Assessing the depolarization capabilities of nonspherical particles in a super-ellipsoidal shape space

Lei Bi; Wushao Lin; Dong Liu; Kejun Zhang

doi:10.1364/OE.26.001726

1. Introduction

Polarization LiDAR is a powerful remote sensing technique for inferring the microphysics of aerosols and hydrometeors in the atmosphere [1,2]. As a critical backscattering quantity, the linear depolarization ratio (LDR) has been commonly used in active polarization LiDAR remote sensing applications, for example, for discriminating cloud phase (e.g., [3,4]) and typing atmospheric aerosols such as dust, polluted dust, and smoke (e.g., [5–7]). The LDR is zero for a homogeneous sphere, such as liquid droplets, because the backscattered waves retain the incident state of polarization. For non-spherical particles, such as ice crystals and aerosols, the LDR is generally non-zero except for special orientations of a nonspherical particle. In addition to the particle shape, the LDR also strongly depends on the particle sizes and refractive indices. However, no simple relationship exists between the LDR and the particle irregularity and refractive index. Sassen [1] reviewed typical depolarization values of various components in the atmosphere, including molecules, water clouds, ice clouds and aerosols. Because of growing interest in polarization LiDAR remote sensing and advances in computational techniques, the LDR studies have been continuously carried out, including studies addressing simple geometries and complex aerosols, such as mixed aerosols and aged soot, commonly occurred in aerosol transport processes [8–11]. Although significant advances in LDR studies have been achieved, understanding the LDR change with respect to the aerosol particle morphology and refractive index is far less completed. This is mostly because of the inability to model exact replicas of actual particle ensembles, and a lack of efficient and sufficiently accurate computational tools to compute the optical properties of arbitrarily shaped nonspherical particles. Moreover, in the laboratory, measuring the LDR for an aerosol sample at the direct backscattering direction is challenging. Recent progress in making laboratory measurements of the LDRs of aerosols and ice particles can be found in [12–14].

Two theoretical approaches exist in modeling the shape dependence of LDRs. One common approach is based on representing aerosols with idealized geometries, such as spheroids, cylinders, and cubes [8–11, 14–16]. This approach can reproduce the LDRs for typical particle systems at a single wavelength and requires less demanding computational resources (e.g., [14]). However, the general representativeness might be called into question due to a limited degree of freedom. The second approach is based on geometrical models constructed from microscopic images, aiming to capture the major detailed characteristics of realistic aerosol samples (e.g., [17]). This approach is advantageous for testing whether the detailed features have importance on the optical properties. However, this “realistic” model might have limited representativeness because the aerosol morphology in ambient atmosphere is relatively diverse. Additionally, unlike idealized geometries, “realistic” models normally require much more computational resources.

To further improve our understanding of the LDR of atmospheric particles, here we systematically investigate the LDRs in a super-ellipsoidal shape space (see Section 2) to assess how the LDR changes with respect to the particle shape. Including spheroids as special scenarios, super-ellipsoids have additional morphology freedom and can model the particle roundness, sharp edges and concavities, which are salient characteristics of most realistic aerosols. Specifically, various shapes can be generated in capturing the diversity of realistic aerosols by varying the aspect ratio and roundness parameters. In addition, super-ellipsoids can even be employed to mimic some realistic aerosol morphologies (e.g., rounded sea salts). Furthermore, a continuous variation of super-ellipsoids particularly from spherical to super-ellipsoidal particles can also be used as a reference to understand the LDR change as the particle morphology changes during phase transition under ambient atmospheric conditions. Although the shape variation in the atmosphere could be different, it is important to understand how the particle change has possible distinct effects on the LDR based on modeling studies. The aforementioned studies are now feasible due to the invariant imbedding T-matrix program (II-TM) [18–20], which allows for extensive computations of the optical properties of arbitrarily shaped nonpsherical particles with reasonable computational resources.

In this paper, we start with a systematic study of the depolarization ratios of spheroids, followed by investigating the LDR change from spheroids to super-ellipsoids. A fine resolution of the aspect ratio and size parameter of spheroids was adopted for modeling the LDRs, because LDRs could be highly sensitive to changes in these parameters. To represent the optical constants of atmospheric aerosols, the refractive index was set up with the real part ranging from 1.1 to 2.0 and the imaginary part from 10⁻⁷ to 0.5. We used a combination of the extended boundary condition method (EBCM) [21] and the II-TM for computing the optical properties of spheroids. Based on the LDR knowledge of spheroids, we analyzed the effect of the shape variation (from spheroids to super-ellipsoids) on the LDR. To compute the optical properties of general super-ellipsoids, we used solely the II-TM, which allows for computing the optical properties of super-ellipsoids in a relatively large size parameter range with reasonable computational resources and time. In Section 2, we describe the super-ellipsoidal model. The representative results as well as the implications in remote sensing are discussed in Section 3. Section 4 is the summary and conclusions of this study.

2. Model, definition and computational methods

The super-ellipsoidal equation is written as [22,23]

{[{(\frac{x}{a})}^{2 / e} + {(\frac{y}{b})}^{2 / e}]}^{e / n} + {[\frac{z}{c}]}^{2 / n} = 1,

where

a

,

b

, and

c

are three semi-axes along the

x

,

y

, and

z

directions in the Cartesian coordinate system, and

e

and

n

are roundness parameters to specify the morphology variation of the particle. When

n = e = 1

, the shape is an ellipsoidal particle. As

n (= e)

increases, the shape becomes octahedron-like until a perfect octahedron is formed at

n = e = 2

. For

n (e) > 2

, the octahedron becomes concave. As

n (= e)

decreases, the shape tends to become cube-like and a perfect cube is formed as

n (= e)

approaches zero. In the present study, we focus on the supper-ellipsoidal particles with

a = b

and

n = e

. The aspect ratio is defined as

a / c

(or

b / c

because

a

is assumed to equal to

b

), which is smaller than unity for prolate particles and larger than unity for oblate particles. The size parameter is defined as

k x_{m}

, where

k

is the wavenumber (

k = 2 π / λ

;

λ

is the wavelength), and

x_{m}

is the maximum of semi-axes (i.e.,

x_{m} = c

for prolate particles;

x_{m} = a

for oblate particles). The advantage of super-ellipsoids compared to spheroids is that the roundness parameters allow flexible choice in modeling the shape variation in addition to the aspect ratio (

a / c

). Figure 1 shows the shape variation from a sphere or spheroids in a super-ellipsoidal space. Four representative examples are highlighted in the dashed-line boxes.

Fig. 1 The geometries of supper-ellipsoidal particles with different aspect ratios and roundness parameters: (a) shape variation from a sphere to prolate and oblate spheroids; (b) shape variation from a prolate spheroid to cube-like and octahedron-like shapes, achieved by changing the parameters $n$ and $e$ ; (c) and (d) are similar to (b) except that the aspect ratios are different.

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We consider randomly oriented super-ellipsoids. The scattering matrix that determines the change in the Stokes vector $[I_{i n c}, Q_{i n c}, U_{i n c}, V_{i n c}]$ of the incident wave to that of scattered waves $[I_{s c a}, Q_{s c a}, U_{s c a}, V_{s c a}]$ is given by [24]:

[\begin{matrix} I_{s c a} (θ) \\ Q_{s c a} (θ) \\ U_{s c a} (θ) \\ V_{s c a} (θ) \end{matrix}] \propto [\begin{matrix} P_{11} (θ) & P_{12} (θ) & 0 & 0 \\ P_{21} (θ) & P_{22} (θ) & 0 & 0 \\ 0 & 0 & P_{33} (θ) & P_{34} (θ) \\ 0 & 0 & P_{43} (θ) & P_{44} (θ) \end{matrix}] [\begin{matrix} I_{i n c} \\ Q_{i n c} \\ U_{i n c} \\ V_{i n c} \end{matrix}],

where

θ

is the scattering angle ranging from 0 to 180 degrees. The off-diagonal elements are zeros because super-ellipsoids have mirror symmetry. Note, both the Stokes parameters are defined with respect to the scattering plane. If the incident light is polarized linearly in one scattering plane, then the Stokes vector of the incident light is:

Q_{i n c} = I_{i n c}

,

U_{i n c} = V_{i n c} = 0

. The co-polarized and cross-polarized scattered intensities are given by

P_{11} (θ) + P_{12} (θ)

and

P_{12} (θ) + P_{22} (θ)

, respectively. The LDR can be accordingly defined as

LDR(θ)= \frac{I_{s c a} (θ) - Q_{s c a} (θ)}{I_{s c a} (θ) + Q_{s c a} (θ)} = \frac{P_{11} (θ) - P_{22} (θ)}{P_{11} (θ) + 2 P_{12} (θ) + P_{22} (θ)} .

The LDR vanishes if

P_{11} (θ) = P_{22} (θ)

, which is the case for a homogeneous sphere. At the direct backscattering direction (namely,

θ = 180 °

),

P_{12} (θ = 180 °) = 0

for any randomly oriented particles, and thus:

LDR (180 °) = \frac{1 - P_{22} (180 °) / P_{11} (180 °)}{1 + P_{22} (180 °) / P_{11} (180 °)} .

By invoking the reciprocal symmetry in electromagnetic wave scattering, the LDR for randomly oriented particle is in the range of [0, 1], because

0 \leq P_{22} (θ = 180 °) / P_{11} (θ = 180 °) \leq 1

[8].

To compute the LDRs of super-ellipsoids, we employ the T-matrix methods which rigorously solve Maxwell’s equations. The T-matrix methods first compute the T-matrix, then compute the phase matrix of randomly oriented super-ellipsoids, and finally compute the LDRs from Eq. (4). For axially symmetric spheroids, we first use the extended boundary condition method (EBCM) to compute the LDR for spheroids if the EBCM converges, and then use the II-TM for the remaining simulations. For general super-ellipsoids, we employ the II-TM. The II-TM method has been well tested through inter-comparisons with the other computational techniques, including the EBCM [21] and discrete-dipole-approximation (DDA) methods [25] as well as the multi-sphere T-matrix (MSTM) [26] for a number of particle shapes (see [18–20, 27–29] for comparisons).

3. Results

3.1 Spheroids

In this subsection, we systematically study the LDRs of spheroids. The top panel of Fig. 2 shows the LDRs of spheroids with the aspect ratio ranging from 0.4 to 2.0, and with the size parameter ranging from 0.1 to 100. The refractive index is assumed to be 1.55 + i0.0003, a typical refractive index of dust aerosol particles [30]. The dark blue central line at an LDR of zero corresponds to spheres ( $a / c = 1$ ). The LDRs (>~60%) are found to be pronounced for nearly-spherical particles with the aspect ratio close to unity for both prolate and oblate spheroids. The remarkable phenomenon of LDRs peaking at aspect ratios of spheroids very close to unity was first discovered by Mishchenko and Hovenier [8] (see also [11]) and illustrates the inherent difficulty of treating the LDR as an indicator of the degree of particle non-sphericity. In [8], the size parameter of spheroid is less than 30 at the refractive index 1.5 + i0.005, slightly different from the present refractive index We found that as the size parameter further increases, the region of smaller depolarization ratio between enhanced LDRs (indicated by red colors) narrows. This feature can be more explicitly observed in the bottom panel of Fig. 2, where the depolarization ratios are depicted as a function of the aspect ratio ranging from 0.9 to 1.1 for three size parameters (40, 60, and 100). This is because, for a fixed close-to-unity aspect ratio, the difference between the semi-major axis and the semi-minor axis increases as the size parameter increases, ultimately achieving sensible nonsphericity. Thus, for the large size parameter, the LDR could be large when the aspect ratio has a slight deviation from unity. Note that a fine resolution of the aspect ratio has to be taken into account to model the depolarization ratio for nearly-spherical particles. In the bottom panel of Fig. 2, a total of 201 aspect ratios with an interval of 0.001 in the range of 0.9 to 1.1 are considered to resolve the sensitivity of the LDR on the particle shape.

Fig. 2 Top: Depolarization ratio (%) as a function of the aspect ratio and the size parameter of randomly oriented spheroids. Bottom: Depolarization ratio (%) of spheroids with close-to-unity aspect ratios at three size parameters. The refractive index is 1.55 + i0.0003.

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To understand which particle orientation contributes most to the pronounced LDRs, Fig. 3 shows the results of prolate and oblate spheroids with fixed orientations depicted as functions of the angle of light incidence. As expected, the depolarization is zero for the axial incidence (the incident angle is zero as indicated with icons in the figure). For both oblate and prolate spheroids, the LDR reaches its maximum at the incident angle of 90 degrees (side on incidence). Interestingly, in the oblate spheroid case, we found that the depolarization ratio has a peak at about 40–50 degrees. Nevertheless, it is clear that the pronounced depolarization ratios identified in Fig. 2 for randomly oriented spheroids are mostly contributed by relatively large incident angles.

Fig. 3 Depolarization ratio with respect to the incident angle for spheroids with close-to-unity aspect ratios ( $a / c = 0.95, a / c = 1.05$ ).

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Aiming to relate this mechanism to the reality in the atmosphere, we need to answer the following question: Is the phenomenon of enhanced LDRs of nearly-spherical particles in Fig. 2 a general property for different aerosol compositions? Figure 4 shows the LDRs of spheroids at different refractive indices with the same imaginary part (0.001). The aspect ratio of spheroids ( $a / c$ ) is from 0.5 to 2.0 and the size parameter ( $k x_{m}$ ) is from 0.1 to 100. We found that the LDR is pronounced for a large range of size parameters for various aspect ratios at the refractive index of 1.1 + i0.001. As the real part increases, large LDR cases occur for nearly-spherical particles, and finally disappear as the real part is larger than 1.7. In Fig. 5, the real part of refractive index is set up as 1.5, the imaginary part of the refractive index increases from 10⁻⁷ to 0.5. We found that the pronounced depolarization phenomenon for nearly-spherical particles starts to disappear when the imaginary part is larger than 0.01. Therefore, the real part of the refractive index should be in a range so that a sufficient portion of the energy can penetrate into the particle and then exit the particle at the direct backscattering direction, thus producing depolarization. As the real part of the complex refractive index increases, a large portion of energy is reflected back towards to the medium. In addition, the imaginary part of the refractive index should be sufficiently small so that the high-order transmission has noticeable contribution to backscattering. For strongly absorptive particles, the diffraction and the external reflection of the spheroids dominate in the scattering, thus producing no backscattering depolarization (the non-zero LDR stems from high order transmitted waves from particles to the medium). The imaginary refractive index value ~0.01 appears to be a rather general threshold above which the LDRs rapidly diminish (for example, see Fig. 10 in [31], where the real part of the refractive index is 1.31).

Fig. 4 Depolarization ratio (%) as a function of the aspect ratio and the size parameter of randomly oriented spheroids at five different refractive indices with the same imaginary part: (a) 1.1 + i0.001; (b) 1.3 + i0.001; (c) 1.5 + i0.001; (d) 1.7 + i0.001; (e) 2.0 + i0.001.

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Fig. 5 Depolarization ratio (%) as a function of the aspect ratio and the size parameter of randomly oriented spheroids at five different refractive indices with the same real part: (a) 1.5 + i10⁻⁷; (b) 1.5 + i0.001; (c) 1.5 + i0.01; (d) 1.5 + i0.1; (e) 1.5 + i0.5.

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To obtain a more accurate understanding of the effect of the refractive index on the LDR for nearly-spherical particles, we focus on the aspect ratio range from 0.9 to 1.2, but with a fine resolution of the real part of the refractive index ranging from 1.1 to 1.8 at an interval of 0.01 (a total of 70 refractive indices). The imaginary part is fixed at 0.001 (the pattern with the imaginary part smaller than 0.001 is similar at least for the size parameter less than 100). For clarity, Fig. 6 shows the results of 12 refractive indices. We found that the LDRs have similar phenomena, like those shown in Fig. 2, for the refractive indices with the real part in the range of (1.3, 1.7). Four distinct features are observed: (1) for the real part of the refractive index from 1.1 to ~1.2, a large domain of the size parameter and aspect ratio have enhanced LDRs and resonance features are evident (namely, for a fixed aspect ratio, a few ranges of size parameters have LDRs larger than 60%); (2) for the real part of the refractive index from ~1.20 to ~1.30, large LDRs are less frequently detected with weak resonance features; (3) for the real part of the refractive index from ~1.3 to ~1.55, pronounced LDRs accumulate for nearly-spherical particles; whereas for refractive index close to 1.3, large LDRs could be observed at the size parameter of ~20 for smaller aspect ratios (~0.8); and (4) for the real part of refractive index from ~1.55 to ~1.7, the feature of pronounced LDRs for nearly-spherical particles becomes increasingly weaker, and finally disappears when the real part of refractive index is larger than 1.7.

Fig. 6 Similar to Fig. 4, but for an aspect ratio range from 0.8 to 1.2 with a fine resolution of refractive index.

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3.2 Super-ellipsoids

The next question that arises is: Is the enhanced LDRs for nearly-spherical particles an instinct property of spheroids? This question has importance because realistic atmospheric particles are not perfect spheroids. Although enhanced LDRs are observed for nearly-spherical Chebyshev particles [11], to further address this, we studied the depolarization of super-ellipsoids by varying the particle shape from spheroids (spheres as special cases) to cube-like and octahedron-like shapes. The aspect ratios of spheroids from Figs. 7(a)–7(e) are 0.5, 0.95, 1.0, 1.05, and 2.0. The refractive index is assumed to be 1.33 + i10⁻⁷. First, we looked at the shape variation shown in Fig. 7(c). We did not observe pronounced LDRs for nearly-spherical particles with the unity aspect ratio. On the contrary, the LDRs for nearly-spherical particle are fairly small. Next, we looked at the shape variation from nearly-spherical spheroids shown in Fig. 7(b) and Fig. 7(c), where the particles at $n (e) = 1$ are spheroids with aspect ratio of 0.95 and 1.05, respectively. From Fig. 6(g), the LDRs are large at the two aspect ratios when the size parameter is larger than 20. In a super-ellipsoidal shape space, we do observe a large area with stronger LDRs when $0.8 < n (e) < 1.2$ . These simulation results show that a small deviation of the aspect ratio of super-ellipsoids is critical to produce the pronounced LDRs. However, for roundness parameters smaller than 0.8 or larger than 1.4, the depolarization ratio patterns in Figs. 7(b)–7(d) are similar. That means, for nonspherical particles (n<0.8, and n>1.4), a slight difference in the aspect ratios has little impact on the LDR. Lastly, we consider the shape variation from spheroids with aspect ratios of 0.5 and 2.0, shown in Figs. 7(a) and 7(e), respectively; the LDR was found to change smoothly as the shape parameters changed. Note that, compared to Figs. 7(b) –7(d), the LDRs in Figs. 7(a) and 7(e) begin to have sizable values at relatively smaller size parameter. This is because the aspect ratio strongly deviates from unity.

Fig. 7 Comparison of depolarization ratios for different shape variations from spheroids to cube-like and octahedron-like particles. The aspect ratios of spheroids are: 0.5 (a), 0.95 (b), 1.0 (c), 1.05 (d), and 2.0 (e). The refractive index is 1.33 + i10⁻⁷.

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Figure 8 is similar to Fig. 7 except that the refractive index is 1.55 + i0.0003. In general, the LDR distribution patterns with respect to size parameter and shape parameter are similar to their counterparts in Fig. 7. However, the magnitudes of the LDRs are much smaller for regular nonspherical particles, and are stronger for nearly-spherical particles. Interestingly, we found that the maximum values in Fig. 8(d) are associated with super-ellipsoids, whereas the maximum value in Fig. 8(b) is from spheroids. Also, we noted that the LDR had pronounced values from both sides ( $e = n > 1$ and $e = n < 1$ ) approaching the spheroid ( $e = n = 1$ ), although the shape variation procedures are different when $n$ is larger than unity compared to when $n$ is smaller than unity.

Fig. 8 Comparison of depolarization ratios for different shape variations from spheroids to cube-like and octahedron-like particles. The aspect ratios of spheroids are: 0.5 (a), 0.95 (b), 1.0 (c), 1.05 (d), and 2.0 (e). The refractive index is 1.55 + i0.0003.

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Next, we consider the shape variation from octahedra to concave super-ellipsoids. Because slight difference of aspect ratio for regular nonspherical particle has little effect on the LDR, we consider three aspect ratios, namely $a / c$ = 0.5, 1.0, and 2.0. The refractive index of the top panel is 1.33 + i10⁻⁷. We found that as the shape parameter ( $n$ or $e$ ) increases, the LDRs are in general smaller for all of the aspect ratios. The LDRs for compact particles are much larger than the other two aspect ratios. This finding was not detected in the bottom panel, where the refractive index is 1.55 + i0.0003. The fundamental reason for this is unclear; however, we notice that the external reflections for convex particles do not induce depolarization, but for concave particles, multiple external reflections could induce depolarization. Thus, the impact of refractive index on the LDRs could be more complex for concave particles. Overall, the LDRs are again less than 60%. To further resolve the dependence of the LDRs on the aspect ratio, Fig. 10 shows the LDRs as a function of aspect ratio for three representative roundness parameters (0.5, 1.8, and 2.5). In the first column, the LDRs of cube-like shapes are relatively insensitive to the aspect ratio when the aspect ratio is larger than 1.4. In the second column, for octahedron-like shapes, the LDRs are relatively larger for the aspect ratio ranging from 0.8 to 1.2. In the third column, for concave particles, the LDR distribution has a u-shaped pattern. As the refractive index changes from 1.33 + i10⁻⁷ to 1.55 + i0.0003 (the first row to the second row), we found that the magnitude of LDRs decreases for cube-like and octahedron-like shapes, but is relatively insensitive to the refractive index for concave particles.

From Figs. 7–10, we only consider two refractive indices. To examine the dependence of the LDRs on the refractive index, we consider nine particle shapes associated with three representative aspect ratios (0.5, 1.0, and 2.0) and three representative roundness parameters (0.5, 1.8, and 2.5). The results are shown in Fig. 11. To reduce the computational burden, we consider the refractive index range [1.33, 1.55] because this range of refractive index is of most interest for atmospheric particles. Five discrete values of the refractive index (1.33, 1.40, 1.45, 1.50, and 1.55) are selected. We found that the LDRs strongly depend on the particle shapes. For the roundness parameter ( $n = e = 0.5$ ) in the first column, the LDRs for compact particle are generally smaller than prolate and oblate particles. For the roundness parameter of 1.8 in the second column, the LDRs for prolate shapes ( $a / c = 0.5$ ) are much smaller than their counterparts in the first column. For concave particles in the third column, the stronger the refractive index, the larger the LDRs. Also, we note that the LDRs change in the refractive index range (1.45–1.55) is weak for these irregular nonspherical particles. Lastly, the LDRs are within an upbound of ~60%.

Fig. 9 Comparison of depolarization ratios for shape variations from octahedral to concave particles. The aspect ratios of spheroids are: 0.5 (a), 1.0 (c), and 2.0 (c). The refractive index is 1.55 + i0.0003.

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Fig. 10 Comparison of depolarization ratios as a function of aspect ratio ranging from 0.5 to 2.0 for three roundness parameters (0.5, 1.8 and 2.5).

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Fig. 11 Comparison of depolarization ratios as a function of the refractive index for nine representative shapes with different aspect ratios and roundness parameters. The imaginary part of refractive index is fixed to be 10⁻⁷.

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From previous results, we can conclude that nearly-spherical particles are different from spherical particles and other nonspherical particles. Different nonspherical particle geometries, except for nearly-spherical particles, might have different depolarization ratios. However, we observed that the depolarization ratios for normal nonspherical particles, excluding nearly-spherical particles, are almost smaller than ~60% when the real part of refractive index is larger than ~1.33. Therefore, for an ensemble of aerosol particles (particularly, for the particle size smaller than ~10 $μ m$ at the wavelength of 0.532 $μ m$ ), if the LDR is larger than ~60% (in this case, the measured LDR is the size-integration result), there is high probability that nearly-spherical particles dominate in the ensemble.

To estimate the LDRs for an ensemble of nonspherical particles with different particle sizes, Fig. 12 shows the size-integration results for nine particle shapes shown in Fig. 11. We use the log-normal function given by

\frac{d N}{d \ln r} = \frac{N_{0}}{\sqrt{2 π} \ln σ} \exp [- \frac{{(\ln r - \ln r_{m})}^{2}}{2 {(\ln σ)}^{2}}],

where

N_{0}

is the total number of particles per volume,

r_{m}

is the mean geometric radius, and

σ

is the geometric standard deviation. In Eq. (5), the radius of super-ellipsoids is assumed to be the radius of an equivalent-projected-area sphere. Because

N_{0}

has no effect on the LDRs, thus

N_{0}

is assumed to be unity. To study the variation of the LDR with respect to the mean geometric radius,

σ

is assumed to be 1.5. The particle size information is obtained from size parameter by assuming the incident wavelength to be 0.532

μ m

. For large mean geometric radii, the results may suffer inaccuracies because the maximum size parameter is 50 in this study. Note that, to clearly see the variation of the LDRs, the range of the color bar of LDRs is modified with a maximum value of 60%. From Fig. 12, it is evident that the LDRs of concave particles have distinct dependence on the refractive index. Figure 13 is similar to Fig. 12, except that the geometric standard deviation (

σ

) is 2.2. It is found that the LDRs begin to have relatively large values even when the effective size is in sub-micron range.

Fig. 12 Similar to Fig. 11, except that the results are size-integration values. The wavelength is assumed to be 0.532 $μ m$ . The geometric standard deviation is 1.5. Note, the maximum value of color bar is 60%.

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Fig. 13 Similar to Fig. 11, except that the results are size-integration values. The wavelength is assumed to be 0.532 $μ m$ . The geometric standard deviation is 2.2. Note, the maximum value of color bar is 60%.

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Figure 12 and Fig. 13 can be used as a general reference for understanding the LDRs from laboratory measurements. Järvinen et al. [12] reported the laboratory measurements of near-backscattering (178°) depolarization ratios of over 200 dust samples at the wavelengths of 0.488 μm and 0.552 μm. The measured LDRs were found to range from 0.03 to 0.36. As seen from Fig. 12 and Fig. 13, this is most probably because the refractive index of dust is generally larger than 1.45. Also, from the measurements, the dust fine- and coarse-mode distributions have distinct LDRs. This feature is found to be consistent with the results shown in Fig. 12. But, as shown in Fig. 13, the dust fine-mode may also have large LDRs in the case of large geometric standard deviation. Furthermore, the LDRs of cube-like hematite particles were found to be much smaller than those of the other dust samples. The measurements are consistent with the present simulations. From Fig. 12(d) and Fig. 13(d), the LDRs of cube-like shapes are much smaller than those of the other super-ellipsoidal geometries, particularly for the refractive index larger than 1.4. Miffre et al. [14] made the measurements of the direct backscattering LDRs for two dust samples at the wavelength of 0.355 μm and 0.532 μm. The LDRs at the wavelength of 0.355 μm are found to be larger than their counterparts at 0.532 μm. Although the refractive index has slight difference at the two wavelengths, the major reason that causes the LDR change may be the size parameter change, because the sensitivity of the LDRs to the refractive index (>1.5) is weak. Progress on measuring the LDRs for ice crystals in a laboratory cloud chamber were reported in Smith et al. [13]. The sizes of ice crystals are much larger than the size of super-ellipsoids in the present simulations. However, the measured LDRs (Fig. 8, Fig. 11, and Fig. 14 in [13]) are exclusively smaller than ~0.6, which is similar to the finding in this study because ice particles are not nearly-spherical particles. Note that, the LDRs of ice columns are close to ~0.6 (Fig. 8 in [13]), most likely because the refractive index of ice particle is close to 1.33 (see the first columns of Fig. 12 and Fig. 13).

4. Summary and conclusion

In this paper, we have systematically studied the LDRs with respect to morphology variation in a super-ellipsoidal shape space. In particular, we focus on quantifying the range of morphology parameters and refractive indices that have enhanced LDRs. Based on the super-ellipsoidal model, high LDR is not an intrinsic property of spheroids, but the aspect ratio is also a sensitive parameter. This finding suggests that the detailed shape change process from spherical particle to nonspherical particle is critical to the backscattering LDRs. To achieve high LDRs for nearly-spherical particles, the real part of the refractive index should be in the region of 1.3–1.7, and the imaginary part of the refractive index should be less than 0.01. The maximum size parameter of the high depolarization ratio depends on the imaginary part of refractive index. Note that the aforementioned refractive index regime covers the typical refractive indices of aerosols in the atmosphere. Two distinct regions of the LDR were identified: (0, 0.6) and (0.6, 1). The case of nearly-spherical particles is most likely associated with the liquid to solid phase transition. Of course, the results presented in this study cannot be employed to reject any other possible mechanisms that might lead to pronounced linear depolarization ratios, because the present study is not exhaustive, particularly for large size parameters. It would be interesting to investigate the depolarization ratio in the regime of geometric optics, although ray-approximation might lead to uncertainties at the direct backscattering direction. Note that, the physical mechanism that leads to pronounced backscattering of nearly-spherical particles is unclear, although it is true as predicted from Maxwell’s equations. To further identify the physical mechanism, the Debye series might be useful to explain the unique depolarization capability of near-spherical particles by separating the contribution to backscattering from diffraction, reflection, and higher-order transmissions [32,33], which will be a future research subject. Finally, although the results presented here are extensive, other complexities, such as the surface roughness, inhomogeneity and aggregation are known to affect the LDRs. How these complexities affect the LDRs of super-ellipsoids also has importance in LiDAR remote sensing applications; however, this is beyond the scope of the present study.

Funding

National Key Research and Development Program of China (2016YFC0200700); National Natural Science Foundation of China (NSFC) (41675025); Fundamental Research Funds for the Central Universities (2017QNA3017).

Acknowledgments

The authors thank Michael I. Mishchenko for using his EBCM T-matrix code and two anonymous reviewers for their comments that improve this manuscript. We acknowledge Ms. Rui Liu from the Training Center of Atmospheric Sciences of Zhejiang University for her effort related to managing computing resources. A portion of the computations was performed on the National Supercomputer Center in Guangzhou (NSCC-GZ) and the cluster at State Key Lab of CAD&CG at Zhejiang University.

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Assessing the depolarization capabilities of nonspherical particles in a super-ellipsoidal shape space

Abstract

1. Introduction

2. Model, definition and computational methods

3. Results

3.1 Spheroids

3.2 Super-ellipsoids

4. Summary and conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (13)

Equations (5)

Optics Express