Polarized radiative transfer in dense dispersed media containing optically soft sticky particles

Lanxin Ma; Lanxin Ma; Cunhai Wang; Linhua Liu; Linhua Liu

doi:10.1364/OE.404121

1. Introduction

Optically soft particles, whose refractive indices are close to that of the surrounding medium, extensively exist in natural environment and industrial production [1]. Meanwhile, the problems of scalar and vector radiative transfer in dense dispersed media composed of optically soft particles are important in many scientific and engineering aspects, such as biomedical sciences, atmospheric and ocean optics, remote sensing, solar energy utilization and so on [2–6]. In the theoretical analysis of radiative transfer process in a dense discrete random medium, two important problems occur and cannot be neglected in many cases. The first problem is that the dependent scattering effects between particles make the traditional radiative transfer equation unreliable [7–10]. The second problem is that densely packed particles have a tendency to form clusters and bond with each other due to the surface adhesive potential [11–13]. Under these circumstances, neglecting the effects of dependent scattering and particle agglomeration may result in great errors in radiation transfer calculations.

At present, the researches on the radiative transfer in dense discrete random media mainly focus on two aspects: to deal with the radiative transfer problems by directly solving the Maxwell equations [14–16] and to improve the validity of RTE by considering the dependent scattering and absorption effects [4,8,17–19]. Mackowski and Mishchenko [14] examined the character of electromagnetic energy transport in a slab target containing thousands of random distributed spheres using the superposition T-matrix method (STMM). By using the finite-difference time-domain (FDTD) method, Bao et al. [15] studied the spectral reflection and emission properties of double-layer coatings composed of densely packed TiO₂, SiO₂, and SiC nanoparticles. Pattelli et al. [20] investigated the role of packing density and spatial correlations in dense discrete random media by using the T-matrix GPU method. However, it should be noted that the methods based on directly solving the Maxwell equations are mainly suitable for the model composed of relatively small size and number particles. Aiming at this issue, many researchers attempted to solve the radiative transfer problems in dense discrete random media by considering the far-field interference and near-field effects between particles [21–23]. For densely packed optically soft particles, due to the weak shadowing between particles, light can easily interact with all particles in the cluster (similar to the case of particles at large distances), thus the near-field effects can be approximated to some extent [2,24]. Moreover, the former research results showed that the far-field interference between spherical particles can be well characterized by the structure factor and hard-sphere model [17,21]. Fraden and Maret [17] investigated the effects of short-range inter-particle correlations on the multiple scattering of light in colloidal suspensions and verified the hard-sphere model by comparing the transport mean free path and width of coherent backscattering cone with the experimental results. Nguyen et al. [4] measured the scattering coefficients of monodisperse silica particle suspensions and compared them with the results of Mie calculations combined with the Percus-Yevick hard-sphere model [25,26]. Excellent agreements between the experimental and theoretical results were observed.

In view of the problem of particle agglomeration, many researches have been conducted, but are mainly concentrated on the optical and radiative properties of particle aggregates for different forming conditions, material composition and fractal structures [11–13,27–29]. Penders and Vrij [27] analyzed the turbidity of colloidal silica dispersions using the sticky hard-sphere model. By comparing with the experimental results, the research demonstrated that the model calculations fit the experimental curves very well up to high volume fractions of 30%∼40%. Based on the research of Otanicar et al. [13], it is found that the coupling effect of temperature and particle agglomeration played a significant role in determining the optical properties of nanoparticle suspensions. Al-Gebory and Mengüç [11] studied the scattering coefficients of titanium dioxide nanoparticle suspensions at various pH values. The results showed that the pH values have a significant impact on the radiative properties and the dependent and independent scattering.

Although many researches on the radiative properties of densely packed particles with considering the particle agglomeration have been carried out, studies on the radiative transfer characteristics are rarely reported, especially when the polarization effects are taken into consideration. Polarization provides more information in the radiative transfer process, which has found broad applications in a variety of traditional and emerging fields, such as biomedical application [2], spectroscopic ellipsometry measurement [30], and optical material characterization [31]. For polarized radiative transfer in dispersed media, multiple scattering is a universal phenomenon which not only causes extensive depolarization but also alters the polarization state of the residual polarization preserving signal. Currently, the effect of multiple scattering on the polarized radiative transfer characteristics of dispersed media has been extensively studied [30–32]. However, few researches have involved in the polarized radiative transfer problems with considering the effects of dependent scattering and particle agglomeration. For instance, the majority of mammalian tissues and cells (i.e., skin, vessel wall, and blood) can be represented as particle systems composed of densely packed optical soft scattering particles [2]. Information about the structure of a tissue can be extracted from the depolarization degree of initially polarized light, or the polarization state transformation of the scattered light. In such cases, neglecting the effects of dependent scattering and particle agglomeration may lead to accurate results.

In this work, the effects of dependent scattering and particle agglomeration on polarized radiative transfer in a dense dispersed layer composed of optically soft sticky particles are investigated as an extension of our previous researches, which mainly focused on the scalar radiative transfer problems [33]. The radiative properties of the particles are calculated using the Lorenz-Mie theory combined with the Percus-Yevick sticky hard sphere model, and the vector radiative transfer equation is solved by using the spectral method [34,35]. The influences of particle size, particle volume fraction, particle agglomeration degree and layer thickness on the polarized radiative transfer characteristics are discussed.

2. Theory and Methods

We consider polarized radiative transfer in a plane-parallel slab composed of densely packed optically soft sticky spherical particles, as shown in Fig. 1. The effects of dependent scattering and particle agglomeration on particle scattering are considered by using the Percus-Yevick sticky hard sphere model. The approach combined Mie theory and Percus-Yevick hard sphere model for both nonsticky and sticky particles has been verified by many researchers (for details, please refer to [4,8,19,27]). For monodispersed spherical particles, the radiative properties, including the dependent scattering efficiency $Q_{\textrm{sca}}^\textrm{d}$, dependent scattering coefficient $\mu _{\textrm{sca}}^\textrm{d}$ and dependent single-scattering Mueller matrix elements $P_{ij}^\textrm{d}$, are calculated as follows [2,4,21]

(1)$$\frac{{Q_{\textrm{sca}}^\textrm{d}}}{{{Q_{\textrm{sca}}}}} = \frac{1}{2}\int_0^\pi {S({{f_\textrm{v}},\theta ,\nu } )} {P_{11}}(\theta )\sin \theta d\theta, $$

(2)$$\mu _{\textrm{sca}}^\textrm{d} = \frac{{3{f_\textrm{v}}{Q_{\textrm{sca}}}}}{{8a}}\int_0^\pi {S({{f_\textrm{v}},\theta ,\nu } )} {P_{11}}(\theta )\sin \theta d\theta, $$

(3)$$P_{ij}^\textrm{d}(\theta )= \frac{{{Q_{\textrm{sca}}}}}{{Q_{\textrm{sca}}^\textrm{d}}}S({{f_\textrm{v}},\theta ,\nu } )P_{ij}^{}(\theta ),\textrm{ }i = 1,2,3,4,\textrm{ }j = 1,2,3,4, $$

where a and f_v are the sphere radius and sphere volume fraction, θ is the scattering angle, ν is a dimensionless parameter whose inverse is a measure of the attraction or stickiness between particles, Q_sca and P_ij are the scattering efficiency and single-scattering Mueller matrix elements determined from the Lorenz-Mie calculations [36,37]. P₁₁ is the (1,1) element of the single-scattering Mueller matrix P and satisfies the following normalization condition

(4)$$\frac{1}{2}\int_0^\pi {{P_{11}}(\theta )\sin \theta d\theta = 1}. $$

S is the static structure factor incorporating the far-field interference effect between different spheres. For a discrete random medium composed of identical spherical particles, the structure factor can be written as [26]

(5)$$S({{f_\textrm{v}},\theta } )= 1\textrm{ + }\frac{{4\pi \rho }}{q}\int_0^\infty {r\sin ({qr} )({g(r )- 1} )} dr$$

where ρ = 3f_v/4πa³ is the number density of particles, q=(4πn_m/λ)sin(θ/2) is magnitude of the scattering wave vector, λ is the incident light wavelength, n_m is the refractive index of the medium, and g(r) is the radial distribution function. Note that Eq. (5) is based on the assumption of point scatterers under the so-called extended Rayleigh-Debye scattering approximation [21,38]

(6)$$2x\left|{\frac{{{n_p}}}{{{n_m}}} - 1} \right|< < 1$$

where x = 2πan_m/λ and n_p are the size parameter and refractive index of the particle, respectively. Based on Eq. (5) and by using the radial distribution function for sticky hard spheres in the Percus-Yevick approximation, the structure factor of can be written as in a closed-form analytic expression in terms of particle volume fraction f_v, stickiness parameter ν, and sphere radius a according to [26,39]

Fig. 1. Scattering geometry of a laterally infinite particulate layer which is confined to the space between the two imaginary horizontal planes. The external medium on both sides of the dispersed layer is same as the host medium of the dispersed layer. The layer is illuminated from above by a polarized plane electromagnetic wave.

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(7a)$$S({{f_\textrm{v}},\theta ,\nu } )= {[{A{{(X )}^2}\textrm{ + }B{{(X )}^2}} ]^{ - 1}},$$

(7b)$$A(X )= \frac{{{f_\textrm{v}}}}{{1 - {f_\textrm{v}}}}\left\{ {\left( {1 - t{f_v} + \frac{{3{f_\textrm{v}}}}{{1 - {f_\textrm{v}}}}} \right)\Gamma (X )+ [{3 - t({1 - {f_\textrm{v}}} )} ]\Psi (X )} \right\} + \cos (X ),$$

(7c)$$B(X )= \frac{{{f_\textrm{v}}}}{{1 - {f_\textrm{v}}}}X\Gamma (X )+ \sin (X ),$$

(7d)$$\Gamma (X )= 3\left[ {\frac{{\sin (X )}}{{{X^3}}} - \frac{{\cos (X )}}{{{X^2}}}} \right],$$

(7e)$$\Psi (X )= \frac{{\sin (X )}}{X},$$

where X = qa and the parameter t is the smallest solution of the following the quadratic equation

(8)$$\frac{{{f_\textrm{v}}}}{{12}}{t^2} - \left( {\nu + \frac{{{f_\textrm{v}}}}{{1 - {f_\textrm{v}}}}} \right)t + \frac{{1 + {{{f_\textrm{v}}} / 2}}}{{{{({1 - {f_\textrm{v}}} )}^2}}} = 0. $$

Note that the structure factor for non-sticky hard spheres without considering the particle agglomeration can be recovered by taking the limit ν → ∞.

In our analysis, we will assume that the vector radiative transfer equation (VRTE) remains valid in application to the dense dispersed particle model. Given the radiative properties of the densely packed particles, the polarized radiative transfer problems are accounted for by solving the VRTE which can be written as [40,41]

(9)$${{\mathrm{\boldsymbol \Omega}} } \cdot \nabla {\textbf I} ={-} ({{\mu_{\textrm{abs}}}\textrm{ + }\mu_{\textrm{sca}}^\textrm{d}} ){\textbf I}(s,{{\mathrm{\boldsymbol \Omega}} }) + \frac{{\mu _{\textrm{sca}}^\textrm{d}}}{{4\pi }}\int\limits_{4\pi }^{} {{{\overline {\bf Z} }^d}({{\mathrm{\boldsymbol \Omega}} ^{\prime}},{{\mathrm{\boldsymbol \Omega}} }){\textbf I}(s,{{\mathrm{\boldsymbol \Omega}} ^{\prime}})\textrm{d}\Omega }, $$

where s is the coordinate along the ray trajectory, ${\overline {\bf Z} ^\textrm{d}}({{\mathrm{\boldsymbol \Omega}} ^{\prime}},{{\mathrm{\boldsymbol \Omega}} })$ is the dependent normalized phase matrix from the incident direction ${{\mathrm{\boldsymbol \Omega}} }$ to the scattering direction ${{\mathrm{\boldsymbol \Omega}} ^{\prime}}$. As the meridian planes of the incident and the scattered beams are not aligned with the scattering plane, the dependent normalized phase matrix ${\overline {\bf Z} ^\textrm{d}}$ is usually expressed based on transformation of the dependent single-scattering Mueller matrix P^d [42]. Note that for radiative transfer calculation in dense discrete random media, the effective refractive indexes of the dispersed medium are required. According to Refs. [43,44], the effective refractive index n_eff is calculated from the known refractive indexes of the particles n_p and the host medium n_m

(10)$${n_{\textrm{eff}}} = {[{{f_\textrm{v}}({n_\textrm{p}^\textrm{2} - n_\textrm{m}^\textrm{2}} )+ n_\textrm{m}^\textrm{2}} ]^{1/2}}. $$

The external medium on both sides of the dispersed layer is same as the host medium of the dispersed layer, which means that the external medium has different refractive index with the dispersed layer and the interface reflection and refraction should be considered. A polarized incident light described by the Stokes parameters vector S = (I, Q, U, V)^T is perpendicularly incident on the upper boundary of the layer by default, as seen in Fig. 1. The reflection and refraction at the boundaries are considered using the reflection matrix and refraction matrix based on the Fresnel formulas [45]. Considering the large amount of calculation, instead of using the Monte Carlo method (used in our previous work [33]), the spectral method, which has the features of higher-order accuracy and fast calculation speed, is used to solve the VRTE in this work [34,35]. By using N_e = 20 elements and M_θ× M_φ= 60 × 80 directions, the Stokes components obtained by the spectral method are grid independent. For brevity, the detailed contents about the spectral method will not be introduced, more information please refer to our previous work [34,35].

For the sake of comparing and analyzing, the 4 × 4 so-called Stokes reflection matrix R which can fully characterize the diffuse reflection properties of the dispersed layer is systematically investigated. The definition of the Stokes reflection matrix R is as follows [46]

(11)$${{\textbf S}_r} = \frac{1}{\pi }\cos {\theta _i}{\textbf R}({{\theta_r},{\theta_i},{\varphi_r},{\varphi_i}} ){{\textbf S}_i}$$

where S_i and S_r define the Stokes parameters vector of the incident and reflected light, θ_i and φ_i define the incident polar angle and incident azimuth angle, θ_r and φ_r define the reflected polar angle and reflected azimuth angle. In order to obtain the full Stokes reflection matrix R, four simulations applying an incident Stokes vector S_i of (1, 1, 0, 0)^T, (1, -1, 0, 0)^T, (1, 0, 1, 0)^T and (1, 0, 0, 1)^T which is labeled by H, V, P and R, respectively, are conducted. After the VRTE calculations, the reflected Stokes vectors S_r obtained in each of the four simulations are denoted as: ($I_\textrm{H}^{\textrm{Rf}}$, $Q_\textrm{H}^{\textrm{Rf}}$, $U_\textrm{H}^{\textrm{Rf}}$, $V_\textrm{H}^{\textrm{Rf}}$)^T, ($I_\textrm{V}^{\textrm{Rf}}$, $Q_\textrm{V}^{\textrm{Rf}}$, $U_\textrm{V}^{\textrm{Rf}}$, $V_\textrm{V}^{\textrm{Rf}}$)^T, ($I_\textrm{P}^{\textrm{Rf}}$, $Q_\textrm{P}^{\textrm{Rf}}$, $U_\textrm{P}^{\textrm{Rf}}$, $V_\textrm{P}^{\textrm{Rf}}$)^T and ($I_\textrm{R}^{\textrm{Rf}}$, $Q_\textrm{R}^{\textrm{Rf}}$, $U_\textrm{R}^{\textrm{Rf}}$, $V_\textrm{R}^{\textrm{Rf}}$)^T, respectively. Then, the Stokes reflection matrix R of the scattering system can be obtained from the following equation [47]

(12)$${\textbf R} = \frac{\pi }{2}\left[ {\begin{array}{{cccc}} {I_\textrm{H}^{\textrm{Rf}} + I_\textrm{V}^{\textrm{Rf}}}&{I_\textrm{H}^{\textrm{Rf}} - I_\textrm{V}^{\textrm{Rf}}}&{2I_\textrm{P}^{\textrm{Rf}} - I_\textrm{H}^{\textrm{Rf}} - I_\textrm{V}^{\textrm{Rf}}}&{2I_\textrm{R}^{\textrm{Rf}} - I_\textrm{H}^{\textrm{Rf}} - I_\textrm{V}^{\textrm{Rf}}}\\ {Q_\textrm{H}^{\textrm{Rf}} + Q_\textrm{V}^{\textrm{Rf}}}&{Q_\textrm{H}^{\textrm{Rf}} - Q_\textrm{V}^{\textrm{Rf}}}&{2Q_\textrm{P}^{\textrm{Rf}} - Q_\textrm{H}^{\textrm{Rf}} - Q_\textrm{V}^{\textrm{Rf}}}&{2Q_\textrm{R}^{\textrm{Rf}} - Q_\textrm{H}^{\textrm{Rf}} - Q_\textrm{V}^{\textrm{Rf}}}\\ {U_\textrm{H}^{\textrm{Rf}} + U_\textrm{V}^{\textrm{Rf}}}&{U_\textrm{H}^{\textrm{Rf}} - U_\textrm{V}^{\textrm{Rf}}}&{2U_\textrm{P}^{\textrm{Rf}} - U_\textrm{H}^{\textrm{Rf}} - U_\textrm{V}^{\textrm{Rf}}}&{2U_\textrm{R}^{\textrm{Rf}} - U_\textrm{H}^{\textrm{Rf}} - U_\textrm{V}^{\textrm{Rf}}}\\ {V_\textrm{H}^{\textrm{Rf}} + V_\textrm{V}^{\textrm{Rf}}}&{V_\textrm{H}^{\textrm{Rf}} - V_\textrm{V}^{\textrm{Rf}}}&{2V_\textrm{P}^{\textrm{Rf}} - V_\textrm{H}^{\textrm{Rf}} - V_\textrm{V}^{\textrm{Rf}}}&{2V_\textrm{R}^{\textrm{Rf}} - V_\textrm{H}^{\textrm{Rf}} - V_\textrm{V}^{\textrm{Rf}}} \end{array}} \right]$$

3. Results and discussion

In this work, we focus on colloidal silica dispersions, which have proven to be suitable model systems for light scattering studies on interactions between the particles [4,27]. Considering the application condition of the Percus-Yevick sticky hard sphere model (based on Eq. (6)), densely packed spherical silica particles embedded in water with radius a = 0.05, 0.1 and 0.2 μm, respectively, are investigated. The wavelength of incident light is set to λ = 0.6 μm. At this wavelength, the absorption of silica particles and water can be neglected, and the refractive indices of silica particles and water are n_p = 1.458 and n_m = 1.333, respectively. For each size of particles, non-sticky particles with ν = ∞ (take values of 1000 in this study) and three kinds of sticky particles with stickiness parameter ν = 0.1, 0.2 and 0.5, respectively, are studied. Owing to the azimuthal symmetry of the illumination-reflection geometry (under the normal incidence condition), the Stokes reflected matrix R is independent of the azimuth angles of the incident and reflect directions, and depends only on the polar angle of the reflection direction.

3.1 Radiative properties of the particles

Tables 1 and 2 list the characteristic parameters of the scattering system for different particle radii and particle volume fractions, respectively. The layer thickness L is set to 0.1 mm by default, and $\tau = {\mu _{\textrm{sca}}}L$ and ${\tau _\textrm{d}} = \mu _{\textrm{sca}}^\textrm{d}L$ are the optical thickness of the layer for independent scattering and dependent scattering, respectively. As can be clearly seen from the tables, the dependent scattering coefficients are less than the independent scattering coefficients for non-sticky particles, as a result of the dependent scattering effect. When considering the particle agglomeration, differences in the scattering coefficients for different stickiness parameters are observed. Compared with the results for dependent scattering without agglomeration, the scattering coefficients under the same particle volume fractions increase with the decrease of the stickiness parameter. When ν = 0.1, the dependent scattering coefficients are even larger than the independent scattering coefficients in some cases. More discussions about the dependent scattering coefficients of the particles refer to Ref. [33].

Table 1. Characteristic parameters of the scattering system (f_v= 0.2, L = 0.1 mm).

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Table 2. Characteristic parameters of the scattering system (a = 0.1 μm, L = 0.1 mm).

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For spherical particles, the 4 × 4 single-scattering Mueller matrix has eight non-zero elements, of which only four are independent. Figure 2 illustrates the single-scattering Mueller matrix elements of non-sticky and sticky spherical particles with particle radius a = 0.1 μm and particle volume fraction f_v = 20%. The olive, magenta, blue and red curves represent the dependent scattering results for ν = 0.1, 0.2, and 0.5, and for non-sticky particles, respectively. Other characteristic parameters of the scattering system refer to Table 2. For comparison, the independent scattering results, which are represented by black curves, are also illustrated in the figures. As shown in Fig. 2, the effects of dependent scattering and particle agglomeration on all the matrix elements are significant. The element P₁₁, which is also called phase function, determines the angular distribution of the scattered intensity for unpolarized incident light. Compared with the results for independent scattering, the dependent scattering effect leads to a reduction of the forward scattering intensity and an increase of the backward scattering intensity. As the particles form agglomerations, the forward scattering intensity increase obviously with the decrease of the stickiness parameter. When ν = 0.1 and 0.2, the scattering intensities in forward directions are even larger than the results for independent scattering. Element P₁₂ refers to the degree of linear polarization of the scattered light, and element P₃₄ displays the transformation of 45^° obliquely polarized incident light into circularly polarized scattered light. It can be seen from Figs. 2(b) and 2(d) that the variations of P₁₂ with the stickiness parameter are similar to those of P₃₄. With the increase of particle agglomeration degree, the maximum absolute values of P₁₂ and P₃₄ and the corresponding scattering angles gradually decrease. Moreover, it is found that the impacts of dependent scattering and particle agglomeration on the absolute values of element P₃₃ are similar to those on the values of element P₁₁. In addition, based on Eq. (3), it is important to note that the dependent Mueller matrix elements (except P₁₁) coincide with the independent Mueller matrix elements if normalization to the first element P₁₁ is used.

Fig. 2. Single-scattering Mueller matrix elements of non-sticky and sticky spherical particles with a = 0.1 μm and f_v = 20%. Black curves represent the independent scattering Mueller matrix elements; olive, magenta, blue, and red curves represent the dependent scattering Mueller matrix elements for ν = 0.1, 0.2, and 0.5, and for non-sticky particles, respectively, which are obtained according to Eq. (3).

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3.2 Effect of Particle Size

According to the calculation results, the reflection matrix elements R₁₃, R₁₄, R₂₃, R₂₄, R₃₁, R₃₂, R₄₁, R₄₂, R₃₄ and R₄₃ take very small values and thus will not be discussed in this study. Figures 3–5 illustrate the normalized Stokes reflection matrix elements of the dispersed plane-parallel layers composed of densely packed spherical particles with f_v = 20%, L = 0.1 mm, and a = 0.05, 0.1 and 0.2 μm, respectively. The black curves represent the independent scattering results, and the olive, magenta, blue and red curves represent the dependent scattering results for ν = 0.1, 0.2, 0.5 and ∞, respectively. For the convenience of discussion, all elements of the reflection matrix (except R₁₁) are normalized to R₁₁ along the given direction to get results within a range from −1 to 1.

Fig. 3. Normalized Stokes reflection matrix elements of non-sticky and sticky spherical particles with a = 0.05 μm, f_v = 20% and L=0.1 mm. Black curves represent the results for independent scattering; olive, magenta, blue, and red curves represent the results for dependent scattering with ν = 0.1, 0.2, and 0.5, and for non-sticky particles, respectively, which are obtained according to Eq. (12).

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Fig. 4. As in Fig. 3, but for particle radius a = 0.1 μm.

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Fig. 5. As in Fig. 3, but for particle radius a = 0.2 μm.

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Element R₁₁ determines the angular distribution of the reflected intensity of the layer for unpolarized incident light, and its integral over the reflection directions is equal to the hemispherical (diffuse) reflectance. As shown in Figs. 3(a)–5(a), due to the inhibition effect of far-field interference on particle scattering, the values of R₁₁ for dependent scattering are significantly less than the results for independent scattering for all the three size particles. As the particles form agglomerations, it is found that the variation tendency of R₁₁ with the agglomeration degree is different for different size particles. For example, when a = 0.05 μm, the values of R₁₁ increase with the decrease of stickiness parameter. This is because the agglomeration behavior leads to an obvious increase in the scattering coefficients, which enhances the multiple scattering and results in more radiation energy being scattered and reflected from the layer. As the particle radius increases to a = 0.1 and 0.2 μm, the values of the R₁₁ decrease with the decrease of stickiness parameter. This is mainly due to the strong forward scattering characteristics of clusters induced by particle agglomeration, which results in more radiation energy being scattered in the forward direction and pass through the layer. Actually, due to the combined interaction of multiple scattering, dependent scattering and particle agglomeration, the variation tendency of R₁₁ is very complex and depends on several different factors. These results are consistent with the result on the directional-hemispherical reflectance of the layer of our previous study, and more discussions refer to Ref. [33].

The ratio R₂₁/R₁₁ refers to the degree of linear polarization for unpolarized incident light. As can be seen from Figs. 3(c)–5(c), the ratio R₂₁/R₁₁ is negative for most view zenith angles. This phenomenon means that the vibrations of the electric field vector occur predominantly in the plane perpendicular to the scattering plane [48]. Meanwhile, it is found that the ratio R₂₁/R₁₁ is equal to about zero at small view zenith angles, which indicates the depolarization effect is more apparent in these reflection directions. As the particle radius increases, the absolute value of R₂₁/R₁₁ for most scattering angles presents a significant decreasing trend. For example, when a = 0.2 μm, the maximum absolute value of R₂₁/R₁₁ is 0.103 at about θ = 80^°, which is obviously smaller than that for the case of a = 0.05 μm. This result indicates that the linear polarization properties could be better maintained by smaller size particles. By comparing the ratio R₂₁/R₁₁ for dependent scattering (without agglomeration) with the result for independent scattering, it is found that the dependent scattering effect leads to an increase in the absolute value of R₂₁/R₁₁ at most view zenith angles for all the three size particles. This result indicates that the dependent scattering effect weakens the process of depolarization. For sticky particles, it is observed that the impact of particle agglomeration on R₂₁/R₁₁ differs depending on the particle radius. For small size particles with a = 0.05 and 0.1 μm, as shown in Figs. 3(c) and 4(c), the absolute value of R₂₁/R₁₁ at most view zenith angles decreases obviously with the increase of agglomeration degree. However, when a = 0.2 μm, the ratio R₂₁/R₁₁ changes little with the decrease of the stickiness parameter.

According to the Lorenz-Mie theory, the single-scattering Mueller matrix of spherical particles has a symmetric structure with P₁₂= P₂₁. However, as shown in Figs. 3(b), 3(c), 5(b), and 5(c), obvious differences between the ratios R₁₂/R₁₁ and R₂₁/R₁₁ are observed. Moreover, the ratio R₁₂/R₁₁ of different particle radius shows different variation patterns with the increase of agglomeration degree. For small particles with a = 0.05 μm, the absolute value of R₁₂/R₁₁ at small view zenith angles increases with the decrease of the stickiness parameter, which is opposite to that of at large view zenith angles. This result indicates that particle agglomeration leads to strong depolarization at large view zenith angles, but weakens the depolarization at small view zenith angles. With the particle radius increases to a = 0.2 μm, the absolute value of R₁₂/R₁₁ in all reflection directions presents a decreasing trend with the increase of agglomeration degree.

R₂₂/R₁₁ displays to the ratio of depolarized light to the total scattering light. For single-scattering Mueller matrix of spherical particles, the ratio P₂₂/P₁₁ is equal to unity at any scattering angle [2]. However, due to the multiple scattering effect, an obvious deviation of R₂₂/R₁₁ from unity is observed from Figs. 3(d)–5(d). Through comparing the results of different reflection directions, it is found that the ratio R₂₂/R₁₁ increases gradually with the increase of the view zenith angle. This indicates that the depolarization effect is more pronounced at small view zenith angles, which is consistent with the result of R₂₁/R₁₁. Meanwhile, we observe that the ratio R₂₂/R₁₁ decreases significantly with the increase of the particle radius. This is understandable because the scattering coefficients of the particles increase remarkably with the increase of particle radius, which leading to an obvious enhancement of the multiple scattering effect, as shown in Table 1. For non-sticky particles, due to the inhibition effect of far-field interference on particle scattering, the ratio R₂₂/R₁₁ for dependent scattering is obviously larger than the result for independent scattering. However, as the particles form agglomerations, an obvious decrease in the ratio R₂₂/R₁₁ is found for all the three size particles. When ν = 0.1, the ratios R₂₂/R₁₁ at most view zenith angles are even less than the results for independent scattering. The main reason for these changes is that particle agglomeration leads to more clusters with stronger scattering properties, which will enhance the depolarization.

The ratios of R₃₃/R₁₁ and R₄₄/R₁₁ describe the reduction of the degree of 45^° linear polarization and circular polarization for 45^° linear polarization incident light and circularly polarized incident light, respectively. For single-scattering Mueller matrix of spherical particles, it satisfies the Lorenz-Mie identity P₃₃ = P₄₄ [36]. However, as shown in Figs. 3(e), 3(f), 5(c), and 5(f), multiple scattering give rise to an obvious difference in the ratios of R₃₃/R₁₁ and R₄₄/R₁₁. Meanwhile, the absolute value of R₃₃/R₁₁ is always less than that of R₄₄/R₁₁, which manifests that the circular polarization survives more scattering events and can be maintained better than the 45° linear polarization [49,50]. Moreover, it is observed that the effects of dependent scattering and particle agglomeration on R₃₃/R₁₁ and R₄₄/R₁₁ are similar to that of R₂₂/R₁₁. Dependent scattering leads to an obvious increase in the absolute values of R₃₃/R₁₁ and R₄₄/R₁₁ at most view zenith angles, while particle agglomeration leads to an obvious decrease in these values. As the particle radius increases, due to the enhancement of multiple scattering, the absolute values of R₃₃/R₁₁ and R₄₄/R₁₁ show a decreasing trend.

To summarize, the impacts of dependent scattering and particle agglomeration on the reflection matrix elements for different size particles are inconsistent due to the difference in the radiative properties. The dependent scattering effect weakens the process of depolarization, while particle agglomeration enhances the depolarization. The linear polarization properties could be better maintained by smaller size particles, and the circular polarization can be maintained better than the 45° linear polarization.

3.2 Effect of Particle Volume Fraction

As the particle volume fraction increases, multiple and dependent scattering effects are enhanced and particles are more easily to form agglomerations, which will have a noticeable impact on the polarized radiative transfer process. Figures 6–8 illustrate the normalized Stokes reflection matrix elements of the dispersed plane-parallel layers composed of densely packed spherical particles with a = 0.1 μm, L=0.1 mm, and f_v = 5%, 10% and 40%, respectively. The representations of the curves are same as in Fig. 3.

Fig. 6. Normalized Stokes reflection matrix elements of non-sticky and sticky spherical particles with a = 0.1 μm, f_v = 5% and L=0.1 mm. Black curves represent the results for independent scattering; olive, magenta, blue, and red curves represent the results for dependent scattering with ν = 0.1, 0.2, and 0.5, and for non-sticky particles, respectively, which are obtained according to Eq. (12).

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Fig. 7. As in Fig. 6, but for particle volume fraction f_v = 10%.

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Fig. 8. As in Fig. 6, but for particle volume fraction f_v = 40%.

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As shown in Figs. 4, 6–8, due to the enhancement of the dependent scattering and its inhibition effect on particle scattering, the differences in the normalized Stokes reflection matrix elements between dependent scattering (without agglomeration) and independent scattering increase with the increase of particle volume fraction. As for the element R₁₁, under the combined interaction of multiple scattering and dependent scattering, the dependent scattering results do not increase monotonically with the particle volume fraction, but increase first and then decrease. For instance, the elements R₁₁ for f_v = 20% at most view zenith angles are larger than the results for f_v = 5%, 10% and 40%. This is consistent with the analysis results of our previous study [33]. Moreover, we observe that the absolute values of R_ij/R₁₁ (except R₁₂) for dependent scattering at most view zenith angles are less than the results for independent scattering, especially for particles with higher volume fractions. This verifies that the dependent scattering weakens the depolarization caused by multiple scattering.

When particles tend to agglomerate and form larger clusters, due to the combined interaction of multiple scattering, dependent scattering, and particle agglomeration, the variation tendencies of the normalized reflection matrix elements with the agglomeration degree are very complex. When f_v = 5%, as shown in Fig. 6, the differences in the normalized reflection matrix elements between different stickiness parameters are relatively small. This is mainly due to the weak dependent scattering effect, which leads to small differences in the dependent scattering coefficients, as indicated in Table 2. However, as the particle volume fraction increases high values, it can be seen from Figs. 4, 7, and 8 that obvious differences in the reflection matrix elements between different agglomeration degrees are observed, especially for elements R₂₂, R₃₃ and R₄₄. Meanwhile, the absolute values of R_ij/R₁₁ (except R₁₂) at most view zenith angles are significantly decreased with increasing particle agglomeration degree. As discussed above, this is mainly due to the effect of particle agglomeration which induces more clusters with relatively larger size parameters and stronger scattering properties, and thus results in strong depolarization. Furthermore, some special results about the elements R₁₁ and R₁₂ are observed. As shown in Fig. 8, when f_v = 40%, the differences in the elements R₁₁ for ν =∞, 0.5 and 0.2 are very small, and the ratio R₁₂/R₁₁ does not change monotonically with decreasing stickiness parameter.

On the whole, for non-sticky particles, because of the enhancement of dependent scattering and its inhibition effect on particle scattering, the errors caused by neglecting the effect of dependent scattering increase obviously with the increase of particle volume fraction. As particles form agglomerations, due to the combined interaction of multiple factors, the variation tendencies of the normalized reflection matrix elements with the agglomeration degree will become very complex, especially for the matrix elements R₁₁ and R₁₂.

3.3 Effect of layer thickness

Because of the proportional relationship between the geometric thickness and optical thickness of the dispersed layer, the layer thickness has an evident influence on the Stokes reflection matrix elements. Considering that the thickness of optical coatings or thin-films is usually 10 µm∼1 mm [51], polarized radiative transfer in the dispersed layer for three layer thicknesses (L = 0.01, 0.1 and 0.5 mm) is investigated in this work. Figures 9 and 10 illustrate the normalized Stokes reflection matrix elements of the dispersed plane-parallel layers composed of densely packed particles with a = 0.1 μm, f_v = 20%, and L=0.01 and 0.5 mm, respectively. The representations of the curves are same as in Fig. 3.

Fig. 9. Stokes reflection matrix elements of non-sticky and sticky spherical particles with a = 0.1 μm, f_v = 20% and L=0.01 mm. Black curves represent the results for independent scattering; olive, magenta, blue, and red curves represent the results for dependent scattering with ν = 0.1, 0.2, and 0.5, and for non-sticky particles, respectively, which are obtained according to Eq. (12).

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Fig. 10. As in Fig. 9, but for layer thickness L = 0.5 mm.

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When the layer thickness is 0.01 mm, the optical thickness of the layer with L=0.01 for independent scattering is 0.307 such that the multiple scattering effect is weak. Thus, it can be seen from Fig. 9 that the values of the element R₁₁ are small. As the layer thickness increases to L=0.1 and 0.5 mm, the multiple scattering effect increases significantly and leads to an obvious increase in the element R₁₁, as shown in Figs. 4 and 10. As the particles form agglomerations, the elements R₁₁ at most view zenith angles gradually decrease with the increase of agglomeration degree for all the three different layer thicknesses. As discussed in Section 3.1, this is due to the strong forward scattering characteristics of clusters induced by particle agglomeration, which results in more radiation energy being scattered in the forward direction and pass through the layer.

By comparing the ratios R_ij/R₁₁ for different layer thicknesses and particle stickiness parameters, it is observed that the absolute values of R_ij/R₁₁ (except R₁₂) at most view zenith angles decrease obviously with the increase of the layer thickness. This is caused by the enhancement of multiple scattering effect, which results in strong depolarization in the radiative transfer process. Moreover, it is found that the variations of R_ij/R₁₁ with the particle agglomeration degree are not entirely consistent under the three different cases of layer thickness. When the layer thickness is relatively thin with L = 0.01 mm, as shown in Fig. 9, the absolute values of R₁₂/R₁₁, R₂₁/R₁₁ and R₄₄/R₁₁ exhibit different variation tendencies with the agglomeration degree in different reflection directions. For instance, the absolute value of R₄₄/R₁₁ for ν = 0.1 at small view zenith angles is the smallest, but it increases to the maximum when the view zenith angle is greater than about 44°. This is mainly caused by the competitive relationship between dependent scattering and particle agglomeration. Dependent scattering weakens the depolarization, while particle agglomeration enhances it. Meanwhile, the effects of dependent scattering and particle agglomeration on the single-scattering Mueller matrix elements of particles are different at different scattering angles, as shown in Fig. 2. When the layer thickness increases to L = 0.5 mm, as illustrated in Fig. 10, the absolute values of R_ij/R₁₁ at most view zenith angles decrease obviously with the increase of agglomeration degree, even for the ratio R₁₂/R₁₁. The reason is that, with the increase of the layer thickness, the increase in scattering coefficients results from particle agglomeration will further lead to the significant enhancement of the multiple scattering and depolarization.

Overall, the increasing of the layer thickness will enhance the multiple scattering and depolarization, which leads to an increase in the element R₁₁ and a decrease of absolute values of R_ij/R₁₁ (except R₁₂) at most view zenith angles. Moreover, the variations of R_ij/R₁₁ with the particle agglomeration degree are not entirely consistent under the three different cases of layer thickness, which is also the comprehensive results of multiple scattering, dependent scattering and particle agglomeration.

4. Conclusion

In this work, we focus on polarized radiative transfer in colloidal silica dispersions with considering the effects of dependent scattering and particle agglomeration. The dependent scattering coefficients and dependent single-scattering Mueller matrix elements of the spherical particles embedded in water are calculated based on the Mie theory and Percus-Yevick sticky hard sphere model, and the VRTE is solved by using the spectral method. The effects of particle size, particle volume fraction and layer thickness on the normalized Stokes reflection matrix elements of the dispersed layers are discussed.

The results indicate that the effects of multiple scattering, dependent scattering and particle agglomeration have different degrees of influence on the normalized Stokes reflection matrix elements of the dispersed layer. As for the element R₁₁, the dependent scattering results do not increase monotonically with the particle volume fraction, but increase first and then decrease. Moreover, the variation tendency of element R₁₁ with the agglomeration degree is different for different size particles. As for the polarization components, the linear polarization properties could be better maintained by smaller size particles, and the circular polarization can be maintained better than the 45° linear polarization. The increasing of particle radius and layer thickness will enhance the multiple scattering and depolarization, which leads to a decrease of absolute values of R_ij/R₁₁ at most view zenith angles. Due to the inhibition effect of far-field interference interaction on particle scattering, the dependent scattering weakens the depolarization caused by multiple scattering. For non-agglomeration particles, as the dependent scattering dominates in the polarized radiative transfer process, the absolute values of R_ij/R₁₁ increase gradually with the increase of particle volume fraction. However, as the particles form agglomerations, the scattering coefficients of the particles obviously increase with the agglomeration degree, leading to the significant enhancement of the multiple scattering and depolarization. Therefore, due to the combined interaction of multiple factors, the variation tendencies of the normalized reflection matrix elements with the agglomeration degree will become very complex, especially for the matrix elements R₁₁ and R₁₂.

This work give quantitative analyses of the impacts of dependent scattering and particle agglomeration on the polarized radiative transfer process, which should be useful for analyzing and optimizing applications of polarization techniques in optically soft particle systems. Future simulation analysis will take into account the polydisperse distributions of scattering particles and different refractive indices of particles and host medium.

Funding

National Natural Science Foundation of China (51806124, 51906014); China Postdoctoral Science Foundation (2019M662353); Fundamental Research Fund of Shandong University (2018GN045); Young Scholars Program of Shandong University.

Disclosures

The authors declare no conflicts of interest.

References

1. S. K. Sharma and D. J. Somerford, Light Scattering by Optically Soft Particles: Theory and Applications (Springer, 2006).

2. V. V. Tuchin, A. Z. Dmitry, and L. V. Wang, Optical Polarization in Biomedical Applications (Springer, 2006).

3. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and Software (SPIE Press, 2010).

4. V. D. Nguyen, D. Faber, E. Van Der Pol, T. Van Leeuwen, and J. Kalkman, “Dependent and multiple scattering in transmission and backscattering optical coherence tomography,” Opt. Express 21(24), 29145–29156 (2013). [CrossRef]

5. P. J. Werdell, L. I. W. Mckinna, E. Boss, S. G. Ackleson, S. E. Craig, W. W. Gregg, Z. Lee, S. Maritorena, C. S. Roesler, and C. S. Rousseaux, “An overview of approaches and challenges for retrieving marine inherent optical properties from ocean color remote sensing,” Prog. Oceanogr. 160, 186–212 (2018). [CrossRef]

6. C.-H. Wang, Y.-Y. Feng, K. Yue, and X.-X. Zhang, “Discontinuous finite element method for combined radiation-conduction heat transfer in participating media,” Int. Commun. Heat Mass Transfer 108, 104287 (2019). [CrossRef]

7. B. L. Drolen and C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. Heat Transfer 1(1), 63–68 (1987). [CrossRef]

8. L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: Comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transfer 187, 255–266 (2017). [CrossRef]

9. M. I. Mishchenko, ““Independent” and “dependent” scattering by particles in a multi-particle group,” OSA Continuum 1(1), 243–260 (2018). [CrossRef]

10. B. X. Wang and C. Y. Zhao, “Modeling radiative properties of air plasma sprayed thermal barrier coatings in the dependent scattering regime,” Int. J. Heat Mass Transfer 89, 920–928 (2015). [CrossRef]

11. L. Al-Gebory and M. P. Mengüç, “The effect of pH on particle agglomeration and optical properties of nanoparticle suspensions,” J. Quant. Spectrosc. Radiat. Transfer 219, 46–60 (2018). [CrossRef]

12. M. Du and G. H. Tang, “Optical property of nanofluids with particle agglomeration,” Sol. Energy 122, 864–872 (2015). [CrossRef]

13. T. Otanicar, J. Hoyt, M. Fahar, X. Jiang, and R. A. Taylor, “Experimental and numerical study on the optical properties and agglomeration of nanoparticle suspensions,” J. Nanopart. Res. 15(11), 2039 (2013). [CrossRef]

14. D. W. Mackowski and M. I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 123(123), 103–112 (2013). [CrossRef]

15. H. Bao, C. Yan, B. X. Wang, X. Fang, C. Y. Zhao, and X. L. Ruan, “Double-layer nanoparticle-based coatings for efficient terrestrial radiative cooling,” Sol. Energy Mater. Sol. Cells 168, 78–84 (2017). [CrossRef]

16. A. Egel, L. Pattelli, G. Mazzamuto, D. S. Wiersma, and U. Lemmer, “CELES: CUDA-accelerated simulation of electromagnetic scattering by large ensembles of spheres,” J. Quant. Spectrosc. Radiat. Transfer 199, 103–110 (2017). [CrossRef]

17. S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65(4), 512–515 (1990). [CrossRef]

18. L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: Effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transfer 196, 94–102 (2017). [CrossRef]

19. M. I. Mishchenko, D. H. Goldstein, J. Chowdhary, and A. Lompado, “Radiative transfer theory verified by controlled laboratory experiments,” Opt. Lett. 38(18), 3522–3525 (2013). [CrossRef]

20. L. Pattelli, A. Egel, U. Lemmer, and D. S. Wiersma, “Role of packing density and spatial correlations in strongly scattering 3D systems,” Optica 5(9), 1037–1045 (2018). [CrossRef]

21. J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 1—theoretical investigation,” J. Heat Transf. 108(3), 608–613 (1986). [CrossRef]

22. N. R. Rezvani, S. Sukhov, J. J. Sáenz, and A. Dogariu, “Near-Field Effects in Mesoscopic Light Transport,” Phys. Rev. Lett. 115(20), 203903 (2015). [CrossRef]

23. D. Baillis and J. F. Sacadura, “Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization,” J. Quant. Spectrosc. Radiat. Transfer 67(5), 327–363 (2000). [CrossRef]

24. Y. Okada and A. Kokhanovsky, “Light scattering and absorption by densely packed groups of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110(11), 902–917 (2009). [CrossRef]

25. J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110(1), 1–13 (1958). [CrossRef]

26. R. J. Hunter, Foundations of Colloid Science (Oxford University, 2001).

27. M. Penders and A. Vrij, “A turbidity study on colloidal silica particles in concentrated suspensions using the polydisperse adhesive hard sphere model,” J. Chem. Phys. 93(5), 3704–3711 (1990). [CrossRef]

28. L. M. Zurk, L. Tsang, K. Ding, and D. P. Winebrenner, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12(8), 1772–1781 (1995). [CrossRef]

29. R. Tazaki, H. Tanaka, T. Muto, A. Kataoka, and S. Okuzumi, “Effect of dust size and structure on scattered-light images of protoplanetary discs,” Mon. Not. R. Astron. Soc. 485(4), 4951–4966 (2019). [CrossRef]

30. B. Kaplan, G. Ledanois, and B. Drévillon, “Mueller matrix of dense polystyrene latex sphere suspensions: measurements and Monte Carlo simulation,” Appl. Opt. 40(16), 2769–2777 (2001). [CrossRef]

31. D. Héran, M. Ryckewaert, Y. Abautret, M. Zerrad, C. Amra, and R. Bendoula, “Combining light polarization and speckle measurements with multivariate analysis to predict bulk optical properties of turbid media,” Appl. Opt. 58(30), 8247–8256 (2019). [CrossRef]

32. N. Riviere, R. Ceolato, and L. Hespel, “Polarimetric and angular light-scattering from dense media: Comparison of a vectorial radiative transfer model with analytical, stochastic and experimental approaches,” J. Quant. Spectrosc. Radiat. Transfer 131, 88–94 (2013). [CrossRef]

33. L. X. Ma, C. Wang, and J. Y. Tan, “Light scattering by densely packed optically soft particle systems, with consideration of the particle agglomeration and dependent scattering,” Appl. Opt. 58(27), 7336–7345 (2019). [CrossRef]

34. J. M. Zhao, L. H. Liu, P.-F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 111(3), 433–446 (2010). [CrossRef]

35. C.-H. Wang, Y.-Y. Feng, Y.-H. Yang, Y. Zhang, and X.-X. Zhang, “Chebyshev collocation spectral method for vector radiative transfer equation and its applications in two-layered media,” J. Quant. Spectrosc. Radiat. Transfer 243, 106822 (2020). [CrossRef]

36. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

37. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

38. H. C. Van De Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic Press, 1980).

39. L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley2001).

40. M. F. Modest, Radiative Heat Transfer (Academic Press, 2013).

41. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles (Cambridge University, 2006).

42. S. Chandrasekhar, Radiative Transfer (Oxford University, 1950).

43. J. M. Beechem, “Global analysis of biochemical and biophysical data,” Methods Enzymol. 210, 37–54 (1992). [CrossRef]

44. R. Hass, M. Münzberg, L. Bressel, and O. Reich, “Industrial applications of Photon Density Wave spectroscopy for in-line particle sizing,” Appl. Opt. 52(7), 1423–1431 (2013). [CrossRef]

45. L. Tsang, J. A. Kong, and R. T. Shin, Theory of microwave remote sensing (Wiley-Interscience, 1985).

46. M. I. Mishchenko and L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res.-Atmos. 102(D14), 16989–17013 (1997). [CrossRef]

47. I. L. Maksimova, S. V. Romanov, and V. F. Izotova, “The effect of multiple scattering in disperse media on polarization characteristics of scattered light,” Opt. Spectrosc. 92(6), 915–923 (2002). [CrossRef]

48. M. I. Mishchenko and L. D. Travis, Polarization and depolarization of light. In: Moreno F, González F, editors. Light scattering from microstructures. (Springer-Verlag, 2000), pp. 159–175.

49. J. D. van der Laan, J. B. Wright, D. A. Scrymgeour, S. A. Kemme, and E. L. Dereniak, “Evolution of circular and linear polarization in scattering environments,” Opt. Express 23(25), 31874 (2015). [CrossRef]

50. S. Zhang, J. Zhan, Q. Fu, J. Duan, Y. Li, and H. Jiang, “Effects of environment variation of glycerol smoke particles on the persistence of linear and circular polarization,” Opt. Express 28(14), 20236–20248 (2020). [CrossRef]

51. M. P. Diebold, Application of Light Scattering to Coatings (Springer, 2014).

a (μm)	ν	n_eff	μ_sca (mm^-1)	τ	$μ_{sca}^{d}$ (mm^-1)	τ_d
0.05	∞	1.3589	6.1570	0.616	1.6882	0.169
0.05	0.5	1.3589	6.1570	0.616	2.7264	0.273
0.05	0.2	1.3589	6.1570	0.616	4.4589	0.446
0.05	0.1	1.3589	6.1570	0.616	12.785	1.279
0.1	∞	1.3589	30.650	3.065	14.846	1.485
0.1	0.5	1.3589	30.650	3.065	17.304	1.730
0.1	0.2	1.3589	30.650	3.065	21.159	2.116
0.1	0.1	1.3589	30.650	3.065	39.008	3.901
0.2	∞	1.3589	87.810	8.781	47.000	4.700
0.2	0.5	1.3589	87.810	8.781	53.140	5.314
0.2	0.2	1.3589	87.810	8.781	61.937	6.194
0.2	0.1	1.3589	87.810	8.781	99.221	9.922

f_v	ν	n_eff	μ_sca (mm^-1)	τ	$μ_{sca}^{d}$ (mm^-1)	τ_d
0.05	∞	1.3395	7.6624	0.766	6.4685	0.649
0.05	0.5	1.3395	7.6624	0.766	6.7523	0.675
0.05	0.2	1.3395	7.6624	0.766	7.2052	0.721
0.05	0.1	1.3395	7.6624	0.766	8.4542	0.845
0.1	∞	1.3460	15.325	1.533	10.862	1.086
0.1	0.5	1.3460	15.325	1.533	11.786	1.179
0.1	0.2	1.3460	15.325	1.533	13.306	1.331
0.1	0.1	1.3460	15.325	1.533	20.997	2.100
0.2	∞	1.3589	30.650	3.065	14.846	1.485
0.2	0.5	1.3589	30.650	3.065	17.304	1.730
0.2	0.2	1.3589	30.650	3.065	21.159	2.116
0.2	0.1	1.3589	30.650	3.065	39.008	3.901
0.4	∞	1.3844	61.299	6.130	9.0127	0.901
0.4	0.5	1.3844	61.299	6.130	14.538	1.454
0.4	0.2	1.3844	61.299	6.130	19.145	1.915
0.4	0.1	1.3844	61.299	6.130	25.239	2.524

a (μm)	ν	n_eff	μ_sca (mm^-1)	τ	$μ_{sca}^{d}$ (mm^-1)	τ_d
0.05	∞	1.3589	6.1570	0.616	1.6882	0.169
0.05	0.5	1.3589	6.1570	0.616	2.7264	0.273
0.05	0.2	1.3589	6.1570	0.616	4.4589	0.446
0.05	0.1	1.3589	6.1570	0.616	12.785	1.279
0.1	∞	1.3589	30.650	3.065	14.846	1.485
0.1	0.5	1.3589	30.650	3.065	17.304	1.730
0.1	0.2	1.3589	30.650	3.065	21.159	2.116
0.1	0.1	1.3589	30.650	3.065	39.008	3.901
0.2	∞	1.3589	87.810	8.781	47.000	4.700
0.2	0.5	1.3589	87.810	8.781	53.140	5.314
0.2	0.2	1.3589	87.810	8.781	61.937	6.194
0.2	0.1	1.3589	87.810	8.781	99.221	9.922

f_v	ν	n_eff	μ_sca (mm^-1)	τ	$μ_{sca}^{d}$ (mm^-1)	τ_d
0.05	∞	1.3395	7.6624	0.766	6.4685	0.649
0.05	0.5	1.3395	7.6624	0.766	6.7523	0.675
0.05	0.2	1.3395	7.6624	0.766	7.2052	0.721
0.05	0.1	1.3395	7.6624	0.766	8.4542	0.845
0.1	∞	1.3460	15.325	1.533	10.862	1.086
0.1	0.5	1.3460	15.325	1.533	11.786	1.179
0.1	0.2	1.3460	15.325	1.533	13.306	1.331
0.1	0.1	1.3460	15.325	1.533	20.997	2.100
0.2	∞	1.3589	30.650	3.065	14.846	1.485
0.2	0.5	1.3589	30.650	3.065	17.304	1.730
0.2	0.2	1.3589	30.650	3.065	21.159	2.116
0.2	0.1	1.3589	30.650	3.065	39.008	3.901
0.4	∞	1.3844	61.299	6.130	9.0127	0.901
0.4	0.5	1.3844	61.299	6.130	14.538	1.454
0.4	0.2	1.3844	61.299	6.130	19.145	1.915
0.4	0.1	1.3844	61.299	6.130	25.239	2.524

Polarized radiative transfer in dense dispersed media containing optically soft sticky particles

Abstract

1. Introduction

2. Theory and Methods

3. Results and discussion

3.1 Radiative properties of the particles

3.2 Effect of Particle Size

3.2 Effect of Particle Volume Fraction

3.3 Effect of layer thickness

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (10)

Tables (2)

Equations (16)

Optics Express