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Abstract
The degree of linear polarization (–P12/P11) of scattered light by particles with a core-shell structure may display a distinct negative minimum at near-backscattering directions. However, the specific range of microphysical parameters within which this phenomenon occurs and the underlying physical mechanism are still unclear. Therefore, this study systematically investigated the impacts of particle size, shell-core ratio and refractive index on the negative minimum of –P12/P11 at near-backscattering angles for both coated spheres and coated super-spheroids. The findings reveal that the pronounced negative minimum at near-backscattering angles mostly appeared when the size parameter defined in terms of the mean radius was smaller than approximately 14.5 (e.g., the mean radius is smaller than approximately 2 μm at 0.865 µm wavelength) and the shell-core ratio was in a range of 1.4–1.9. The presence of weakly- and moderately-absorptive shells would lead to pronounced negative polarization at near backscattering directions. However, as the core absorption increased, the amplitude of negative minimum decreased and then stabilized. As for coated super-spheroids, the non-sphericity of the shell tended to suppress the negative polarization at near-backscattering directions. As a result, the pronounced negative minimum (<–0.4) mostly appeared when the aspect ratio and roundness of the shell were close to unity (the overall shape of the particle was nearly-spherical). However, the negative minimum of –P12/P11 showed little dependence on the shape of the core. Furthermore, the Debye series approach was employed to investigate the underlying mechanism of the negative minimum of –P12/P11 for coated spheres. The results demonstrated that the interference among the partial waves underwent one internal reflection on the shell-medium interface and, without internal reflection on the core-shell interface, led to the pronounced negative polarization at near-backscattering angles. When the core absorption was significant, the interference became negligible and the amplitude of the negative minimum was suppressed. This study enhances our understanding the scattering characteristic of coated particles and has implications in aerosol classification and polarized remote sensing.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Particles with a core-shell structure are commonly found in nature, such as mixed aerosols, melting hydrometeors, mammalian cells, and nanocapsules [1–3]. Understanding the light scattering properties of these core-shell particles have important applications in optical characterization, remote sensing, and climate studies [4–6]. In the past few decades, extensive research has been conducted on light scattering by core-shell or coated particles in various fields [7–9].
As an extension of the Lorenz–Mie theory developed for a homogeneous dielectric sphere, the solution of light scattering by a coated sphere was first obtained by Aden and Kerker in 1951 [10]. Lock et al. [11] studied the properties of coated sphere rainbows with various radii of the core and thicknesses of the coating using the Aden-Kerker approach. The interior electromagnetic energy density of coated sphere has been examined, which is affected by the physical parameters of either the core or coating [7–14]. With the development of computational electromagnetics methods, such as the discrete dipole approximation [15,16], the finite difference time domain [17–19], and the T-matrix methods [20–22], the optical properties of non-spherical core-shell particles can now be efficiently obtained. For example, Bi et al. [4] investigated the backscattering linear depolarization by coated super-spheroids using the invariant imbedding T-matrix method with a graphics processing unit (GPU) implementation.
Recently, Li et al. [6] investigated the optical properties of wet sea salt aerosols and a pronounced negative minimum at near-backscattering directions (170°–175°) was found for the –P12/P11 of the coated sphere. Interestingly, such salient characteristics were not found for the homogeneous sphere or non-spherical particle. This distinct feature may be important in remote sensing studies. This study also demonstrated a pronounced negative minimum at the scattering angle of ∼175° can also be observed for the simulated top-of-the-atmosphere (TOA) polarized radiances. However, the physical mechanism leading to the negative minimum of –P12/P11 of the coated spheres remains unclear. Additionally, the ranges of particle size, shell-core ratio and refractive index within which this phenomenon could occur require to be quantified because core-shell particles, such as black carbon aerosol, are commonly observed in the atmosphere [5,23,24].
In this paper, we provide a comprehensive analysis of the linear polarization of both spherical and non-spherical coated particles with different sizes, refractive indices of core and shell, and shell-core ratios. The ranges of these parameters within which the extreme negative polarization can be observed are quantified. Innovatively, to provide additional insight into the aforementioned optical phenomenon, the Debye series is employed to investigate the scattering mechanism leading to the pronounced negative values of the linear polarization from the perspective of a fundamental scattering process. Section 2 introduces the methods and particle models used in this study. The computational methods utilized include the Lorenz-Mie theory [1], the invariant imbedding T-matrix (IITM) method [4,20,26], and the Debye series of coated sphere particles [37]. In Section 3, we present the findings obtained from analyzing coated sphere with varying radii, shell-core ratios, and refractive indices. Additionally, we investigate the effects of the shell roundness and aspect ratio on linear polarization of coated super-spheroids. Moreover, we analyze the physical mechanism of the negative polarization using Debye series. Section 4 serves as a summary, outlining the key points of our research and highlighting potential applications of our findings.
2. Method and model
2.1 Method
The Lorenz-Mie theory [25] and the invariant imbedding T-matrix (IITM) method [4,20,26] were employed to compute the optical properties of coated spheres and non-spherical core-shell particles, respectively. In particular, we used a computational program called CMIE developed by Cai et al [27], which is proved to be numerically stable. Currently, the state-of-the-art IITM can accurately and efficiently compute the optical properties of arbitrarily shaped and inhomogeneous particles with a broad range of size parameters. The maximum size parameter depends on the computational resources.
Different from the Lorenz-Mie theory, the Debye series approach is often utilized to analyze the scattering mechanism. Specifically, it decomposes each of the partial wave scattering amplitudes into an infinite series of terms that describe diffraction, external reflection, and transmission after making p–1 (p ≥ 1) internal reflections [28–30]. The Debye series was first introduced by Peter Debye for an infinite circular cylinder and can also be applied to a sphere [31–34]. The Debye series approach has been further developed to compute the optical properties of coated spheres [11,35–39] and non-spherical particles [29,40,41]. In particular, Xu et al. [41] innovatively applied the extended boundary condition method for computing the Debye series of nonspherical particles. Lin et al. [42] used the Debye series to determine the optimal number of edge-effect terms for accurately computing the extinction efficiencies of spheroids. Bi et al. [29] used the Debye series approach to address the depolarization of nearly spherical particles and found the enhanced depolarization for optically soft particles was mostly induced by the transmission after one internal reflection. Bi and Gouesbet [30] developed a new formulation of the Debye series with expansion of T-matrix to compute electromagnetic wave scattering by non-spherical particles. The Debye series approach is a valuable method for understanding various light scattering physical phenomena such as rainbows and glories [43,44] because the contributions from diffraction, reflection and higher-order transmissions can be separated. Therefore, we employed the Debye series approach in this study to assess the underlying mechanism of the pronounced negative linear polarization of coated particles. However, the application of the Debye series is currently limited to coated spheres due to the absence of a computational program for coated super-spheroids.
Following the formalism given by Laven and Lock [37], a python computational program has been developed in this study and validated for investigating the linear polarization of coated spheres.
In the single-scattering process, the Stokes parameters of the incident light (I0, Q0, U0, and V0) and the scattered light (Is, Qs, Us, and Vs) are related by the scattering matrix [21]:
If the incident light is unpolarized, the stokes vector Is and Qs are proportional to P11 and P12, respectively. The I and Q can be described as follows [21]:
where ${E_\parallel }$ and ${E_ \bot }$ are the parallel and perpendicular components of the electric field with respect to the scattering plane, respectively:In this study, we focused on the degree of linear polarization, which is defined as
The Mueller matrix elements are related to the complex amplitude scattering matrix elements [45]:
According to the Lorenz–Mie theory, the two elements in the complex amplitude scattering matrix are
In the context of Debye series, the decomposition of the coated sphere partial wave scattering amplitudes an and bn can be written as [37]:
In Eqs. (15) and (16), q–1 represents the number of the reflections on the internal surface of shell-medium interface, and p–1 represents the number of the reflections on the internal surface of shell-core interface. The core, shell, and medium are indicated by the subscript “1”, “2”, and “3”, respectively. For example, R323 represents the reflection on the external surface of shell-medium interface, and T32 represents the transmission coefficient from the medium to the shell. The other terms can be understood in the same way. The different scattering processes corresponding to these terms are illustrated in Fig. 1. Note that the “T32” on the right side of the second equal sign in Eq. (15) was mistyped as “T23” in Eq. (7) in Ref. [37].
The partial wave Fresnel reflection and transmission coefficients on the shell-medium and core-shell interfaces are given as follows [37]:
2.2 Model
In this study, both coated spheres and coated super-spheroids were adopted to investigate the linear polarization of particles with a core-shell structure. The equation of a super-ellipsoid is written as [46]:
Figure 2 shows three examples of the particle models used in this study. The coated sphere model (Fig. 2(a)) is extensively used in climate and chemical models due to its simplicity [47–49]. Figure 2(b) shows an example of coated super-spheroids. The core is concave with a roundness of 2.6, which is considered as a typical shape parameter for optical modeling of dust aerosols [50–52], while the shell is convex with a roundness of 1.2. This model has been adopted for internally-mixed dust aerosols in Ref. [5]. Another example of coated super-spheroids whose core and shell have consistent shape parameters is shown in Fig. 2(c). Such convex coated super-spheroid can be used to characterized the inhomogeneity of wet sea salt particles [53].
Fig. 2. Examples of geometry structure of three models. (a) Coated sphere with a shell-core ratio of 1.5; (b) Coated super-spheroid with a shell-core ratio of 1.0, both the core and shell have an aspect ratio of 1.0 and the roundness parameters of the core and shell are 2.6 and 1.2, respectively; (c) Coated super-spheroid with a shell-core ratio of 1.5, the aspect ratio and roundness parameter of both the core and shell are 1 and 1.5, respectively.
In analysis of the coated sphere model (Fig. 2(a)), we examined the negative polarization feature for various parameters. These parameters included the mean radius (ranging from 0.01 to 5 µm), the shell-core ratio (ranging from 1.1 to 2.0), the real part of the complex refractive index (ranging from 1.1 to 1.8), and the refractive index of common aerosols such as sea salt, dust and black carbon. For the coated super-spheroid model (Fig. 2(b)), we considered the following parameters: mean radius (ranging from 0.01 to 5 µm), shell roundness (ranging from 1.0 to 2.4), core roundness (fixed at 2.6), aspect ratio (ranging from 0.5 to 2.0), and the imaginary part of the shell refractive index (ranging from 0.001 to 0.5). In addition, for the coated super-spheroid model (Fig. 2(c)), we specifically investigated the effect of shell-core ratios (1.3, 1.5, and 1.7). All of these simulations were carried out at a wavelength of 0.865 µm, which is commonly used for polarization measurements in instruments such as POLDER/PARASOL (Polarization and Directionality of the Earth's Reflectances/Polarization and Anisotropy of Reflectances for Atmospheric Sciences Coupled with Observations from a LiDAR).
3. Results
3.1 Degree of linear polarization of coated spheres
The coated sphere is the simplest inhomogeneous particle model used to study mixed aerosols. In this section, an in-depth analysis of the linear polarization of coated spheres with different lognormal size distributions, shell-core ratios, and refractive indices was conducted. For the lognormal size distributions, the standard deviation parameter was fixed to be 2.0, and the mean radius varied.
Firstly, the impact of mean radius on the –P12/P11 of coated spheres at backward scattering angles (150° – 180°) was investigated (shown in Fig. 3). A single-mode lognormal size distribution with mean radius ranging from 0.01 to 5 µm (0.01, 0.1, 0.3, 0.5, 0.8, 1, 2, 3, and 5 µm) was adopted. The wavelength was 0.865 µm and the refractive indices of shell and core were selected to be 1.324 and 1.48, respectively, corresponding to the case of sea salt particle with water coating. Figure 3 displays the results, with the y-axis shown logarithmically to better illustrate the changes. The negative minimum values of –P12/P11 generally appeared at the mean radius between 0.1 and 2 µm (the size parameter is between 0.73 and 14.5, correspondingly), indicating that the negative minimum is primarily induced by small size particles. When the coating was relatively thin (with a shell-core ratio of 1.1 or 1.3), the negative minimum (around –0.4) appeared at the scattering angle between 160° and 170°, while a positive maximum appeared as the mean radius increased from 1 µm to 5 µm. When the shell-core ratio was 1.5 or 1.7 (namely thicker coating), much more pronounced negative minimum values of –P12/P11 (can reach –0.6) were found. Additionally, the location of the negative minimum of –P12/P11 was found to move to larger scattering angles as the mean radius increased from 0.1 µm to 5 µm. Interestingly, a positive maximum appeared at the scattering angle between 170° and 180° as the mean radius was larger than approximately 3 µm. The negative minimum became less pronounced for thickly-coated particles with a shell-core ratio of 1.9 or 2.0.
Fig. 3. The –P12/P11 of coated spheres with different size distributions. The lognormal size distribution with mean radius ranging from 0.1 to 5 µm and a standard deviation (σ) of 2.0 was adopted. The wavelength was 0.865 µm and the refractive indices of the core and shell were 1.48 and 1.324, respectively. Six shell-core ratios, i.e., 1.1, 1.3, 1.5, 1.7, 1.9, and 2.0, were selected.
To further assess the impact of the shell-core ratio on the negative polarization, computations of –P12/P11 were conducted for coated spheres with fine-resolution shell-core ratios ranging from 1.1 to 2.0 with an interval of 0.1 at six selected mean radii (0.1, 0.5, 1.0, 2.0, 3.0, and 5.0 µm). The results are shown in Fig. 4. When the mean radius was small (i.e., 0.1 µm), the negative minimum of –P12/P11 was found at scattering angles between 160° and 170° for coated spheres with shell-core ratios ranging from 1.1 to 2.0, but was not prominent. However, as the mean radius increased (0.5 or 1.0 µm), the negative minimum became much more pronounced for the shell-core ratios ranging from approximately 1.4 to 1.9, especially when the shell-core ratio was around 1.6, which was consistent to the finding from Fig. 3. When the mean radius was larger than 2 µm, the negative minimum disappeared. Moreover, large sized particles tended to produce positive –P12/P11 at near-backscattering directions, especially when the shell-core ratio was small (see the bottom panels in Fig. 4, and also the top panels in Fig. 3).
Fig. 4. The –P12/P11 of coated spheres with shell-core ratios ranging from 1.1 to 2.0. The wavelength was 0.865 µm and the refractive indices of the core and shell were 1.48 and 1.324, respectively. Six mean radii, i.e., 0.1, 0.5, 1.0, 2.0, 3.0, and 5.0 µm, were selected and the standard deviation was 2.0.
From Figs. 3 and 4, we learnt that the pronounced negative values of –P12/P11 were mostly induced by particles with a limited range of sizes (when the mean radius was smaller than 2 µm at wavelength of 0.865 µm, and the corresponding size parameter was smaller than 14.5) and appeared mostly when the shell-core ratio was in a range of 1.4–1.9. However, the refractive indices of core and shell were fixed in these simulations.
Figure 5 shows the –P12/P11 of the coated spheres with a shell-core ratio of 1.5. The refractive indices of shell and core varied from 1.1 to 1.8 with an interval of 0.1 (the shell refractive index is fixed for each subplot) and the mean radius was set to be 0.3 µm. Generally, pronounced negative values of –P12/P11 tended to appear when the refractive index of the core was larger than 1.4. For particles with optically soft shell (mshell = 1.1), the pronounced negative minimum appeared at smaller scattering angle as the refractive index of the core increased from 1.5 to 1.8. When the refractive index of the shell was 1.2, pronounced negative values were observed for the refractive index of the core ranging from 1.35 to 1.45 and from 1.6 to 1.8. Similar results were found for the case of mshell = 1.3, but the ranges of the refractive index of the core were slightly different (i.e., from 1.4 to 1.5 and from 1.7 to 1.8). The negative minimum of –P12/P11 tended to appear at the scattering angles ranging from 160° to 175° for the cases of mshell = 1.2, 1.3 and 1.4. As the refractive index of the shell increased from 1.5 to 1.8, the negative values of –P12/P11 became less pronounced and the scattering angle where the minimum occurred became smaller. It was noted that the positive value of –P12/P11 at near-backscattering angles (150° – 180°) disappeared when the shell was optically hard (mshell = 1.8).
Fig. 5. The –P12/P11 of coated super-spheroids with varying shape parameters. (a-d): Results for coated super-spheroids with a fixed shell aspect ratio of 1.0; (e-h): Results for coated super-spheroids with a fixed shell roundness of 1.2. The core roundness was fixed at 2.6 and the aspect ratio matched that of the shell. The core had a refractive index of 1.5 + i0.001, while the shell had four selected refractive indices (1.3 + i10−5, 1.4 + i10−5, 1.5 + i10−5, and 1.6 + i10−5). All super-spheroids had a mean radius of 0.3 µm with a standard deviation of 2.0.
In the previous simulations (Figs. 3–5), both the shell and core of the particles were assumed non-absorptive. However, absorption is a significant property that can affect optical properties of atmospheric aerosols. For example, dust is weakly absorptive and black carbon has strong absorption. To investigate the impact of absorption on the negative minimum of –P12/P11, four typical refractive indices of atmospheric aerosols (i.e., 1.48 + i10−6 for almost non-absorptive sea salt, 1.50 + i0.001 for weakly-absorptive dust, and 1.75 + i0.45 and 1.95 + i0.79 for weakly-absorptive and strongly-absorptive black carbon) were selected for the cores of coated spheres [6,51,54,55,56]. The real part of the refractive indices of the shells was set as 1.55 and the imaginary part varied from 0.001 to 0.5. As can be seen in Fig. 6, the absorption of the shell generally dominated the occurrence of the negative minimum of –P12/P11 at near-backscattering directions. Specifically, when the shell was weakly absorptive (mi,shell = 0.001 and 0.01), the negative minimum values were found at scattering angle between 160° to 170°. However, the values of –P12/P11 were positive when the shell was highly absorptive (mi,shell = 0.5); the scattering was mainly from the external reflection. Most pronounced negative minimum values, which could be smaller than –0.4, were found for weakly-absorptive cores (mcore = 1.48 + i10−6 and 1.50 + i0.001) with moderately-absorptive coatings (mi,shell = 0.05 and 0.1). As the absorption of the cores became stronger, the negative minimum values became less pronounced for the cases of mi,shell =0.05 and disappeared for the cases of mi,shell = 0.1. We can see from Fig. 6 that strong absorption (for both the core and shell) tended to produce positive values of –P12/P11. These findings may bring new insight into polarized remote sensing of atmospheric aerosols and be useful for aerosol identification and classification. For example, it might be possible to identify inhomogeneous mixed aerosols or to distinguish dust coated with strongly-absorbing aerosols from that with other non- or weakly-absorbing aerosols using the polarized signals at near-backscattering directions received by satellite instruments.
Fig. 6. The –P12/P11 of coated spheres with different refractive indices. The refractive indices of the cores were (a) 1.48 + i10−6, (b) 1.50 + i0.001, (c) 1.75 + i0.45, and (d) 1.95 + i0.79, respectively. The real part of the refractive index of the shell was 1.55 and the imaginary parts were 0.001, 0.01, 0.05, 0.1, and 0.5. The shell-core ratio was 1.5. A lognormal size distribution with a mean radius of 0.3 µm and a standard deviation of 2.0 was adopted.
3.2 Degree of linear polarization of coated super-spheroids
Non-spherical particles with a core-shell structure are commonly observed, for example, dust mixed with other aerosols [5,55,57] and sea salt with water coating [6,53,58,59,60]. Therefore, it is necessary to investigate the impact of non-sphericity on the degree of linear polarization at near-backscattering directions.
In this section, we employed the IITM to compute the degree of linear polarization of coated super-spheroids, which were assumed to be randomly oriented. The wavelength was again set as 0.865 µm. The refractive index of the core was 1.5 + i0.001, the real part of the refractive index of the shell was 1.3, and the imaginary part of the shell refractive index ranged from 10−5 to 0.05. The roundness of the super-spheroidal core was 2.6, which is considered as an optimal shape parameter for dust [50,51]. The shell roundness varied from 1.0 to 2.4. The core and the shell were assumed to have identical aspect ratio and radius, indicating that the shell-core ratio was 1.0. An example of the coated super-spheroidal model was given in Fig. 2(b). The maximum size parameter of coated super-spheroids was 50. It should be pointed out that the scattering properties data reported in a previous study [4] were adopted in this study.
Figure 7 shows the –P12/P11 for coated super-spheroids with different size distributions. Lognormal distributions with a standard deviation of 2.0 were adopted. We changed the size distributions by varying the mean radius. As seen from Fig. 7, when the mean radius was 0.01 µm, there was no negative minimum of –P12/P11 at the near-backscattering angles. As expected, –P12/P11 was similar to the case of Rayleigh scattering. Similar to the cases of coated spheres (shown in Fig. 3), negative linear polarization appeared at near-backscattering angles for mean radius ranging from 0.1 µm to 1.0 µm. When the mean radius was 0.3 µm, the most pronounced negative minimum at the scattering angle of around 170° was found for the case of rm = 0.3 µm. As the radius increased from 0.3 µm to 1.0 µm, the amplitude of the minimum decreased, and the location of the minimum moved to larger scattering angles.
Fig. 7. The –P12/P11 of coated super-spheroids with different size distributions. A lognormal size distribution was adopted. The mean radius (rm) ranged from 0.01 µm to 1.0 µm. The standard deviation was 2.0. The aspect ratio and roundness of the core were 1.0 and 2.6, and were 1.0 and 1.2 for the shell, respectively. The refractive indices of the core and shell were 1.5 + i0.001 and 1.30 + i10−5, respectively.
The impact of the shell roundness on the negative polarization of the coated super-spheroids was illustrated in Fig. 8. Generally, negative minimum values of –P12/P11 were found at the near-backscattering angles for all the cases. The most pronounced negative polarizations were found when the roundness of the shell was 1.0 or 1.2 (spherical or nearly-spherical shell). As the roundness of the shell increased from 1.2 to 2.4 (the thickness of the shell became smaller), the amplitude of the minimum decreased and the location of the minimum moved to smaller scattering angles.
Fig. 8. The –P12/P11 for of coated super-spheroids with different with shell roundness of 1.0, 1.2, 1.4, 1.6, 1.8, and 2.4. The aspect ratio and roundness of the core were 1.0 and 2.6, respectively. The refractive indices of the core and shell were 1.5 + i0.001, and 1.30 + i10−5, respectively. The mean radius was 0.3 µm and the standard deviation (σ) was 2.0.
To further investigate the impact of non-sphericity and the refractive index on the degree of linear polarization at near-backscattering directions, the –P12/P11 of coated super-spheroids with different shape parameters and refractive indices were computed. The results are depicted in Fig. 9. Figures 9(a)–9(d) (upper four panels) show the –P12/P11 of coated super-spheroids at four selected shell refractive indices with a fixed aspect ratio of 1.0, but with various shell roundness parameters ranging from 1.0 to 2.4. Pronounced negative minimum values at near-backscattering directions were found when the shell roundness was close or equal to unity (the shell was nearly-spherical or spherical). For larger real part of the refractive index of the shell, the negative minimum values became less pronounced (see Figs. 9(c) and 9(d)). Figures 9(e)–9 h (bottom four panels) show the –P12/P11 of coated super-spheroids at four selected shell refractive indices with a fixed shell roundness of 1.2, but with various aspect ratios ranging from 0.5 to 2.0. Note that the aspect ratio of the core was identical to that of the shell. As is shown in Fig. 9(e)–9 h, the negative minimum values were pronounced when the aspect ratio was around unity (0.9–1.2) and became less pronounced as the aspect ratio deviated from unity. The minimum disappeared when the aspect ratio was close to 0.5 and 2.0. Similar to Figs. 9(a)–9(d), the negative polarization was less pronounced for larger refractive index of the shell. From Fig. 9, we can learn that coated super-spheroids with a nearly-spherical shell would produce pronounced negative polarization at near-backscattering directions. In other words, the non-sphericity of the shell (which determines the overall shape of the coated particle) would suppress the negative polarization at near-backscattering directions.
Fig. 9. The –P12/P11 of coated super-spheroids with different shape parameters. (a-d): Results for coated super-spheroids with a fixed shell aspect ratio of 1.0; (e-h): Results for coated super-spheroids with a fixed shell roundness of 1.2. The roundness of the core was fixed as 2.6 and the aspect ratio was identical to that of the shell. The refractive index of the core was 1.5 + i0.001. Four selected refractive indices for the shell were indicated in the figure. The mean radius was 0.3 µm, and the standard deviation was 2.0.
Figure 10 presents the –P12/P11 values for different imaginary parts of the refractive index of the shell. The curves for mi,shell = 0.001 and 0.01 in Fig. 10 show little difference. Pronounced negative minimum values at near-backscattering angles were found for mi,shell ≤ 0.05, which was in line with the finding from Fig. 6(b) for coated spheres that weakly- or moderately-absorptive shell led to pronounced negative polarization at near-backscattering angles. Similar to the results presented in Fig. 6(b), strongly-absorptive shell (mi,shell = 0.5) resulted in positive polarization. But different from Fig. 6(b), the negative minimum of –P12/P11, in contrast, became less pronounced as the imaginary part of the shell increased from 0.001 to 0.05. Moreover, mi,shell = 0.1 produced much less pronounced negative minimum compared to the corresponding results in Fig. 6(b). This might be due to the non-sphericity, the difference in the real part of the complex refractive index of the shell, or the shell-core ratio. Nevertheless, we can still draw a “universal” conclusion that weakly- or moderately-absorptive shell would lead to pronounced negative polarization at near-backscattering angles.
Fig. 10. Variation of –P12/P11 for coated super-spheroids with different imaginary part of the shell refractive index ranging from 0.001 to 0.05. The real part of the shell refractive index was fixed at 1.3. The core refractive index was 1.5 + i0.001. The super-spheroids had a mean radius of 0.3 µm with a standard deviation of 2.0.
Now we turn to the coated super-spheroid whose core and shell have the same geometrical shape. Figure 11 shows the –P12/P11 of coated super-spheroids with different roundness parameters and shell-core ratios. It should be noted that the shell-core ratios referred to the long axis of the shell and core of the coated super-spheroid. The negative values at near-backscattering directions were most pronounced for the case of n = 0.8 and were least pronounced for the case of n = 2.5, which was consistent to the finding from Fig. 9 that roundness of the shell close to unity (the overall shape was nearly-spherical) produced pronounced negative polarization. As the shell-core ratio increased (the coating became thicker), the negative minimum of –P12/P11 became more pronounced, except for the case of n = 2.5. This is consistent to the finding from Fig. 4 that the pronounced negative values of –P12/P11 appeared mostly when the shell-core ratio was in a range of 1.4–1.9.
Fig. 11. The –P12/P11 of coated super-spheroids with different roundness and shell-core ratios. The core and shell had the same shape parameters. The aspect ratio ($\alpha $) was set as 1.0. The refractive index of the core was 1.5, that of the shell was 1.3. The mean radius was 0.3 µm and the standard deviation (σ) was 2.0.
3.3 Scattering mechanism of the negative minimum of –P12/P11 for a coated sphere
Now we turn to investigate the underlying scattering mechanism of the negative polarization at near-backscattering angles using the Debye series approach. Note that the analysis was limited to coated spheres because the computation of non-spherical coated particles remained challenging. Figure 12 shows the P11 and –P12/P11 of a coated sphere with a shell-core ratio of 1.5 using the computational program of the Debye series (refer to Section 2.1) and the Lorenz-Mie theory (CMIE code, in particular), respectively. The size parameter of the shell was set at 15 and the refractive index of the core and shell were 1.48 and 1.324, respectively. As can be seen from Fig. 12, excellent agreement has been obtained between the results computed from different methods, validating the reliability of the Debye series computational program.
Fig. 12. Comparison of the P11 and –P12/P11 of a coated sphere computed from the Debye series and CMIE. The refractive indices of the shell and core were 1.324 and 1.48, respectively. The particle size parameter of the shell was 15, and the shell-core ratio was 1.5.
The P11 (i.e., phase function) and –P12/P11 with different times of internal reflections on the shell-medium interface (q–1 in Eq. (15)) were computed and compared with the total scattering to investigate the contributions from various orders of reflection and transmission to the total scattering. The refractive indices of the shell and core were 1.324 and 1.48, respectively. As is shown in Figs. 13(a) and 13(b), a pronounced negative minimum was found for the total –P12/P11 at the scattering angle between 170° and 175°. When q ≤ 1 (i.e., diffraction, external reflection and transmission without internal reflection on the shell-medium interface), the scattering at backward directions (>90°) was significantly underestimated (Fig. 13(a)) and no extreme negative –P12/P11 value appeared at the scattering angle of around 175°. However, q ≤ 2 yielded close results with the total scattering and led to a negative extreme value of –P12/P11 at the scattering angle between 170° and 175°. This indicates that the q = 2 term (i.e., waves undergo one internal reflection on the shell-medium interface) dominated the scattering contributions and the occurrence of the negative extreme –P12/P11 value at backward directions. Moreover, the summation of wave contributions with q ≤ 3 could reasonably reproduce the total phase function and –P12/P11. To further investigate the contributions of internal reflections on the core-shell interface (indicated by p–1 in Eq. (16)) to the total scattering, the phase function and –P12/P11 were computed with q ≤ 2 and various values of p. As can be seen in Figs. 13(c) and 13(d), the internal reflections on the core-shell interface only had minor influence on the P11 and –P12/P11. It should be pointed out that the negative extreme value had already appeared at the near-backscattering angle (around 175°) with q ≤ 2 and p ≤ 1 (without internal reflections on the core-shell interface). The above analysis was based on the simulations that the core had no absorption. For additional analysis, we conducted simulations for absorptive cores with varying imaginary part of the core refractive index (denoted as mi,core). The results of mi,core values of 0.01, 0.05, 0.1 and 0.5 are shown in Fig. 14. For the case of mi,core = 0.01, there was a pronounced minimum at the near backscattering angle, and q ≤ 2 and p ≤ 1 can reproduce the whole scattering curve to a large extent, especially at near backscattering directions. However, for the cases of highly-absorptive cores (mi,core =0.05, 0.1 and 0.5), the amplitude of the negative minimum decreased. The simulations with q ≤ 2 and p ≤ 1 could not reproduce the total scattering curves at scattering angles ranging from 90° to 165°, but can produce the negative polarization at near-backscattering directions. It was noted that q ≤ 3 can reproduce the whole scattering curve. From Fig. 13 and 14, we learnt that the negative extreme –P12/P11 value of a coated sphere at near-backscattering angles was primarily induced by waves undergone one internal reflection on the shell-medium interface without internal reflection on the core-shell interface.
Fig. 13. The contribution of Debye’s series of various order of q (a-b) and p (c-d) to P11 and –P12/P11 for a coated sphere. q-1 and p-1 indicated the number of internal reflections on the shell-medium interface and the core-shell interface, respectively. The refractive indices of the shell and core were 1.324 and 1.48, respectively. The shell-core ratio was 1.5. The mean radius was 0.8 µm, and the standard deviation was 2.0.
Fig. 14. The contribution of Debye’s series of various order p to P11 and –P12/P11 for a coated sphere. The refractive indices of the shell were 1.342 for all the simulations. The core refractive indices were 1.48 + i0.01 (a, b), 1.48 + i0.05 (c, d), 1.48 + i0.1 (e, f) and 1.48 + i0.5 (g, h), respectively. p-1 indicated the number of internal reflections on the core-shell interface. The shell-core ratio was 1.5, the mean radius was 0.8 µm, and the standard deviation was 2.0.
According to Eqs. (15) and (16), the Debye series decomposition of the scattering amplitudes of the partial waves undergone one internal reflection on the shell-medium interface without internal reflection on the core-shell interface can be written as:
Figures 15(b)–15(d) show the different ray paths corresponding to the three terms (${T_a}$, ${T_b}$, and ${T_c}$, as shown in Eqs. (39–41)). Actually, ${T_b}$ contains two terms (Figs. 15(b) and 15(c)) and their electromagnetic scattering amplitudes are identical. ${T_a}$ represented the ray undergoing two external reflections on the core-shell interface and one internal reflection on the shell-medium interface. ${T_b}$ represented the ray undergoing one external reflection on the core-shell interface and one internal reflection on the shell-medium interface, then transmitting through the core and exiting. ${T_c}$ represented the ray transmitting through the shell and core with one internal reflection on the shell-medium interface. To further analyze the contribution of ${T_a}$, ${T_b}$, and ${T_c}$ to the total scattering, the ${a_n}$ and ${b_n}$ coefficients in Eqs. (15) and (16) were computed by keeping an individual term (${T_a}$, ${T_b}$, and ${T_c}$), and then the scattering matrix for different terms was computed by the corresponding ${a_n}$ and ${b_n}$ coefficients. The individual contribution of the three terms to the –P12/P11 is presented in Fig. 15(e) and pronounced negative values were not found at near-backscattering angles. Then, the summations of every two of the three terms were computed to further investigate the impact of the interference among every two terms (Fig. 15(f)). For the case of “${T_a}$+${T_c}$”, a negative minimum similar to the total degree of polarization (see the dashed line in Fig. 15(g)) was found at the near-backscattering direction. Furthermore, we investigated the influence of the interference among the three terms on the linear polarization. If the ${a_n}$ and ${b_n}$ coefficients were calculated by Eq. (38), the interference among the three terms are included. For the case without interference, three scattering phase matrices were computed from the ${a_n}$ and ${b_n}$ coefficients with an individual ${T_a}$, ${T_b}$, or ${T_c}$ term, and then they are weighted averaged by scattering cross sections of these three terms. As can be seen in Fig. 15(g), no negative extreme values were found when the interference effect was excluded. However, the summation of three terms with interference incorporated successfully reproduced the negative minimum at near-backscattering angle of around 175° and the curve captured the features of the total degree of linear polarization. Thus, we can conclude that the negative minimum of –P12/P11 at near-backscattering angles for coated spheres is induced by the interference effect among the partial waves after making one internal reflection on the shell-medium interface without internal reflection on the core-shell interface. The interference between the ${T_a}$ and ${T_c}$ terms are more influential.
Fig. 15. (a)-(d): Ray paths for q = 2 and p = 1; (e) The contributions of the three expanded terms (${T_a}$, ${T_b}$, and ${T_c}$) to the –P12/P11; (f) The summation of two of the three terms with interference; (g) The summation of the three terms with or without interference. ${T_a}$ represents ray path (a), ${T_b}$ represents ray path (b), and ${T_c}$ represents ray path (c) and (d). The refractive indices of the shell and core were 1.324 and 1.48, respectively. The shell-core ratio was 1.5. The mean radius was 0.8 µm, and the standard deviation was 2.0.
Further calculations have been conducted with varying imaginary parts of the core refractive index. The results, shown in Fig. 16, were obtained for mi,core values of 0.01, 0.05, 0.1, 0.3, 0.4, and 0.5. When mi,core ≤ 0.1, the results were similar to those of Fig. 15(e)–15(g). However, as the imaginary part of core refractive index increased, the amplitude of the negative minimum decreased. Additionally, it is observed that interference (see Fig. 16(c), Fig. 16(f), and Fig. 16(i)) played a significant role in reproducing the negative minimum of the total linear polarization. For large mi,core values (mi,core ≥ 0.3), the amplitude of negative minimum polarization was relatively small and almost independent of the increase in mi,core. In these cases, the interference effect (see Fig. 16(l), Fig. 16(o), and Fig. 16(r)) was found to be almost negligible. It was observed that the ${T_a}$ term dominated, and the near-backscattering polarization could be solely reproduced by the ${T_a}$ term.
Fig. 16. Each row resembles the second row in Fig. 15, but with varying imaginary parts of the core refractive index. The imaginary parts of the core refractive index for the panels from top to bottom were 0.01, 0.05, 0.1 and 0.5, respectively.
To gain a better understanding of this phenomenon, we compared the scattering coefficients associated with ${T_a}$, ${T_b}$, and ${T_c}$ terms, as shown in Fig. 17. When the core was non-absorptive or weakly-absorptive, the scattering coefficients for all three terms were comparable. However, as the absorption of core increased, the scattering coefficients for the ${T_b}$ and ${T_c}$ terms decreased significantly, while the dominance shifted towards ${T_a}$. This can be attributed to the fact that an increase in mi,core led to a greater external reflection at the core-shell interface and the ${T_b}$ and ${T_c}$ rays experienced stronger absorption in the core. It should be noted that the scattering coefficient associated with ${T_c}$ was smaller than that of ${T_b}$ because the ${T_c}$ ray experienced two-way absorption, while the ${T_b}$ ray experienced one-way absorption. This indicates that pronounced negative polarization occurred when the core had lower absorption and was caused by interference among ${T_a}$, ${T_b}$, and ${T_c}$ terms, with the interference between the ${T_a}$ and ${T_c}$ terms being more influential.
Fig. 17. A comparison of scattering coefficients associated with ${T_a}$, ${T_b}$ and ${T_c}$ terms at varying imaginary part of the core refractive index (mi,core).
4. Summary
In this study, we systematically investigated the degree of linear polarization (–P12/P11) at near-backscattering angles of core-shell particles. In particular, the range of refractive index, particle size and shell-core ratios within which distinct negative polarization can be observed have been quantified. For coated spheres, we found that the pronounced negative minimum at near-backscattering angle was mostly induced by particles with a limited range of sizes (e.g., the mean radius was smaller than 2 µm at 0.865 µm wavelength) and mostly appeared when the shell-core ratio was in a range of 1.4–1.9. The absorption of both the shell and core showed impact on the negative minimum. Specifically, weakly- and moderately-absorptive shell would lead to pronounced negative polarization at near backscattering directions. The amplitude of negative minimum decreases as the core absorption increases. Similar results were found for coated super-spheroids, indicating that the pronounced negative minimum of –P12/P11 at near-backscattering angle could be a “universal” phenomenon for particles with a core-shell structure. The pronounced negative minimum (<–0.4) mostly appeared when the aspect ratio and roundness of the shell were close to unity (the overall shape of the particle was nearly-spherical) for coated super-spheroids. The non-sphericity of the shell tended to suppress the negative polarization at near-backscattering directions. The shape of the core had little impact on the negative minimum of –P12/P11, which is consistent to the finding in Ref. [6] (see Fig. 2 therein). Further investigation on the underlying mechanism of the negative minimum of –P12/P11 for coated spheres using the Debye series approach found that such a phenomenon resulted from the interference among the partial waves undergone one internal reflection on the shell-medium interface and without internal reflection on the core-shell interface.
The Debye series, which can separate the diffraction, external reflection, and transmission from the total scattering, is considered as a useful approach to analyze the scattering mechanism. It should be pointed out that the scattering mechanism of –P12/P11 in Section 3.3 is only limited to the case of coated spheres because the Debye series approach applied to non-spherical coated particle remains challenging. However, our investigation on the case of coated spheres showed that the pronounced negative polarization at near-backscattering angles were mostly related to the partial waves that had not experienced internal reflection on the core-shell interface. Moreover, it was found in this study (also in Ref. [6]) that the shape of the core had little impact on the pronounced negative polarization at near-backscattering angles. Thus, the explanation of the pronounced negative polarization at near-backscattering angles for coated spheres might also be valid for non-spherical coated particles, or at least, for coated particles with spherical shells. More efforts could be made on the development of the Debye series for further understanding of the underlying mechanism of pronounced negative polarization at near-backscattering angles for non-spherical coated particles. However, it is beyond the scope of this study.
In the atmosphere, particles could have a core-shell structure under certain circumstances, for example, transported dust coated with soot, soot coated with sulfate, and sea salt coated with water during the deliquescence and crystallization processes. The light scattering intensity and polarization state differ between different particle structures and thus can significantly impact the top of atmospheric radiance, polarized radiance and radiative forcing [5,6,53,58]. The difference between the degree of linear polarization (–P12/P11) of coated particles and that of homogeneous particles could provide a possible means to identify the internal structure of particles and classify aerosols. This research improves our understanding of the light scattering characteristics of coated particles, which is valuable for interpreting polarized signals obtained from aerosol remote sensing. It should be pointed out that the negative polarization of non-spherical particles has been an important research direction [61–65], but the amplitude of the negative polarization for non-spherical particles (normally <20%) is smaller than the distinct polarization of coated spheres (could reach 60%) discussed in the present study. Additionally, the physical mechanism of the negative polarization of core-shell particles and homogeneous particles is fundamentally different.
Funding
National Natural Science Foundation of China (42022038, 42090030).
Acknowledgments
The authors acknowledge Maxim Yurkin for his helpful advice to improve this manuscript and Weiwei Cai for providing the CMIE code. A portion of the computations was performed on the cluster at State Key Lab of CAD&CG at Zhejiang University and the computing facilities at East China HPC Cloud Computing Center.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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