Linearization of light scattering properties based on the physical-geometric optics method

Dongbin Liang; Dongbin Liang; Bingqiang Sun; Bingqiang Sun

doi:10.1364/OE.460404

1. Introduction

Cirrus clouds have been identified as one of the most important factors influencing climate [1,2]. The optical properties of cirrus clouds directly or indirectly affect the radiative balance of the Earth atmosphere system and climate [3,4]. The distance between cirrus ice crystals is much larger than the geometric dimensions of the crystals themselves, which makes it meaningful to study the single-scattering properties (SSPs) of individual ice crystals [5,6]. However, the non-spherical nature of ice crystals and orientation issues make it difficult to study their SSPs [7].

Various ice crystal shapes have been discovered during in situ observations [8,9], and they tend to exhibit hexamorphic symmetry under ideal conditions and irregular shapes under complex atmospheric conditions [10–12]. Collisions and adhesion between multiple ice crystals can form complex aggregates and the SSPs of the aggregates are dominated by those of individual particles [13,14]. For small non-spherical particles, the SSPs can be obtained using the numerically rigorously methods, such as finite-difference time-domain method [15,16], the pseudo-spectral time-domain method [17], the discrete dipole approximations [18–20], and the T-matrix methods [21–25]. The computational resources required by the numerically rigorous methods increase dramatically with particle size parameters, so these methods perform poorly with large particles [26]. The SSPs of terrestrial ice cloud crystals are generally calculated by the geometric optics method (GOM) owing to their large size [27]. Geometries such as hexagonal prisms, plates, cylinders, droxtals, bullets, droplet and capped columns have been widely used in the GOMs for SSPs studies of non-spherical particles [11,28–34].

Light scattering by large particles involves mainly surface reflection, refraction inside particles, and far-field mapping [35]. Ray-tracing process using Snell's law and Fresnel's formula, and diffraction using Babinet’s principle are used to describe these processes in the conventional geometric optics method (CGOM) [3,11,36]. The CGOM can be improved by using the Fraunhofer diffraction to accomplish the far-field mapping [37,38]. Furthermore, the ray-tracing process inside particles can be analytically obtained for facet particles, where ray-tracing are replaced by beam-tracing to improve computational efficiency, and the mapping process can be accurately done using Maxwell’s equations in the physical-geometric optics method (PGOM) [31,39–41]. Beam splitting technique is essential to the beam-tracing process to make one beam only incident on a single facet and determines the computational efficiency of the geometric optics method. The details can be found in section 2.2. Mathematical methods and the “divide-and-conquer” algorithm of computer graphics have been used in beam splitting [31,40].

The linearization of SSPs, (i.e., the partial derivative of SSPs with respect to scattering parameters) is critical to remote sensing and retrieval [42]. For small non-spherical particles, the extended-boundary-condition method [43] and the invariant-imbedding T-matrix method [44] provide the linearization of SSPs for axisymmetric or asymmetric particles. The algorithm for the linearization of SSPs is lacking for large non-spherical particles. In this study, the linearization of the SSPs for convex facet particles is derived based on the PGOM and applied to regular hexagonal prisms for an example. Moreover, the winding number method is introduced to improve the beam splitting process [45]. For regular hexagonal prisms, the numerical results of the SSP linearization are verified by the results of the finite-difference method, and the sensitivities of SSPs to scattering parameters such as aspect ratio, effective radius, and refractive index are correspondingly discussed.

The paper is divided into four sections. The derivation of SSPs based on PGOM and the beam-splitting method are given in Sections 2.1 and 2.2, respectively. Section 2.3 provides the derivation of linearization of SSPs. In Section 3, the linearization is verified by comparison with the finite-difference method, and its sensitivity to scattering parameters is analyzed. The conclusion is presented in Section 4.

2. Method

To simplify the algorithm, only linearization for the convex and homogeneous facet particles are considered. The linearization of SSPs is the partial derivative of SSPs to the aspect ratio $\sigma $ (for regular prisms defined as 2a/h, a is the radius of the circumscribed circle of the prism bottom and h is the prism height), the effective radius ${r_e}$ (defined as the radius of spheres of equal volume), and the real and imaginary parts of the complex refractive index m. Table 1 summarizes the main variables used in this study.

Table 1. Main variables

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2.1 Single-scattering properties

The volume-integral equation from Maxwell’s equations is used to do the far-field mapping, where the far-field ${\vec{E}^s}$ can be expressed an integration of the near-field $\vec{E}$ as [23]:

(1)$${ {{{\vec{E}}^s}({\vec{r}} )} |_{r \to \infty }} = \frac{{k_0^2\exp ({i{k_0}r} )}}{{ - 4\pi r}}\mathop{\int\!\!\!\int\!\!\!\int}\limits_{V} {({1 - {m^2}} )\{{\vec{E}({\vec{r}^{\prime}} )- \hat{r}[{\hat{r} \cdot \vec{E}({\vec{r}^{\prime}} )} ]} \}\exp ({ - i{k_0}\hat{r} \cdot \vec{r}^{\prime}} ){d^3}\vec{r}^{\prime}} .$$

The near-field $\vec{E}$ in Eq. (1) is obtained using the beam-tracing process. For convenience, the process entering from the outside to the inside of a particle is denoted as p = 0. Incident light is denoted as either ${\hat{e}^i}$ or ${\hat{e}_0}$. Light refracted into the interior of a particle at p = 0 undergoes continuous refraction and reflection. Some sub-prisms are formed by the reflected light inside the particle. For example, the p-th order and (p + 1)-th order geometric cross-sections, and p-th order light direction ${\vec{e}_p}$ form a closed p-order prism as shown in Fig. 1. The near electric-field $\vec{E}$ inside the particle can be considered as the sum of electric fields in all sub-prisms, and the electric field vector ${\vec{E}_p}({\vec{r}} )$ for any position $\vec{r}$ inside the p-th order sub-prism can be written as:

(2)$${\vec{E}_p}({\vec{r}} )= {\textbf{U}_p}\left[ {\begin{array}{c} {E_\alpha^i}\\ {E_\beta^i} \end{array}} \right]\exp \left\{ {i{k_0}\left[ \begin{array}{l} {\phi_{p,1}} + ({{N_{p,r}}{{\hat{e}}_p} + i{{\vec{A}}_p}} )\cdot {{\vec{\omega }}_p}\\ + l({{N_{p + 1,r}}{{\hat{e}}_{p + 1}} + i{{\vec{A}}_{p + 1}}} )\cdot {{\hat{e}}_{p + 1}} \end{array} \right]} \right\},$$

(3)$$\vec{r} = {\vec{r}_{p,1}} + {\vec{\omega }_p} + l{\hat{e}_{p + 1}},$$

(4)$${\vec{A}_p} = \left\{ {\begin{array}{ll} \textbf{0}&{p = 0}\\ {{{\vec{A}}_{p - 1}} + ({{N_{p,i}} - {{\hat{e}}_p} \cdot {{\vec{A}}_{p - 1}}} )\frac{{{{\hat{n}}_{p - 1}}}}{{{{\hat{e}}_p} \cdot {{\hat{n}}_{p - 1}}}}}&{p \ne 0} \end{array}} \right.,$$

(5)$${\phi _{p,j}} = {\varphi _{p,j}} + i{\psi _{p,j}},$$

(6)$${\varphi _{p,j}} = \left\{ {\begin{array}{ll} {{{\hat{e}}^i} \cdot {{\vec{r}}_{p,j}}}&{p = 0}\\ {{\varphi_{p - 1,j}} + {N_{p,r}}|{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |}&{p \ne 0} \end{array}} \right.,$$

(7)$${\psi _{p,j}} = \left\{ {\begin{array}{ll} 0 &{p = 0}\\ {{\psi_{p - 1,j}} + {N_{p,i}}|{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |}&{p \ne 0} \end{array}} \right.,$$

where $E_\alpha ^i$ and $E_\beta ^i$ are the parallel and perpendicular parts of ${\vec{E}^i}$; and ${\vec{\omega }_p}$ and l respectively describe the position of $\vec{r}$ inside the p-th order geometric cross-section and the distance between corresponding points in adjacent geometric cross-sections. Equation (4) describes the variation of the absorption properties of the particles in each order. The recurrence relations of the complex phase are shown in Eq. (6) and Eq. (7).

Fig. 1. p-th order sub-prisms. Dash-dot, dashed, and solid lines indicate the geometric cross-section, reflection direction, and facet edge, respectively.

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Combining Eq. (1) and Eq. (2), ${\vec{E}^s}$ can be written as [31]

(8)$${ {{{\vec{E}}^s}({\vec{r}} )} |_{r \to \infty }} = \frac{{\exp ({i{k_0}r} )}}{{ - i{k_0}r}}({1 - {m^2}} )\left( {\sum\limits_{p = 0}^\infty {{T_p}} } \right)\left[ {\begin{array}{c} {E_\alpha^i}\\ {E_\beta^i} \end{array}} \right],$$

(9)$${T_p} = ({|{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |{D_{p + 1}} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |{D_p}} )\frac{{{\textbf{K}_p}{\textbf{U}_p}}}{{{{\vec{\kappa }}_p} \cdot {{\hat{e}}_p}}},$$

(10)$${D_p} = \frac{{{k_0}}}{{4\pi }}\sum\limits_{j = 1}^{{N_{p,v}}} {\left\{ {\left[ {\frac{{({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot ({i{{\vec{\kappa }}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{\kappa }}_p} \cdot {{\vec{\kappa }}_p} - {{({{{\vec{\kappa }}_p} \cdot {{\hat{n}}_p}} )}^2}}}} \right]{F_p}} \right\}} ,$$

(11)$${\vec{\kappa }_p} = {k_0}({{N_{p,r}}{{\hat{e}}_p} + i{{\vec{A}}_p} - \hat{r}} ),$$

(12)$${F_p} = \frac{{\exp ({i{k_0}{\Phi _{p,j + 1}}} )- \exp ({i{k_0}{\Phi _{p,j}}} )}}{{{\Phi _{p,j + 1}} - {\Phi _{p,j}}}},$$

(13)$${k_0}{\Phi _{p,j + 1}} = {k_0}{\Phi _{p,j}} + {\vec{\kappa }_p} \cdot ({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} ),$$

(14)$${\Phi _{p,j}} = {\phi _{p,j}} - \hat{r} \cdot {\vec{r}_{p,j}},$$

where ${\vec{\kappa }_p}$ is the complex wave vector difference; ${\Phi _{p,j}}$ denotes the complex phase difference, and its definition and recurrence relation are shown in Eq. (13) and Eq. (14).

The amplitude-scattering matrix S connects the incident and scattering fields as

(15)$${\left. {\left[ \begin{array}{c} E_\alpha^s\\ E_\beta^s \end{array} \right]} \right|_{r \to \infty }} = \frac{{\exp ({i{k_0}r} )}}{{ - i{k_0}r}}\left( {\begin{array}{cc} {{S_{11}}}&{{S_{12}}}\\ {{S_{21}}}&{{S_{22}}} \end{array}} \right)\left[ {\begin{array}{c} {E_\alpha^i}\\ {E_\beta^i} \end{array}} \right].$$

Combining Eq. (8) and Eq. (15), S can be written as:

(16)$$S = ({1 - {m^2}} )\left( {\sum\limits_{p = 0}^\infty {{T_p}} } \right)\Gamma ,$$

(17)$$\Gamma = \left( {\begin{array}{rr} {{{\hat{\beta }}^i} \cdot {{\hat{\beta }}^s}}&{{{\hat{\alpha }}^i} \cdot {{\hat{\beta }}^s}}\\ { - {{\hat{\alpha }}^i} \cdot {{\hat{\beta }}^s}}&{{{\hat{\beta }}^i} \cdot {{\hat{\beta }}^s}} \end{array}} \right),$$

where $\Gamma $ represents the frame rotation matrix from incident direction to the scattering plane.

The scattering matrix M is generated by the combination of the elements of S. Parts of M and some integral SSPs can be expressed as:

(18)$$\left( {\begin{array}{cc} {{M_{11}}}&{{M_{12}}}\\ {{M_{21}}}&{{M_{22}}} \end{array}} \right) = \frac{1}{2}\left( {\begin{array}{cc} {({{{|{{S_{11}}} |}^2} + {{|{{S_{12}}} |}^2} + {{|{{S_{21}}} |}^2} + {{|{{S_{22}}} |}^2}} )}&{({{{|{{S_{11}}} |}^2} - {{|{{S_{12}}} |}^2} + {{|{{S_{21}}} |}^2} - {{|{{S_{22}}} |}^2}} )}\\ {({{{|{{S_{11}}} |}^2} + {{|{{S_{12}}} |}^2} - {{|{{S_{21}}} |}^2} - {{|{{S_{22}}} |}^2}} )}&{({{{|{{S_{11}}} |}^2} - {{|{{S_{12}}} |}^2} - {{|{{S_{21}}} |}^2} + {{|{{S_{22}}} |}^2}} )} \end{array}} \right),$$

(19)$$\left( {\begin{array}{cc} {{M_{33}}}&{{M_{34}}}\\ {{M_{43}}}&{{M_{44}}} \end{array}} \right) = \left( {\begin{array}{rr} {{\textrm{Re}} ({{S_{11}}{{\bar{S}}_{22}} + {S_{12}}{{\bar{S}}_{21}}} )}&{{\mathop{\textrm{Im}}\nolimits} ({{S_{11}}{{\bar{S}}_{22}} - {S_{12}}{{\bar{S}}_{21}}} )}\\ { - {\mathop{\textrm{Im}}\nolimits} ({{S_{11}}{{\bar{S}}_{22}} - {S_{12}}{{\bar{S}}_{21}}} )}&{{\textrm{Re}} ({{S_{11}}{{\bar{S}}_{22}} - {S_{12}}{{\bar{S}}_{21}}} )} \end{array}} \right),$$

(20)$${C_{ext}} = \frac{{2\pi }}{{k_0^2}}\textrm{Re} [{{S_{11}}({{{\hat{e}}^i}} )+ {S_{22}}({{{\hat{e}}^i}} )} ],$$

(21)$${C_{abs}} = \frac{1}{2}\sum\limits_{p = 0}^\infty {{N_{p,r}}{{|{{\textbf{U}_p}} |}^2}[{|{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |{{\tilde{D}}_p} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |{{\tilde{D}}_{p + 1}}} ]} ,$$

(22)$${\tilde{D}_p} ={-} \frac{1}{{4k_0^2}}\sum\limits_{j = 1}^{{N_{p,v}}} {\left\{ {\left[ {\frac{{({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot ({{{\vec{A}}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{A}}_p} \cdot {{\vec{A}}_p} - {{({{{\vec{A}}_p} \cdot {{\hat{n}}_p}} )}^2}}}} \right]{f_p}} \right\}} ,$$

(23)$${f_p} = \frac{{\exp ({ - 2{k_0}{\varphi_{p,j + 1}}} )- \exp ({ - 2{k_0}{\varphi_{p,j}}} )}}{{{\varphi _{p,j + 1}} - {\varphi _{p,j}}}},$$

where ${\bar{S}_{ij}}$ represents the complex conjugation of ${S_{ij}}$. This section is a rough introduction to the PGOM and other details can be referred to Sun et al. [31].

2.2 Beam-splitting method

The beam-splitting method completes the top and bottom of the sub-prism segmentation (Fig. 1). The schematic figure for a regular hexagonal prism hit by light perpendicular to its side facet is shown in Fig. 2. For convenience, only one incident facet (the red facet in Fig. 2) is discussed here. The refracted portion ${\hat{e}_1}$ of the incident light ${\hat{e}^i}$ can be divided into two parts (black and green shades; Fig. 2(a)). Beam-splitting technique ensures the two parts of ${\hat{e}_1}$ hit the black and green facets independently (Fig. 2(b)). Segmentation of the incident facet is shown in Fig. 2(c). This process can be divided into three parts: (1) projection of the next facet to the current facet; (2) polygon clipping; and (3) back-projection of the clipped polygon to the next facet. Steps 1 and 3 can be implemented with plane equations.

Fig. 2. (a) Top view of a series of reflections and refractions of light parallel to the bottom of a regular hexagonal prism. Solid arrows indicate incident light and its refracted part inside the prism. Dashed and dash-dot arrows indicate light outside the prism and normal to the facet, respectively. Red shading indicates incident light, and black and green shading are refracted light. Bold solid lines indicate corresponding sides. (b) Front view of (a). Red shading indicates the incident facet; black and green shading indicate facets hit by ${\hat{e}_1}$. (c) View of the incident light (shading as in (b)).

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For step 2, simple mathematical methods can be used, but they are efficient for only those polyhedrons with a small number of facets. The Weiler-Atherton (WA) algorithm is employed in the state-of-the-art PGOM. It accomplishes polygon clipping by determining the state of the intersection points of subject polygon ${S^ \ast }$ and clipping polygon ${C^ \ast }$. There are two states for the intersection points: entry and exit. The WA requires that the vertices of both ${S^ \ast }$ and ${C^ \ast }$ are arranged clockwise or counterclockwise. For convenience, counterclockwise arrangement is used here. For example, the entry point ${I_1}$ of ${S^ \ast }$ in Fig. 3(a) means that from this point, vertices along with ${S^ \ast }$ belong to the intersecting polygon until the exit point (here is ${I_2}$). In WA algorithm, the exit point of ${S^ \ast }$ strictly corresponds to the entry point of ${C^ \ast }$. Consequently, the next point belonging to the intersecting polygon is located in another polygon ${C^ \ast }$. Since ${I_2}$ and ${I_1}$ are entry and exit points for ${C^ \ast }$, respectively, ${I_2}$, $C_1^ \ast $ and ${I_1}$ are the vertices of the intersecting polygon. The polygon clipping ends as ${I_1}$ is the entry point of ${S^ \ast }$, which has already been added to the intersecting polygon, and the vertices of the intersecting polygon are ${I_1} \to {I_2} \to C_1^ \ast $. The consecutive entry and exit points are the basis of the WA algorithm. However, the state of points could be both in enter and exit for the degradation case as shown in Fig. 3(b), where the complete-edge representation (CEP) is necessary to ensure accuracy [46]. The variety of degradation cases makes CEP complex and tedious. Since the current algorithm does not take the CEP into account, erroneous results and even program crashes occur in degradation cases and the accuracy of the PGOM is affected.

Fig. 3. (a) An example of the WA algorithm. (b) A degradation case. $C_1^ \ast $ is on the edge of ${S^ \ast }$.

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A method based on the winding number effectively handles the degradation case. The winding number $wn$ is often used to determine relationships between points and polygons, and is applied to convex, concave, and self-intersecting polygons. An efficient algorithm for winding number is presented in Appendix. For 2D polygon and an arbitrary point P, condition $wn \ne 0$ indicates that P is inside the polygon (excluding the edge) and condition $wn = 0$ means an outside point. If the vertices of polygons are arranged counterclockwise $wn$ > 1 and $wn \le - 1$ apply only to self-intersecting polygons. Examples of the winding number $wn$ are shown in Fig. 4. To simplify the calculation, only non-self-intersecting polygons are considered and the vertices of polygons are arranged counterclockwise, condition $wn ={-} 1$ was defined as the point on the edge or vertex of a polygon. Without considering self-intersecting or concave polygons, there are four relationships between the subject polygon ${S^ \ast }$ and clipping polygon ${C^ \ast }$. For convenience, $w{n_{S_i^ \ast }}$ denotes the position relationship of vertex $S_i^ \ast $ with respect to ${C^ \ast }$. Similarly, $w{n_{C_i^ \ast }}$ denotes the position relationship of vertex $C_i^ \ast $ with respect to ${S^ \ast }$. After finding the intersecting points of the two polygons, condition $w{n_{S_i^ \ast }} = w{n_{C_i^ \ast }} = 0$ indicates that the two polygons are separated (Fig. 5(a)). Conditions $w{n_{S_i^ \ast }} = 0$ and $w{n_{C_i^ \ast }} \ne 0$ imply the intersecting polygon is ${C^ \ast }$ (Fig. 5(b)), and the intersecting polygon is ${S^ \ast }$ with $w{n_{S_i^ \ast }} \ne 0$ and $w{n_{C_i^ \ast }} = 0$ (Fig. 5(c)). $w{n_{S_i^ \ast }}$ or $w{n_{C_i^ \ast }}$ is a combination of zero and non-zero, with the intersecting polygon being the common part of two polygons (Fig. 5(d)).

Fig. 4. Winding number examples. The vertices are arranged counterclockwise.

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Fig. 5. Clip solid polygon with dashed polygon. The gray area indicates the intersecting polygon. $S_i^ \ast $ and $C_i^ \ast $ are vertices of two polygons. $C_3^ \ast $ is on the edge of ${S^ \ast }$ in (d) and ${I_i}$ are intersection points. Solid and the dash-dot arrows are alignment directions for the vertices of polygons ${S^ \ast }$ and ${C^ \ast }$, respectively.

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In the process of obtaining the intersecting polygons, it is necessary to ensure that the vertices of two polygons are arranged in the same order; i.e., they are both clockwise or counterclockwise. The vertices with $w{n_{S_i^ \ast }} \ne 0$ and $w{n_{C_i^ \ast }} \ne 0$ enclose the intersecting polygon, assuming that the vertices are arranged counterclockwise and all intersecting points are calculated. Taking Fig. 5(d) as an example and starting from the first vertex of the subject polygon ${S^ \ast }$, ${I_1}$ is the first vertex with $wn \ne 0$, and $wn$ of the next point ${I_2}$ is also non-zero. Consequently, both ${I_1}$ and ${I_2}$ are vertices of intersecting polygons, and ${I_1}$ is the entry point. Along with ${S^ \ast }$, condition $w{n_{S_4^ \ast }} = 0$ means that the next vertex of the intersecting polygon is on the other polygon ${C^ \ast }$, so it is necessary to jump from polygon ${S^ \ast }$ to polygon ${C^ \ast }$ at ${I_2}$. Along with ${C^ \ast }$, $w{n_{C_3^ \ast }}$, $w{n_{C_1^ \ast }}$ and $w{n_{C_{_{{I_1}}}^ \ast }}$ are all non-zero and belong to the vertices of the intersecting polygon. As ${I_1}$ is a duplicate of the entry point, all vertices of the intersecting polygon have been found. In the example in Fig. 5(d), the vertices of the intersecting polygons are ${I_1} \to {I_2} \to C_3^ \ast \to C_1^ \ast $. The specific $wn$ is shown in Table 2.

Table 2. Specific winding number of Fig. 5(d). $w{n_{s_i^\ast }}$ is the $wn$ of vertex $S_i^ \ast $ with respect to polygon ${C^ \ast }$.

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An example of two intersecting polygons is shown in Fig. 6. The order of the vertices of both ${S^ \ast }$ and ${C^ \ast }$ is counterclockwise. Starting from $S_1^ \ast $, ${I_1}$ and ${I_2}$ satisfy the condition of entering at an intersecting point. As $w{n_{S_3^ \ast }} = 0$, it must jump from ${S^ \ast }$ to ${C^ \ast }$ in ${I_2}$. In polygon ${C^ \ast }$, the $wn$ of $C_2^ \ast $ and ${I_1}$ are 1 and −1, respectively, and ${I_1}$ is duplicated. Hence, the vertices of the first intersecting polygon have all been found, namely ${I_1} \to {I_2} \to C_2^ \ast $. Similarly, another intersecting polygon is ${I_3} \to {I_4} \to C_1^ \ast $. The details of $wn$ are given in Table 3.

Fig. 6. Example of two intersecting polygons. The meanings of ${I_i}$ and the arrows are as in Fig. 5(d).

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Table 3. Specific winding numbers for Fig. 6.

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2.3 Linearization of single-scattering properties

The amplitude scattering matrix S is the key to solving for the linearization of the SSPs. With $\zeta $ representing $\sigma \textrm{ and }{r_e}$, and m represent ${m_r}\textrm{ and }{m_i}$, and using Eq. (16), the linearization of S with respect to x can be written as:

(24)$$\frac{{\partial S}}{{\partial x}} = \left\{ {\begin{array}{ll} {({1 - {m^2}} )\left( {\sum\limits_{p = 0}^\infty {\frac{{\partial {T_p}}}{{\partial x}}} } \right)\Gamma }&{x = \zeta }\\ {\left[ { - 2m\left( {\sum\limits_{p = 0}^\infty {{T_p}} } \right) + ({1 - {m^2}} )\left( {\sum\limits_{p = 0}^\infty {\frac{{\partial {T_p}}}{{\partial x}}} } \right)} \right]\Gamma }&{x = m} \end{array}} \right.,\textrm{ }$$

where

(25)$$\frac{{\partial {T_p}}}{{\partial x}} = \left\{ {\begin{array}{ll} {\left( {|{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |\frac{{\partial {D_{p + 1}}}}{{\partial x}} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |\frac{{\partial {D_p}}}{{\partial x}}} \right)\frac{{{\textbf{K}_p}{\textbf{U}_p}}}{{{{\vec{\kappa }}_p} \cdot {{\hat{e}}_p}}}}&{x = \zeta }\\ \begin{array}{l} \left( \begin{array}{l} \frac{{\partial |{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |}}{{\partial x}}{D_{p + 1}} + |{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |\frac{{\partial {D_{p + 1}}}}{{\partial x}}\\ - \frac{{\partial |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |}}{{\partial x}}{D_p} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |\frac{{\partial {D_p}}}{{\partial x}} \end{array} \right)\frac{{{\textbf{K}_p}{\textbf{U}_p}}}{{{{\vec{\kappa }}_p} \cdot {{\hat{e}}_p}}} + \\ \frac{{({|{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |{D_{p + 1}} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |{D_p}} )}}{{{{\vec{\kappa }}_p} \cdot {{\hat{e}}_p}}}\frac{{\partial ({{\textbf{K}_p}{\textbf{U}_p}} )}}{{\partial x}} - \\ ({|{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |{D_{p + 1}} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |{D_p}} )\frac{{{\textbf{K}_p}{\textbf{U}_p}}}{{{{({{{\vec{\kappa }}_p} \cdot {{\hat{e}}_p}} )}^2}}}\left( {\frac{{\partial {{\vec{\kappa }}_p}}}{{\partial x}} \cdot {{\hat{e}}_p} + {{\vec{\kappa }}_p} \cdot \frac{{\partial {{\hat{e}}_p}}}{{\partial x}}} \right) \end{array}&{x = m} \end{array}} \right.,\textrm{ }$$

(26)$$\frac{{\partial {D_p}}}{{\partial \zeta }} = \frac{{{k_0}}}{{4\pi }}\sum\limits_{j = 1}^{{N_{p,v}}} {\left[ {\frac{{\frac{{\partial ({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )}}{{\partial \zeta }} \cdot ({i{{\vec{\kappa }}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{\kappa }}_p} \cdot {{\vec{\kappa }}_p} - {{({{{\vec{\kappa }}_p} \cdot {{\hat{n}}_p}} )}^2}}}{F_p} + \frac{{({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot ({i{{\vec{\kappa }}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{\kappa }}_p} \cdot {{\vec{\kappa }}_p} - {{({{{\vec{\kappa }}_p} \cdot {{\hat{n}}_p}} )}^2}}}\frac{{\partial {F_p}}}{{\partial \zeta }}} \right]} ,\textrm{ }$$

(27)$$\frac{{\partial {D_p}}}{{\partial m}} = \frac{{{k_0}}}{{4\pi }}\sum\limits_{j = 1}^{{N_{p,v}}} {\left[ \begin{array}{l} \frac{{\frac{{\partial ({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )}}{{\partial m}} \cdot ({i{{\vec{\kappa }}_p} \times {{\hat{n}}_p}} )+ ({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot \frac{{\partial ({i{{\vec{\kappa }}_p} \times {{\hat{n}}_p}} )}}{{\partial m}}}}{{{{\vec{\kappa }}_p} \cdot {{\vec{\kappa }}_p} - {{({{{\vec{\kappa }}_p} \cdot {{\hat{n}}_p}} )}^2}}}{F_p}\\ - \frac{{2({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot ({i{{\vec{\kappa }}_p} \times {{\hat{n}}_p}} )}}{{{{({{{\vec{\kappa }}_p} \cdot {{\vec{\kappa }}_p} - {{({{{\vec{\kappa }}_p} \cdot {{\hat{n}}_p}} )}^2}} )}^2}}}\left[ {{{\vec{\kappa }}_p} \cdot \frac{{\partial {{\vec{\kappa }}_p}}}{{\partial m}} - ({{{\vec{\kappa }}_p} \cdot {{\hat{n}}_p}} )\left( {\frac{{\partial {{\vec{\kappa }}_p}}}{{\partial m}} \cdot {{\hat{n}}_p}} \right)} \right]{F_p}\\ + \frac{{({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot ({i{{\vec{\kappa }}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{\kappa }}_p} \cdot {{\vec{\kappa }}_p} - {{({{{\vec{\kappa }}_p} \cdot {{\hat{n}}_p}} )}^2}}}\frac{{\partial {F_p}}}{{\partial sp}} \end{array} \right]} ,\textrm{ }$$

(28)$$\begin{aligned} \frac{{\partial {F_p}}}{{\partial x}} &= \left[ {\exp ({i{k_0}{\Phi _{p,j + 1}}} )\frac{{\partial {\Phi _{p,j + 1}}}}{{\partial x}} - \exp ({i{k_0}{\Phi _{p,j}}} )\frac{{\partial {\Phi _{p,j}}}}{{\partial x}}} \right]\frac{{i{k_0}}}{{{\Phi _{p,j + 1}} - {\Phi _{p,j}}}}\\ &- \frac{{\exp ({i{k_0}{\Phi _{p,j + 1}}} )- \exp ({i{k_0}{\Phi _{p,j}}} )}}{{{{({{\Phi _{p,j + 1}} - {\Phi _{p,j}}} )}^2}}}\frac{{\partial ({{\Phi _{p,j + 1}} - {\Phi _{p,j}}} )}}{{\partial x}}. \end{aligned}$$

The linearization associated with ${F_p}$ can be treated in two parts. With Eq. (13), the linearization of ${\Phi _{p,j + 1}} - {\Phi _{p,j}}$ is

(29)$$\frac{{\partial ({{\Phi _{p,j + 1}} - {\Phi _{p,j}}} )}}{{\partial x}} = \left\{ {\begin{array}{ll} {\frac{{{{\vec{\kappa }}_p}}}{{{k_0}}} \cdot \left( {\frac{{\partial {{\vec{r}}_{p,j + 1}}}}{{\partial x}} - \frac{{\partial {{\vec{r}}_{p,j}}}}{{\partial x}}} \right)}&{x = \zeta }\\ {\frac{{\partial {{\vec{\kappa }}_p}}}{{\partial x}} \cdot \frac{{({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )}}{{{k_0}}} + \frac{{{{\vec{\kappa }}_p}}}{{{k_0}}} \cdot \left( {\frac{{\partial {{\vec{r}}_{p,j + 1}}}}{{\partial x}} - \frac{{\partial {{\vec{r}}_{p,j}}}}{{\partial x}}} \right)}&{x = m} \end{array}} \right.,\textrm{ }$$

where

(30)$$\frac{{\partial {{\vec{\kappa }}_p}}}{{\partial x}} = \left\{ {\begin{array}{ll} \textbf{0}&{x = \zeta }\\ {{k_0}\left( {\frac{{\partial {N_{p,r}}}}{{\partial x}}{{\hat{e}}_p} + {N_{p,r}}\frac{{\partial {{\hat{e}}_p}}}{{\partial x}} + i\frac{{\partial {{\vec{A}}_p}}}{{\partial x}}} \right)}&{x = m} \end{array}} \right.,\textrm{ }$$

(31)$$\frac{{\partial {{\vec{A}}_p}}}{{\partial x}}\textrm{ = }\left\{ {\begin{array}{ll} \textbf{0}&{p = 0\textrm{ or }x = \zeta }\\ \begin{array}{l} \frac{{\partial {{\vec{A}}_{p - 1}}}}{{\partial x}} + \left[ {\frac{{\partial {N_{p,i}}}}{{\partial x}} - \frac{\partial }{{\partial x}}({{{\hat{e}}_p} \cdot {{\vec{A}}_{p - 1}}} )} \right]\frac{{{{\hat{n}}_{p - 1}}}}{{{{\hat{e}}_p} \cdot {{\hat{n}}_{p - 1}}}} - \\ ({{N_{p,i}} - {{\hat{e}}_p} \cdot {{\vec{A}}_{p - 1}}} )\frac{{{{\hat{n}}_{p - 1}}}}{{{{({{{\hat{e}}_p} \cdot {{\hat{n}}_{p - 1}}} )}^2}}}\left( {\frac{{\partial {{\hat{e}}_p}}}{{\partial x}} \cdot {{\hat{n}}_{p - 1}}} \right) \end{array}&{p \ne 0\textrm{ and }x = m} \end{array}} \right..$$

With Eq. (14), $\frac{{\partial {\Phi _{p,j}}}}{{\partial x}}$ can be written as

(32)$$\frac{{\partial {\Phi _{p,j}}}}{{\partial x}} = \frac{{\partial {\varphi _{p,j}}}}{{\partial x}} + i\frac{{\partial {\psi _{p,j}}}}{{\partial x}} - \hat{r} \cdot \frac{{\partial {{\vec{r}}_{p,j}}}}{{\partial x}},\textrm{ }$$

where

(33)$$\frac{{\partial {\varphi _{p,j}}}}{{\partial x}} = \left\{ {\begin{array}{ll} {{{\hat{e}}^i} \cdot \frac{{\partial {{\vec{r}}_{p,j}}}}{{\partial x}}}&{p = 0,\textrm{ }x = \zeta \textrm{ or }m}\\ {\frac{{\partial {\varphi_{p - 1,j}}}}{{\partial x}} + {N_{p,r}}\frac{{\partial |{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |}}{{\partial x}}}&{p \ne 0,\textrm{ }x = \zeta }\\ {\frac{{\partial {\varphi_{p - 1,j}}}}{{\partial x}} + \frac{{\partial {N_{p,r}}}}{{\partial x}}|{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |+ {N_{p,r}}\frac{{\partial |{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |}}{{\partial x}}}&{p \ne 0,\textrm{ }x = m} \end{array}} \right.,\textrm{ }$$

(34)$$\frac{{\partial {\psi _{p,j}}}}{{\partial x}} = \left\{ {\begin{array}{ll} 0&{p = 0,\textrm{ }x = \zeta \textrm{ or }m}\\ {\frac{{\partial {\psi_{p - 1,j}}}}{{\partial x}} + {N_{p,i}}\frac{{\partial |{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |}}{{\partial x}}}&{p \ne 0,\textrm{ }x = \zeta }\\ {\frac{{\partial {\psi_{p - 1,j}}}}{{\partial x}} + \frac{{\partial {N_{p,i}}}}{{\partial x}}|{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |+ {N_{p,i}}\frac{{\partial |{{{\vec{r}}_{p,j}} - {{\vec{r}}_{p - 1,j}}} |}}{{\partial x}}}&{p \ne 0,\textrm{ }x = m} \end{array}} \right..$$

The linearization of some SSPs can be written as:

(35)$$ \frac{\partial}{\partial x}\left(\begin{array}{ll} M_{11} & M_{12} \\ M_{13} & M_{22} \end{array}\right)=\frac{1}{2}\left(\begin{array}{cc} \frac{\partial}{\partial x}\left(\left|S_{11}\right|^{2}+\left|S_{12}\right|^{2}+\left|S_{21}\right|^{2}+\left|S_{22}\right|^{2}\right) & \frac{\partial}{\partial x}\left(\left|S_{11}\right|^{2}-\left|S_{12}\right|^{2}+\left|S_{21}\right|^{2}-\left|S_{22}\right|^{2}\right) \\ \frac{\partial}{\partial x}\left(\left|S_{11}\right|^{2}+\left|S_{12}\right|^{2}-\left|S_{21}\right|^{2}-\left|S_{22}\right|^{2}\right) & \frac{\partial}{\partial x}\left(\left|S_{11}\right|^{2}-\left|S_{12}\right|^{2}-\left|S_{21}\right|^{2}+\left|S_{22}\right|^{2}\right) \end{array}\right) $$

(36)$$\frac{\partial }{{\partial x}}\left( {\begin{array}{cc} {{M_{33}}}&{{M_{34}}}\\ {{M_{43}}}&{{M_{44}}} \end{array}} \right) = \left( {\begin{array}{rr} {\frac{\partial }{{\partial x}}{\textrm{Re}} ({{S_{11}}{{\bar{S}}_{22}} + {S_{12}}{{\bar{S}}_{21}}} )}&{\frac{\partial }{{\partial x}}{\mathop{\textrm{Im}}\nolimits} ({{S_{11}}{{\bar{S}}_{22}} - {S_{12}}{{\bar{S}}_{21}}} )}\\ { - \frac{\partial }{{\partial x}}{\mathop{\textrm{Im}}\nolimits} ({{S_{11}}{{\bar{S}}_{22}} - {S_{12}}{{\bar{S}}_{21}}} )}&{\frac{\partial }{{\partial x}}{\textrm{Re}} ({{S_{11}}{{\bar{S}}_{22}} - {S_{12}}{{\bar{S}}_{21}}} )} \end{array}} \right),\textrm{ }$$

(37)$$\frac{{\partial {C_{ext}}}}{{\partial x}} = \frac{{2\pi }}{{k_0^2}}\textrm{Re} \left[ {\frac{\partial }{{\partial x}}{S_{11}}({{{\hat{e}}^i}} )+ \frac{\partial }{{\partial x}}{S_{22}}({{{\hat{e}}^i}} )} \right],\textrm{ }$$

(38)$$\frac{{\partial {C_{abs}}}}{{\partial x}} = \left\{ {\begin{array}{ll} {\frac{1}{2}\sum\limits_{p = 0}^\infty {{N_{p,r}}{{|{{\textbf{U}_p}} |}^2}\left[ {|{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |\frac{{\partial {{\tilde{D}}_p}}}{{\partial x}} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |\frac{{\partial {{\tilde{D}}_{p + 1}}}}{{\partial x}}} \right]} }&{x = \zeta }\\ {\frac{1}{2}\sum\limits_{p = 0}^\infty {\left\{ \begin{array}{l} \left( {\frac{{\partial {N_{p,r}}}}{{\partial x}}{{|{{\textbf{U}_p}} |}^2} + 2{N_{p,r}}{\textbf{U}_p}\frac{{\partial {\textbf{U}_p}}}{{\partial x}}} \right)[{|{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |{{\tilde{D}}_p} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |{{\tilde{D}}_{p + 1}}} ]\\ + {N_{p,r}}{|{{\textbf{U}_p}} |^2}\left[ \begin{array}{l} \frac{{\partial |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |}}{{\partial x}}{{\tilde{D}}_p} + |{{{\hat{e}}_p} \cdot {{\hat{n}}_p}} |\frac{{\partial {{\tilde{D}}_p}}}{{\partial x}}\\ - \frac{{\partial |{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |}}{{\partial x}}{{\tilde{D}}_{p + 1}} - |{{{\hat{e}}_p} \cdot {{\hat{n}}_{p + 1}}} |\frac{{\partial {{\tilde{D}}_{p + 1}}}}{{\partial x}} \end{array} \right] \end{array} \right\}} }&{x = m} \end{array}} \right.,\textrm{ }$$

where

(39)$$\frac{{\partial {{\tilde{D}}_p}}}{{\partial \zeta }} ={-} \frac{1}{{4k_0^2}}\sum\limits_{j = 1}^{{N_{p,v}}} {\left\{ {\left[ {\frac{{\frac{{\partial ({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )}}{{\partial \zeta }} \cdot ({{{\vec{A}}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{A}}_p} \cdot {{\vec{A}}_p} - {{({{{\vec{A}}_p} \cdot {{\hat{n}}_p}} )}^2}}}} \right]{f_p} + \left[ {\frac{{({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot ({{{\vec{A}}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{A}}_p} \cdot {{\vec{A}}_p} - {{({{{\vec{A}}_p} \cdot {{\hat{n}}_p}} )}^2}}}} \right]\frac{{\partial {f_p}}}{{\partial \zeta }}} \right\}} ,\textrm{ }$$

(40)$$\frac{{\partial {{\tilde{D}}_p}}}{{\partial m}} ={-} \frac{1}{{4k_0^2}}\sum\limits_{j = 1}^{{N_{p,v}}} {\left\{ \begin{array}{l} \left[ {\frac{{\frac{{\partial ({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )}}{{\partial m}} \cdot ({{{\vec{A}}_p} \times {{\hat{n}}_p}} )+ ({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot \left( {\frac{{\partial {{\vec{A}}_p}}}{{\partial m}} \times {{\hat{n}}_p}} \right)}}{{{{\vec{A}}_p} \cdot {{\vec{A}}_p} - {{({{{\vec{A}}_p} \cdot {{\hat{n}}_p}} )}^2}}}} \right]{f_p} + \\ \left[ {\frac{{({{{\vec{r}}_{p,j + 1}} - {{\vec{r}}_{p,j}}} )\cdot ({{{\vec{A}}_p} \times {{\hat{n}}_p}} )}}{{{{\vec{A}}_p} \cdot {{\vec{A}}_p} - {{({{{\vec{A}}_p} \cdot {{\hat{n}}_p}} )}^2}}}} \right]\frac{{\partial {f_p}}}{{\partial m}} \end{array} \right\}} ,\textrm{ }$$

(41)$$\begin{aligned} \frac{{\partial {f_p}}}{{\partial x}} =&{-} \left[ {\exp ({ - 2{k_0}{\varphi_{p,j + 1}}} )\frac{{\partial {\varphi_{p,j + 1}}}}{{\partial x}} - \exp ({ - 2{k_0}{\varphi_{p,j}}} )\frac{{\partial {\varphi_{p,j}}}}{{\partial x}}} \right]\frac{{2{k_0}}}{{{\varphi _{p,j + 1}} - {\varphi _{p,j}}}}\\ &- \frac{{\exp ({ - 2{k_0}{\varphi_{p,j + 1}}} )- \exp ({ - 2{k_0}{\varphi_{p,j}}} )}}{{{{({{\varphi_{p,j + 1}} - {\varphi_{p,j}}} )}^2}}}\left( {\frac{{\partial {\varphi_{p,j + 1}}}}{{\partial x}} - \frac{{\partial {\varphi_{p,j}}}}{{\partial x}}} \right). \end{aligned}$$

Using Eqs. (24)–(41), the linearization of the SSPs based on the PGOM is analytically obtained. Correspondingly, the SSPs and its linearization can be calculated together and the PGOM is further extended to the linearized PGOM.

3. Results and discussion

The regular hexagonal prism is used as an example to verify the linearization and discuss the sensitivity of the SSPs. The linearization of SSPs is verified by using the finite-difference method. The SSPs such as the extinction, absorption, and scattering cross-sections, single scattering albedo, and Mueller matrix, and their corresponding linearization are discussed. For convenience, the incident wavelength is assumed as $2\pi $ µm and the unit for all radii is µm. All SSPs are computed in the condition of random orientations. The linearization of SSPs $\partial (\textrm{SSP)}/\partial x$ is marked as ${\partial _x}(\textrm{SSP})$ in the figures for simplification.

3.1 Comparison with the finite-difference method

For the finite-difference method, the partial derivatives of n scattering parameters can be asymptotically obtained by calculating (n + 1) times of the PGOM. However, for the geometric optics method, a tiny change of scattering parameters can cause significant changes of ray paths. Correspondingly, the difference of scattering parameters has to be extremely small to ensure that all ray paths are invariant. In the other side, the computer's round-off error might overwhelm the changes of scattering parameters. To obtain accurate results, more than n + 1 times are required in practice. Several attempts are required for the choice of suitable perturbations. In this study, the absolute increment ${10^{ - 8}}$ is used in the finite-difference method to verify the linearization results. For one orientation and four scattering parameters, the linearization method takes roughly 10 seconds while the finite-difference method totally takes around 12.5 seconds, where all simulations are conducted on a cluster with Intel Xeon Platinum 8260 CPU @2.4GHZ.

Figure 7–Fig. 10 show the comparisons between the linearization and the finite-difference results of the normalized Mueller matrix P with respect to scattering parameters $\sigma ,{r_e},{m_r},\textrm{ and }{m_i}$, respectively, where the scattering parameters are all $\sigma = 1.0$, ${r_e} = 50.0$, ${m_r} = 1.1$, and ${m_i} = 0.0001$. The solid black and dashed red lines represent the linearization and finite-difference results, respectively. The normalized Mueller matrix P is defined relative to the Mueller matrix M as follows:

(42)$$\left\{ \begin{array}{l} {P_{11}} = 2{M_{11}}/\int_{ - 1}^1 {{M_{11}}d\mu } ,\\ {P_{ij}} = {M_{ij}}/{M_{11}},\;ij \ne 11. \end{array} \right.$$

There are six non-zero elements for a regular prism in random orientations. The differences between the two results are negligible and not shown here for simplification.

Fig. 7. Finite-difference and linearization distributions of the normalized Mueller matrix P with respect to the aspect ratio $\sigma $. $\sigma $, ${r_e}$, ${m_r}$ and ${m_i}$ are 1.0, 50.0, 1.1, and 0.0001, respectively.

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Fig. 8. Finite-difference and linearization distributions of the normalized Mueller matrix P with respect to the effective radius ${r_e}$. Scattering parameters are the same as those in Fig. 7.

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Fig. 9. Finite-difference and linearization distributions of the normalized Mueller matrix P with respect to the real part of refractive index ${m_r}$. Scattering parameters are the same as those in Fig. 7.

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Fig. 10. Finite-difference and linearization distributions of the normalized Mueller matrix P with respect to the imaginary part of refractive index ${m_i}$. Scattering parameters are the same as those in Fig. 7.

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Figure 11 shows the application of the linearization and finite-difference methods to the scattering cross-section ${C_{sca}}$ and absorption cross-section ${C_{abs}}$, where the solid black and dashed red lines represent the linearization and finite-difference results, respectively. The difference between the results of the linearization algorithm and the finite-difference method is described by the relative difference, which is defined as

(43)$$\textrm{Relative Difference} = \left|{\frac{{\textrm{F(finite difference)} - \textrm{L(linearization)}}}{{\textrm{L(linearization)}}}} \right|\times 100.$$

Fig. 11. Linearization and finite-difference results of the absorption and scattering cross-sections with respect to four scattering parameters. Other scattering parameters are the same as those in Fig. 7 except the linearization parameter.

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The panels (a1)-(h1) show the comparisons of two results and the panels (a2)-(h2) are the corresponding relative differences. These lines fit extremely well, where the relative differences are much less than 1%.

Thus, the linearization algorithm of the SSPs is verified using the finite-difference method for both the differential scattering properties – the non-zero elements of the normalized scattering matrix P and the integrated scattering properties – the scattering and absorption cross-sections in Fig. 7–Fig. 11.

3.2 Sensitivity analysis

Figure 12 shows the distribution of the absorption and scattering cross-sections and their linearization results as functions of the scattering parameters. The aspect ratio $\sigma $ and equal volume radius ${r_e}$ represent the geometric parameters of a scattering particle while the real part ${m_r}$ and the imaginary part ${m_i}$ of the refractive index represent the optical parameters of the scattering particle. The particle absorption is determined by the absorption ability and path, which are related to the imaginary part of the refractive index and the particle size. Panels (a) and (b) show the absorption cross-sections ${C_{abs}}$ are sensitive to ${m_i}$ and ${r_e}$, and the increase rate can be quantitatively obtained from the corresponding linearization. The scattering cross-sections ${C_{sca}}$ are sensitive to ${m_r}$ as shown in panel (c), which fluctuates with the increasing ${m_r}$ because of the interference between diffracted and reflected-refracted rays. ${C_{sca}}$ increases with ${r_e}$ as shown in panel (d) because the geometric scattering cross-section is increased with ${r_e}$. The variations of $\sigma $ and ${r_e}$ as well cause the interference between diffracted and reflected-refracted parts. Correspondingly, holes appear in panel (d) due to large interference fluctuations.

Fig. 12. Distributions of absorption and scattering cross-sections and their linearization results as functions of scattering parameters. For (a) and (c), $\sigma \textrm{ } = \textrm{ }1.0$, ${r_e}\textrm{ } = \textrm{ }50.0$. For (b) and (d), ${m_r}\textrm{ } = \textrm{ }1.33$, ${m_i}\textrm{ } = \textrm{ }0.01$.

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The angular distributions of ${P_{11}}$ and its linearization as a function of scattering parameters are shown in Fig. 13. The backward scattering of ${P_{11}}$, especially ${P_{11}}({{{180}^\circ }} )$, is strongly oscillated due to interference of refracted rays, which is influenced by geometric parameters and the real part of the refractive index. The backward scattering of ${P_{11}}$ is reduced due to the absorption enhancement with the increase of the imaginary part of the refractive index. The forward scattering of ${P_{11}}$ is mainly from the diffraction contribution and the diffraction is proportional to the geometric cross-section of a scattering particle. Correspondingly, the forward scattering is sensitive to ${r_e}$ as shown in panel (b). The sensitivities and changing rate can be straightforwardly captured form the linearization results.

Fig. 13. ${P_{11}}$ and its linearization as a function of scattering angle and scattering parameters. ${r_e}$ is 50.0 for (a), (c) and (d). $\sigma $ is 1.0 for (c) and (d), 0.5 for (b). ${m_r}$ is 1.33 for (a) and (b), 1.5 for (d). ${m_i}$ is 0.001 for (a) and (b), 0.01 for (c).

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Figure 14 shows the extinction cross-sections ${C_{ext}}$, the single scattering albedo $SSA$, the exactly backward scattering ${P_{11}}({{{180}^\circ }} )$, and their linearization. With increasing ${r_e}$, ${C_{ext}}$ exhibits a monotonically increase but an oscillated increase rate, and $SSA$ displays a decrease tendency due to the increase of the absorption path, while ${P_{11}}({{{180}^\circ }} )$ oscillates because of the ray interference. With increasing ${m_r}$, the ${C_{ext}}$ displays an oscillation due to the interference between diffraction and refracted-reflected rays, $SSA$ shows similar features as increasing ${r_e}$, and ${P_{11}}({{{180}^\circ }} )$ shows an increase tendency because of scattering enhancement. With increasing ${m_i}$, ${C_{ext}}$, $SSA$, and ${P_{11}}({{{180}^\circ }} )$ all display significant decreasing trends due to the absorption enhancement.

Fig. 14. ${C_{ext}}$, $SSA$, ${P_{11}}({{{180}^\circ }} )$ and their linearization with respect to scattering parameters. The scattering parameters of (a), (b) and (c) are the same as those in Fig. 13(b), (c) and (d), respectively. Solid black lines represent the variable itself corresponding to the left y-axis; red dashed lines represent the linearized value with respect to the parameter in the horizontal axis corresponding to the right y-axis.

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4. Conclusion

The linearization of the single scattering properties is systematically derived based on the physical-geometric optics method (PGOM) and analytically obtained. Moreover, the winding number method is introduced to simplify and improve the accuracy of the beam-splitting process. The PGOM is generalized to the linearized physical-geometric optics method (LPGOM). Regular hexagonal prisms are taken as an example to study the linearization algorithm. The linearization algorithm is verified to the finite-difference method, where the relative differences of two simulations are much less than 1%. The sensitivities of extinction, absorption, scattering cross-sections, single scattering albedo, and scattering phase function with respect to aspect ratio, equal volume radius, and complex refractive index are discussed based on the LPGOM. For future work, the LPGOM can be used to generate a linearized single-scattering-property database for the potential applications in remote sensing.

Appendix: determination of winding number

If a 2D polygon is given, the winding number of any point inside the 2D plane is uniquely determined. For convenience, only the counterclockwise arrangement is discussed. Suppose the polygon consists of vertices ${V_j}({j = 1,n} )$ and ${V_1} \ne {V_n}$. An arbitrary point P can form vectors ${\vec{v}_i}({i = 1,n} )$ with the vertices of the given polygon. If the angle between vectors ${\vec{v}_i}$ can be defined as

(44)$${\theta _i} = \left\{ {\begin{array}{ll} {\frac{{{{\vec{v}}_i} \times {{\vec{v}}_{i + 1}}}}{{|{{{\vec{v}}_i}} ||{{{\vec{v}}_{i + 1}}} |}}\arccos \left( {\frac{{{{\vec{v}}_i} \cdot {{\vec{v}}_{i + 1}}}}{{|{{{\vec{v}}_i}} ||{{{\vec{v}}_{i + 1}}} |}}} \right)}&{i < n}\\ {\frac{{{{\vec{v}}_i} \times {{\vec{v}}_1}}}{{|{{{\vec{v}}_i}} ||{{{\vec{v}}_1}} |}}\arccos \left( {\frac{{{{\vec{v}}_i} \cdot {{\vec{v}}_1}}}{{|{{{\vec{v}}_i}} ||{{{\vec{v}}_1}} |}}} \right)}&{i = n} \end{array}} \right.,$$

the winding number can be expressed as

(45)$$wn = \frac{1}{{2\pi }}\sum\limits_{i = 1}^n {{\theta _i}} .$$

In Fig. 15(a1), the point P is inside the polygon so the sum of ${\theta _i}$ is $2\pi $, i.e., wn = 1. In Fig. 15(b1), the sign of ${\theta _4}$ is negative so the sum of ${\theta _i}$ is 0, i.e., wn = 0. However, the function arccos is time-consuming. Another more efficient method is preferred. Suppose the polygon consists of directed edges and the length of each edge is defined as

(46)$$|{{{\vec{l}}_i}} |= \left\{ {\begin{array}{ll} {|{{{\vec{v}}_{i + 1}} - {{\vec{v}}_i}} |}&{i < n}\\ {|{{{\vec{v}}_1} - {{\vec{v}}_i}} |}&{i = n} \end{array}} \right.$$

For convenience, suppose there is a ray L extending to infinity with point P as its endpoint. The relation of L and ${\vec{l}_i}$ can be measured by ${\rho _i}$ [47]:

(47)$$ \rho_{i}=\left\{\begin{aligned} 1 & \vec{l}_{i} \text { upward cross } L \text { except upper endpoint } \\ -1 & \vec{l}_{i} \text { downward cross } L \text { except upper endpoint }, \\ 0 & \text { else } \end{aligned}\right. $$

where the upper endpoint refers to the vertex with maximum of the second coordinate. Consequently, the winding number can be rewritten as

(48)$$wn = \sum\limits_{i = 1}^n {{\rho _i}} .$$

Fig. 15. Two methods for the determination of the winding number. For (a1) and (b1), dashed arrows mean the vector consisting of point P and the vertex ${V_i}$ of the polygon. For (a2) and (b2), the red and blue arrows mean the upward and downward directed edges, respectively.

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In Fig. 15(a2), only ${\vec{l}_2}$ upward crosses L so wn = 1. In Fig. 15(b2), ${\vec{l}_1}$ and ${\vec{l}_4}$ upward and downward cross L, respectively, so wn = 0. The results are the same as those using the calculated angle. It should be noted that wn = 0 means that the point is outside the polygon, and wn ≠ 0 means that the point is inside the polygon. For convex particles, the polygons involved in the beam-splitting process do not contain self-intersecting polygons. If the polygon vertices are arranged counterclockwise, wn = 1 when the point is inside the polygon, and it is reasonable to set wn = −1 to represent the point on the edge of the polygon.

Funding

National Key Research and Development Program of China (2021YFB3900401); National Natural Science Foundation of China (41975021); Natural Science Foundation of Shanghai (19ZR1404100); Shanghai Science and Technology Development Foundation (20dz1200700).

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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$s$	Superscript denoting scattering variables
$i$	Superscript denoting incident variables
$p$	Subscript denoting reflection–refraction order
$r$	Distance from the origin
$\vec{r}$	Position vector of a facet vertex
$\hat{r}$	Observed scattering direction
${\hat{n}}_{p}$	Unit vector normal to the plane outward
${\hat{e}}_{p}$	Propagation direction of a ray
$\hat{α}, \hat{β}$	Unit vector parallel and perpendicular to the plane of a ray
$k_{0}$	Wavenumber
${\vec{κ}}_{p}$	Complex wave vector difference
$m$	Complex refractive index
$m_{r}, m_{i}$	Real and imaginary refractive index
$N_{p, r}, N_{p, i}$	Real and imaginary effective complex refractive index
$K_{p}$	Transformation matrix to the reference frame
$U_{p}$	Matrix associated with Fresnel coefficients and corresponding rotations
$ϕ_{p}$	Complex phase
$φ_{p}, ψ_{p}$	Real and imaginary phase
$Φ_{p}$	Complex phase difference
${\vec{A}}_{p}$	The p-th order variation of the amplitude
$σ, r_{e}$	Aspect ratio and equal volume radius
$j$	Enumeration vertex number
$N_{p, v}$	Number of geometric cross-section vertices
$C_{e x t}$	Extinction cross-section
$C_{a b s}$	Absorption cross-section
$C_{s c a}$	Scattering cross-section
$S$	Amplified scattering matrix
$P$	Normalized Mueller matrix
$w n$	Winding number

$S^{*}$	$S_{1}^{*}$	$S_{2}^{*}$	$S_{3}^{*}$	$I_{1}$	$I_{2}$	$S_{4}^{*}$	$C_{3}^{*}$
$w n_{S_{i}^{*}}$	0	0	0	−1	−1	0	−1
$C^{*}$	$C_{1}^{*}$	$I_{1}$	$C_{2}^{*}$	$I_{2}$	$C_{3}^{*}$
$w n_{C_{i}^{*}}$	1	−1	0	−1	−1

$S^{*}$	$S_{1}^{*}$	$S_{2}^{*}$	$I_{1}$	$I_{2}$	$S_{3}^{*}$	$I_{3}$	$I_{4}$	$S_{4}^{*}$	$S_{5}^{*}$
$w n_{S_{i}^{*}}$	0	0	−1	−1	0	0	−1	0	0
$C^{*}$	$C_{1}^{*}$	$I_{3}$	$I_{2}$	$C_{2}^{*}$	$I_{1}$	$C_{3}^{*}$	$I_{4}$
$w n_{C_{i}^{*}}$	1	−1	−1	1	−1	0	−1

$s$	Superscript denoting scattering variables
$i$	Superscript denoting incident variables
$p$	Subscript denoting reflection–refraction order
$r$	Distance from the origin
$\vec{r}$	Position vector of a facet vertex
$\hat{r}$	Observed scattering direction
${\hat{n}}_{p}$	Unit vector normal to the plane outward
${\hat{e}}_{p}$	Propagation direction of a ray
$\hat{α}, \hat{β}$	Unit vector parallel and perpendicular to the plane of a ray
$k_{0}$	Wavenumber
${\vec{κ}}_{p}$	Complex wave vector difference
$m$	Complex refractive index
$m_{r}, m_{i}$	Real and imaginary refractive index
$N_{p, r}, N_{p, i}$	Real and imaginary effective complex refractive index
$K_{p}$	Transformation matrix to the reference frame
$U_{p}$	Matrix associated with Fresnel coefficients and corresponding rotations
$ϕ_{p}$	Complex phase
$φ_{p}, ψ_{p}$	Real and imaginary phase
$Φ_{p}$	Complex phase difference
${\vec{A}}_{p}$	The p-th order variation of the amplitude
$σ, r_{e}$	Aspect ratio and equal volume radius
$j$	Enumeration vertex number
$N_{p, v}$	Number of geometric cross-section vertices
$C_{e x t}$	Extinction cross-section
$C_{a b s}$	Absorption cross-section
$C_{s c a}$	Scattering cross-section
$S$	Amplified scattering matrix
$P$	Normalized Mueller matrix
$w n$	Winding number

$S^{*}$	$S_{1}^{*}$	$S_{2}^{*}$	$S_{3}^{*}$	$I_{1}$	$I_{2}$	$S_{4}^{*}$	$C_{3}^{*}$
$w n_{S_{i}^{*}}$	0	0	0	−1	−1	0	−1
$C^{*}$	$C_{1}^{*}$	$I_{1}$	$C_{2}^{*}$	$I_{2}$	$C_{3}^{*}$
$w n_{C_{i}^{*}}$	1	−1	0	−1	−1

Linearization of light scattering properties based on the physical-geometric optics method

Abstract

1. Introduction

2. Method

2.1 Single-scattering properties

2.2 Beam-splitting method

2.3 Linearization of single-scattering properties

3. Results and discussion

3.1 Comparison with the finite-difference method

3.2 Sensitivity analysis

4. Conclusion

Appendix: determination of winding number

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (15)

Tables (3)

Equations (48)

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