1. Introduction

Interpretation of the light backscattered out of natural waters requires understanding the backscattering properties of their constituents [1]. However, the backscattering coefficient of marine particles is arguably the poorest known of the inherent optical properties of natural waters [2], and much effort is being focused on remedying this situation. The backscattering properties of marine particles are most-often modeled as homogeneous spheres using Mie theory. The advent of computer codes capable of handling more complex shapes [3], and the increased computational speeds now available, suggest that particle modeling employing simple non-spherical shapes, e.g., disks, rods, etc., will become routine. For example, Gordon and Du [4] used a two-disk model to try to reproduce the backscattering by coccoliths detached from E. huxleyi, which has a well-defined shape (resembling a disk or two roughly parallel disks) and a known composition (Calcite, refractive index relative to water ~1.20). (See Ref. 5 for scanning electron micrographs of E. huxleyi coccoliths.) However, E. huxleyi in fact has a rather complex fine structure that might influence backscattering and should be addressed. I tried to examine this in an earlier paper [5], in which I computed the backscattering of light from thin micrometer-sized disks with periodic angular fine structure using the discrete-dipole approximation [6, 7]. The periodic fine structure was achieved by dividing the disk into equal-angle sectors of angle Δα=2π/2ⁿ, and removing the dipoles from alternate sectors. In that study the diameter of the disks ranged from 1.50 to 2.75 µm and the thickness from 0.05 to 0.15 µm. The values used for n were 4, 5, 6, and 7, providing pinwheel-looking objects (Fig. 1) with 8, 16, 32, and 64 vanes, respectively. The principal result of the study was that when the scale of the periodicity, s (defined to be 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 (water) the backscattering was found to be nearly identical to that of a homogeneous disk possessing a reduced refractive index. In contrast, significant increases in backscattering were observed when the scale of the periodicity was greater than λ/4, reaching a maximum when the scale becomes ~λ/2 [8]. However images of individual coccoliths [5] suggests that their “angular periodic” structure is not precisely periodic. This raises the question: how much does this aperiodicity influence the backscattering, or alternatively, how large must the deviation from periodic be in order to significantly influence the backscattering? I examine this question here by comparing the backscattering cross section of pinwheels with precisely periodic structure with that for pinwheels in which random variations in the angle Δα of individual sectors are introduced.

2. Model of an aperiodic pinwheel

The aperiodic pinwheel is formed by a perturbation of the purely periodic pinwheel effected in the following manner. First, the disk is divided into purely periodic sectors, the angular boundaries of which are designated by the 2ⁿ angles αP. The individual boundary angles are then perturbed to α_I according to

${\alpha }_I={\alpha }_P+\epsilon \frac{2\pi }{2^n}\rho ,$

where 0≤ε≤1 is a constant and -1/2≤ρ≤1/2 is a random number with a uniform probability density. Then, the material of the disk is removed from ever other sector, yielding a pinwheel with a quasi-periodic structure. Four realizations (each based on a difference sequence of pseudorandom numbers) of such pinwheels for n=5 are provided in Fig. 1 for ε=0.5 and 1.0. Defining ∑_I to be the standard deviation in the angle α_I, we find ∑_I=¹²-½εΔα≈0.3εΔα, where Δα=2π/2ⁿ. Likewise defining ∑_Δα to be the standard deviation of the removed (or occupied) sector angles, ∑_Δα=6-½εΔα≈0.4εΔα. Thus, for ε=0.5 and 1.0, the relative standard deviation in angle of the removed (or occupied) sectors is 20 and 40%, respectively. I examined two aperiodic pinwheels. The first has a diameter (D) of 1.50 µm, a thickness (t) of 0.15 µm, and n=5 (Fig. 1). The second has D=2.75 µm, t=0.05 µm, and n=6. The larger disk is similar in size to the distal shield of individual E. huxleyi coccoliths [5].

This method of creating aperiodic pinwheels does not yield structures with the same volume (mass) as the associated periodic pinwheel. The volumes of the four realizations of aperiodic pinwheels studied here are provided in Table 1. The individual realizations are labeled by the one minus the position in a string of pseudorandom numbers where the sampling for ρ begins. Thus, for realization 0000 the sampling begins with the first number, for realization 1000 it begins with the 1001^th number, etc. The variation of the volume for a given ε can be as much as 25% for the smaller (D=1.5 µm, n=5) and 7% for the larger (D=2.75 µm, n=6) pinwheels. The reduction in dispersion from the smaller to the larger is due to the increase in n, which doubles the number of sectors, increasing the probability that the individual realizations have a volume closer to the mean.

 

Fig. 1. The individual rows provide four realizations of the aperiodic pinwheels for n=5 and various values of ε. Periodic pinwheels are shown in the first column.

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3. Operation of the discrete-dipole scattering code

The scattering computations were carried out using the discrete-dipole approximation (DDA) [6, 7]. The accuracy of the DDA for randomly oriented particles is governed by two issues: (1) employing a sufficient number of dipoles to solve the electromagnetic scattering problem for a given orientation; and (2) employing a sufficient number of orientations for performing the orientational average. A measure of the number of dipoles is related to d, the spacing between the dipoles. One wants d to be substantially smaller than the wavelength.

Table 1. The volume (µm³) of the two aperiodic pinwheels examined. The associated volumes for the periodic pinwheel (ε=0) are 0.1325 µm³ and 0.1485 µm³ for the 1.5 and 2.75 µm diameter pinwheels, respectively. D is the diameter and t is the thickness of the associated disk.

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The smaller d, the more dipoles are required to fill the volume of the particle. A convenient measure of the spacing in regard to the wavelength is |m|kd, where m is the refractive index and k=2π/λ. The orientation of a disk-like object is specified by three angles: θ the angle the axis of the disk makes with the incident beam, ϕ, the azimuth of the axis relative to a laboratory-fixed plane containing the incident beam, and β, the angle of rotation around the axis required to place the disk in a specified orientation given θ and ϕ. In general, for an object with no rotational symmetry, e.g., an aperiodic pinwheel, 0°≤θ≤180°, 0°≤ϕ≤360° and 0°≤β≤360° ; however, the high symmetry of a uniform disk reduces these to 0°≤θ≤90°, 0°≤ϕ≤180°, and requires only one value of β, e.g., β=0. For periodic pinwheels (ε=0) the angle β is required, however, its range need only be enough to completely cover one open sector and one adjacent occupied sector. The DDA code performs orientational averaging by computing the scattering at discrete angles equally spaced in ϕ and β, while the angle θ is divided in uniform increments of cosθ. An important consideration in the averaging is that the computation time is roughly proportional to the number of angles (N_θ×N_ϕ×N_β) used in the averaging.

Gordon and Du [8] showed that for a homogeneous disk with D=2.7 µm using ~5000 orientations (N_θ=51, N_ϕ=99, N_β=1) for the orientational average, the error in the backscattering cross section (σb) was of the order of 5% for |m|kd <0.5, and decreased rapidly for smaller values of |m|kd. In the present work, I have always used enough dipoles to keep |m|kd <0.5 (and often <0.4) and a number of orientations that would provide approximately the same averaging accuracy as the uniform disk with D=2.7 µm. For the periodic pinwheels I used N_θ=51, N_ϕ=99 and N_β=4. The four β increments were equally spaced over one open sector and an adjacent occupied sector. Because of the symmetry, this would be equivalent to choosing N_β=128 and 0°≤β≤360°. For the aperiodic pinwheels (with ε=1), I used N_θ=101, N_ϕ=99 and N_β=9. These latter orientations were chosen by extensive testing in which I examined the differential scattering cross section for scattering angles Θ>90°. The region 120°≤Θ≤180°, the phase functions of periodic and aperiodic pinwheels (D=2.75 µm) are highly oscillatory in Θ, and for the above choice of the orientational averaging, the observed oscillations in the periodic and aperiodic cases were similar, facilitating a fair comparison between the two. Actually, σ_b appears to become stable with smaller values of N_θ, N_φ and N_β, where the orientationally-averaged phase function becomes more strongly dependent on the number of orientations, providing confidence that a sufficient number of orientations have been used. In the final analysis, the choice of the number of orientations must be balanced against computational time required.

4. Results of the computations

Figures 2 and 3 provide the results of the backscattering cross section computations for the smaller and larger pinwheels, respectively. Also shown for comparison is the result for a homogeneous disk of the same size (D and t). Note that the homogeneous disk has twice the volume (mass) of the periodic pinwheel (ε=0) and approximately twice the volume of the aperiodic pinwheels (ε>0).

Consider first the periodic pinwheels. For these, s=λ/4 occurs when t/λ=0.255 for the smaller and 0.093 for the larger pinwheel. For t/λ larger than these values, the backscattering increases rapidly with decreasing λ and then undergoes a series of maxima and minima with progressively increasing backscattering at each maximum. The backscattering at the maxima for the smaller pinwheel is, in magnitude, approximately that at the maxima for the homogeneous disk (twice the volume or mass of the pinwheel). For the larger pinwheel, it is approximately 75% of the maxima for a uniform disk. These pinwheel maxima are the result of interference of the fields scattered by the individual vanes of the pinwheel as they occur in the Rayleigh-Gans approximation as well (although only the first maximum occurs at the same position [8]). In the case of the smaller aperiodic pinwheel, when ε=0.5 its backscattering closely follows the periodic pinwheel, with the dispersion of backscattering reaching 20% at the smallest wavelength, while when ε=1.0 there is more significant deviation from the periodic case, and the dispersion is somewhat larger. Recalling that ∑Δα≈0.4εΔα=0. 4×2πε/2ⁿ, we note that the smaller pinwheel with ε=0.5 (n=5) has the same value of ∑_Δα as the larger pinwheel with ε=1.0 (n=6). Interestingly, Figs. 2 (Left Panel) and 3 show that the behavior of σ_b with decreasing λ up to its first minimum are similar in the two cases (ε=0.5, n=5 and ε=1.0, n=6): there are only minor deviations from the periodic pinwheels; and there is small dispersion between the various realizations of the aperiodic pinwheels. In contrast, when ε is increased from 0.5 to 1.0 for the smaller pinwheel, the dispersion increases, and σ_b near its first maximum (t/λ≈0.4) shows a significant decrease from the ε=0 and 0.5 cases. This behavior would be expected under the hypothesis that the maxima in the periodic case results from constructive interference of light interacting with the individual vanes of the pinwheel — when the spacing and angular size of the vanes becomes random the constructive interference is reduced.

 

Fig. 2. The backscattering cross section of four realizations of the aperiodic pinwheels shown in Fig.1 (D=1.5 µm) compared with that for a periodic pinwheel and for a homogeneous disk of the same size. The left panel is for ε=0.5 and the right panel is for ε=1.0.

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Fig. 3. The backscattering cross section of four realizations of the aperiodic pinwheels with ε=1.0, D=2.75 µm and n=6 compared with that for the associated periodic pinwheel.

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Close examination of Fig. 2 at the smaller wavelengths reveals that σ_b for the several realizations is ordered with increasing volume, i.e., the realization with the smallest σ_b has the smallest volume, etc. However, with increasing λ (decreasing t/λ) the order can reverse, particularly near the first maximum near t/λ≅0.4. Thus, there is no way to try to reduce the dispersion throughout the whole wavelength range by normalizing for volume differences. In the Rayleigh-Gans domain, t/λ<0.2 [8], for particles with identical shape, the volume effect on σ_b is proportional to the square of the volume. This would explain the envelope of the variation near t/λ=0.2 in Fig. 2.

Although the volume effect cannot be removed, it is important to understand that real biological particles (e.g., E huxleyi coccoliths) would display similar variations in volume. In fact, if pinwheels were to represent real biological particles, samples would be expected to consist of a number of realizations of their aperiodicity. In this regard, the average σ_b (denoted by 〈σ_b〉) is more important than that for any given realization. Figure 4 compares the 〈σ_b〉 for the four realizations of the aperiodicity examined here with the associated periodic pinwheel. It clearly shows that the main difference between 〈σ_b〉 for the small periodic and aperiodic pinwheels (left panel) occurs near the maxima in the backscattering, and that near the first maximum (but not the second) the difference increases as deviation from periodicity increases (i.e., as ε increases). Considering the large departures from periodicity for the ε=1 realizations in Fig. 1, it is remarkable that, when averaged over realizations, their 〈σ_b〉 is so close to that of periodic pinwheels (ε=0). As suggested in Fig. 3, Fig. 4 (right panel) shows that the 〈σ_b〉 for the large aperiodic pinwheel is very close to its periodic counterpart.

 

Fig. 4. The backscattering cross section 〈σ_b〉 of an equal-number mixture of the four realizations of the aperiodic pinwheels compared with that for a periodic pinwheel (ε=0): left panel D=1.5 µm, t=0.15 µm and n=5; right panel D=2.75 µm, t=0.05 µm and n=6.

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5. Discussion

With only two examples of periodic pinwheels and their aperiodic counterparts, it is difficult to make concrete conclusions regarding the effects of aperiodicity. However, it does appear that some general conclusions are possible. First, Figs. 2 and 3 show that in the transition from the periodic to the aperiodic pinwheel σ_b changes much less than compared to the transition from uniform disk to periodic pinwheel. Second, the dispersion in σ_b among realizations of the aperiodic pinwheels is associated with the dispersion in Δα (or s), which increases with increasing ε. Third, the aperiodic 〈σ_b〉 will usually be somewhat smaller than the periodic σ_b, at least near the position of the first (long-wave) maximum, and this decrease increases with increasing aperiodicity (ε).

Finally, I examined the deviation in the angular spacing of the “spokes” in the distal shield of the individual coccoliths provided in Fig. 2 of reference [8]. For this particular coccolith, ∑Δα/Δα~0.27, and there were 40 open angular sectors. This coccolith shield is similar in size and shape to the larger (2.75 µm) pinwheel (n=6, 32 open sectors) examined here. The computations for the larger pinwheel show that, for the purpose of computing backscattering, the periodic pinwheel is a good approximation to aperiodic pinwheel as long as ∑_Δα/Δα≤0.4 (ε≤1). This suggests that replacing the aperiodic fine structure of the distal shield of E. huxleyi coccoliths with a strictly periodic fine structure will not degrade the modeling of their backscattering, especially for natural samples containing large numbers of coccoliths.