MR	2.75	1.38	1.68	0.2
BR	2.75	1.38	1.58	0.3

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

Abstract

1. Introduction

2. The Electromagnetic scattering problem

3. The differential scattering cross section

4. The Rayleigh-Gans approximation

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

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

7. Concluding Remarks

8. Appendix: scattering by a birefringent disk

A. Anisotropic case

B. Isotropic case

Acknowledgments

References and links

Cited By

Figures (12)

Tables (1)

Equations (62)

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