-
1973年,由Purcell和Pennypacke[9]提出了离散偶极子近似(Discrete Dipole Approximation,DDA)方法,后经发展成为了一种能对任意几何形状、非均匀和各向异性粒子光散射进行计算的方法,从而被广泛应用于大气气溶胶、水滴、冰晶等粒子的光学散射特征的研究上。相较于Mie理论应用于多层球的散射计算,DDA法更精确地用于复杂核壳结构粒子的散射计算,特别是针对非球形核壳结构粒子的散射有一定的计算优势。离散偶极子近似的基本思想就是将目标散射体用有限个离散且相互作用的偶极子阵列来替代,以其中任何一个点对局域电场的响应获来取偶极矩,这些点的辐射总和就构成了总的散射场[10]。假设第
$j$ 个偶极子取自于${r_j}\left( {j = 1,2,...,N} \right)$ ,偶极子的极化率为${\alpha _j}$ ,若电场设为${E_j}$ ,偶极矩${P_j}$ ,则极化强度可表示为:$${P_j} = {\alpha _j} \cdot {E_j} = {\alpha _j}\left( {{E_{inc,j}} - } \right.\begin{array}{*{20}{c}} {\displaystyle\sum_{k \ne j} {{A_{jk}}{P_k}\left. {} \right)} }&{} \end{array}$$ (1) 式中:
${E_j}$ 为入射场${E_{inc,j}}$ 在${r_j}$ 处总的散射场;系数${A_{jk}}$ 是一个$3 \times 3$ 矩阵,表达式有:$${A_{jk}} = \dfrac{{\exp \left( {ik{r_{jk}}} \right)}}{{{r_{jk}}}}\left[ {{k^2}\left( {{{\hat r}_{jk}}{{\hat r}_{jk}} - {I_3}} \right) + } \right.\dfrac{{ik{r_{jk}} - 1}}{{r_{jk}^2}}\left. {\left( {3{{\hat r}_{jk}}{{\hat r}_{jk}} - {I_3}} \right)} \right]$$ (2) 式中:
${r_{jk}} = \left| {{r_j} - {r_k}} \right|$ ;${\hat r_{jk}} = \left( {{r_j} - {r_k}} \right)/{r_{jk}}$ ;$k = {{2{\text{π}} } / \lambda }$ ,$\lambda $ 为入射光波长;${I_3}$ 为$3 \times 3$ 的单位矩阵。在离散偶极子近似法的计算中,解出参量${P_k}$ 后就可求解簇团核壳粒子所有的散射参量。 -
冰晶粒子的核壳结构会对冰晶粒子的消光、吸收和散射效率产生影响。通过对云层的监测发现,云层底部烟煤粒子、沙尘粒子、水溶性粒子以及海洋性粒子是冰晶异质核化的主要成分[11],尤其是烟煤粒子作为晶核介质时吸收效率最为显著。为研究这四种粒子作为核壳介质时非球形簇团粒子的散射特性,文中以烟煤粒子作为核壳介质,建立了如图1(c)、(d)、(e)所示的椭球形、六角平板和六角棱柱3种特殊核壳结构冰晶粒子模型,分别是由图(a)和图(b)构成的不同粒子形状。
设核壳结构冰晶粒子的内外层折射率分别为
${n_1}$ 和${n_2}$ ,利用参考文献[12]中计算均匀混合光学介质膜的折射率公式,得到簇团核壳结构冰晶粒子中间均匀混合层的折射率计算公式:$$\begin{split} n =& \left\{ {{k_x}n_x^2 + {k_y}n_y^2} \right. - {1 / 2}\left[ {} \right.\left( {n_x^2 + n_y^2} \right) - \\ & \sqrt {{{\left( {n_x^2 + n_y^2} \right)}^2} - 4{k_x}{k_y}{{\left( {n_x^2 - n_y^2} \right)}^2}} {\left. {\left. {} \right]} \right\}^{\dfrac{1}{2}}} \\ \end{split} $$ (3) 式中:
${n_x}$ 和${n_y}$ 分别为不考虑介质吸收性的核壳结构冰晶粒子的内外层介质折射率,即${n_x} = {\rm{Re}} \left( {{n_1}} \right)$ 和${n_y} = {\rm{Re}} \left( {{n_2}} \right)$ ;其中有${k_x} = \dfrac{{{n_x}}}{{\left( {{n_x} + {n_y}} \right)}}$ 、${k_y} = \dfrac{{{n_y}}}{{\left( {{n_x} + {n_y}} \right)}}$ 。文中设入射波长
$\lambda = 1.06\;{\text{μ}} {\rm m}$ ,烟煤、冰晶两种介质的折射率分别为${n_1} = 1.75 + 0.44i$ 和${n_2} = 1.300\;5 + 1.69 \times {10^{ - 6}}i$ ,计算可得烟煤−冰晶2种介质的中间混合层折射率为$n = 1.566\;4$ 。 -
自然界中冰晶粒子形状不一、结构复杂,多数学者研究不规则粒子以椭球、六角平板、六角棱柱形状为主[13-14]。为了解激光在簇团核壳结构冰晶粒子中的散射特性受粒子形状及粒子密度的影响情况,基于Cluster-Cluster Aggregation (CCA)模型[15],建立了如图2所示的6种特殊簇团形核壳结构的冰晶粒子模型。模型主要选取2个粒子和10个粒子组成的簇团形结构,各簇团形冰晶粒子模型是模拟近似椭球形、六角平板和六角棱柱形状的特定形状。在同一形状的结构模型中,单个核壳结构中冰晶粒子的偶极子个数相同,粒子的空间分布也相同。
图 2 6种簇团形核壳结构冰晶粒子示意图
Figure 2. Schematic diagram of 6 kinds of agglomerated nucleation core-shell structure ice crystal grains
其中,簇团结构中单个冰晶粒子的核壳结构由4部分组成,从外到内分别为真空介质、冰晶、中间混合层和晶核。其中
$R$ 、$r$ 和$d$ 分别表示核壳结构冰晶粒子的粒子半径、晶核半径、中间混合层厚度。如图3所示。自然界中簇团粒子的空间取向是随机的,对于随机取向的簇团粒子缪勒矩阵元素的统计平均值可以用公式(4)给出[16]:
$$\left\langle {S_{ij}} \right\rangle {\rm{ = }}\dfrac{1}{{8{{\text{π}} ^2}}}\displaystyle\int_0^{2{\text{π}}} {\displaystyle\int_{{\rm{ - }}1}^1 {\int_0^{2{\text{π}}} {{S_{{ij}}}} } } \left( {\beta ,\varTheta ,\varPhi } \right){\rm{d}}\beta {\rm{d(cos}}\varTheta {\rm{)d}}{\varPhi} $$ (4) 式中:
$\left\langle {{S_{{{ij}}}}} \right\rangle$ 为簇团粒子某缪勒矩阵元素的统计平均值;${S_{ij}}\left( {\beta ,\varTheta ,\varPhi } \right)$ 为系统坐标中某特定取向簇团粒子缪勒矩阵元素值,$\beta ,\varTheta ,\varPhi$ 为空间方位角。理论上讲,方位角的取值应该尽可能地取尽所有的值,但是受计算条件限制,文中的计算方案均取了1 000个方位角,即方位角${{{n}}_\beta },{{{n}}_\varTheta },{{{n}}_\varPhi }$ 分别取10,10,10,选取方位角的个数满足计算要求[17]。 -
由公式(1)解出参量
${P_k}$ 后,就可计算出偶极子的电偶极矩${P_j}$ ,相应就可计算出核壳结构粒子的消光截面${C_{\rm ext}}$ 、散射截面${C_{\rm sca}}$ 和吸收截面${C_{\rm abs}}$ [18-19]。吸收截面
${C_{\rm abs}}$ 计算公式为:$${C_{\rm abs}} = \dfrac{{4{\text{π}}k}}{{|{E_0}{|^2}}}\sum_{j = 1}^N {\left\{ {{\rm{Im}} } \right.\left[ {{P_j} \cdot {{\left( {\alpha _j^{ - 1}} \right)}^*}{P_j}^*} \right] - \dfrac{2}{3}{k^3}\left. {|{P_j}{|^2}} \right\}} $$ (5) 散射截面
${C_{\rm sca}}$ 计算公式为:$$\begin{split} {C_{\rm sca}} = {C_{\rm ext}} - {C_{\rm abs}} =& \dfrac{{{k^4}}}{{|{E_{\rm inc}}{|^2}}}\int {{\rm{d}}\varOmega |\displaystyle\sum_{j = 1}^N {\left[ {{P_j} - \hat n\left( {\hat n \cdot {P_j}} \right)} \right]} } \times \\ &\exp \left( { - ik\hat n \cdot {r_j}} \right){|^2}\end{split} $$ (6) 消光截面
${C_{\rm ext}}$ 计算公式:$$ {C_{\rm ext}} = \dfrac{{4{\text{π}} k}}{{|{E_0}{|^2}}}\sum_{j = 1}^{N} {\left\{ {{\rm{Im}} } \right.} \left( {E_{{\rm {inc}},j}^* \cdot {P_j}} \right) = {C_{\rm abs}} + {C_{\rm sca}} $$ (7) 由此可得相应的吸收效率因子、散射效率因子、消光效率因子的计算公式。
$$ \begin{array}{l}{\text{吸收效率}}:\;\;{Q_{\rm abs}} = \dfrac{{{C_{\rm abs}}}}{{{\text{π}} a_{\rm eff}^{^2}}}\\ {\text{散射效率}}:\;\;{Q_{\rm sca}} = \dfrac{{{C_{\rm \rm sca}}}}{{{\text{π}} a_{\rm eff}^{^2}}}\\ {\text{消光效率}}:\;{Q_{\rm ext}} = \dfrac{{{C_{\rm ext}}}}{{{\text{π}} a_{\rm eff}^{^2}}} = {Q_{\rm sca}} + {Q_{\rm abs}} \end{array} $$ (8) 式中:
$\hat n$ 为单位散射矢量;*为取复数的共轭;${a_{\rm eff}}$ 为粒子的等效半径;$\rm{d}\varOmega$ 为立体角微分量。为研究DDA计算中散射光强随入射光强的变化情况,得到入射光强和散射光强[20]的计算公式为:$${I_s}\left( \theta \right) = \dfrac{1}{{{x^2}}}\left( {{S_{11}}} \right){I_0}$$ (9) 式中:
${S_{11}}$ 为一个散射矩阵元素;$x = k{a_{\rm eff}}$ ,$k = 2{\text{π}} /\lambda$ ,$\lambda $ 为入射波长,${a_{\rm eff}}$ 为有效半径:$${{a}_{\rm eff}}={{\left( {}^{3V}\!\!\diagup\!\!{}_{4{\text{π}}}\; \right)}^{{1}/{3}\;}}$$ (10) 式中:
$V$ 为目标散射体的体积。散射振幅矩阵元素为:$$\left( \begin{array}{l} {S_2}\begin{array}{*{20}{c}} {}&{{S_3}} \end{array} \\ {S_4}\begin{array}{*{20}{c}} {}&{{S_1}} \end{array} \\ \end{array} \right)$$ (11) 散射特性可用入射参数(
${I_i},{Q_i},{U_i},{V_i}$ )和散射参数(${I_s},{Q_s},{U_s},{V_s}$ )的Mueller散射矩阵来描述,计算公式如下:$$\left( \begin{array}{l} {I_s} \\ {Q_s} \\ {U_s} \\ {V_s} \\ \end{array} \right) = \dfrac{1}{{{k^2}{r^2}}}\left( \begin{array}{l} {S_{11}}\begin{array}{*{20}{c}} {}&{{S_{12}}\begin{array}{*{20}{c}} {}&{{S_{13}}} \end{array}\begin{array}{*{20}{c}} {}&{{S_{14}}} \end{array}} \end{array} \\ {S_{21}}\begin{array}{*{20}{c}} {}&{{S_{22}}\begin{array}{*{20}{c}} {}&{{S_{23}}} \end{array}\begin{array}{*{20}{c}} {}&{{S_{24}}} \end{array}} \end{array} \\ {S_{31}}\begin{array}{*{20}{c}} {}&{{S_{32}}\begin{array}{*{20}{c}} {}&{{S_{33}}} \end{array}\begin{array}{*{20}{c}} {}&{{S_{34}}} \end{array}} \end{array} \\ {S_{41}}\begin{array}{*{20}{c}} {}&{{S_{42}}\begin{array}{*{20}{c}} {}&{{S_{43}}} \end{array}\begin{array}{*{20}{c}} {}&{{S_{44}}} \end{array}} \end{array} \\ \end{array} \right)\left( \begin{array}{l} {I_i} \\ {Q_i} \\ {U_i} \\ {V_i} \\ \end{array} \right)$$ (12) 式中:4×4矩阵为散射矩阵,Muller散射矩阵中各元素
${S_{ij}}$ 由公式(12)给出。粒子的Muller散射矩阵由16个元素组成,它们体现了散射体所有的散射特性和极化特性,对于随机取向具有某种对称性的粒子,缪勒矩阵中有8个元素不为零,有
${S_{21}} = {S_{12}},{S_{43}} = - {S_{34}}$ ,因此只有6个矩阵元是独立的,分别由振幅散射矩阵的4个元素的模和它们之间的相位差决定,另外${S_{22}}$ 和${S_{11}}$ ,${S_{44}}$ 和${S_{33}}$ 的变化趋势相似,所以散射矩阵只有4个独立的矩阵元[21]。散射特性用入射参数(${I_i},{Q_i},{U_i},{V_i}$ )和散射参数(${I_s},{Q_s},{U_s},{V_s}$ )的Mueller散射矩阵来描述。$$\left( \begin{array}{l} {I_s} \\ {Q_s} \\ {U_s} \\ {V_s} \\ \end{array} \right) = \dfrac{1}{{{k^2}{r^2}}}\left( \begin{array}{l} {S_{11}}\begin{array}{*{20}{c}} {}&{{S_{12}}\begin{array}{*{20}{c}} {}&{{}^{}0} \end{array}\begin{array}{*{20}{c}} {}&{{}^{}0} \end{array}} \end{array} \\ {S_{12}}\begin{array}{*{20}{c}} {}&{{S_{11}}\begin{array}{*{20}{c}} {}&{{}^{}0} \end{array}\begin{array}{*{20}{c}} {}&{{}^{}0} \end{array}} \end{array} \\ {}_{}0\begin{array}{*{20}{c}} \;\;\;{}&{_{}^{}0\begin{array}{*{20}{c}} \;\;\;{}&{{}^{}{S_{33}}} \end{array}\begin{array}{*{20}{c}} \!\!\!\!\! {}&{{S_{34}}} \end{array}} \end{array} \\ {}_{}0\begin{array}{*{20}{c}} \;\;\;{}&{{}_{}^{}0\begin{array}{*{20}{c}} {}&{ - {S_{34}}} \end{array}\begin{array}{*{20}{c}} \!\!\!\!\! {}&{{S_{33}}} \end{array}} \end{array} \\ \end{array} \right)\left( \begin{array}{l} {I_i} \\ {Q_i} \\ {U_i} \\ {V_i} \\ \end{array} \right)$$ (11) $$ \begin{array}{l} {S_{11}} = {{\left( {{{\left| {{S_1}} \right|}^2} + {{\left| {{S_2}} \right|}^2} + {{\left| {{S_3}} \right|}^2} + {{\left| {{S_4}} \right|}^2}} \right)} / 2}\\ {S_{12}} = {{\left( {{{\left| {{S_2}} \right|}^2} - {{\left| {{S_2}} \right|}^2} + {{\left| {{S_4}} \right|}^2} - {{\left| {{S_3}} \right|}^2}} \right)} / 2}\\ {S_{33}} = {\rm{Re}} \left( {{S_1}S_2^* + {S_3}S_4^*} \right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ {S_{34}} = {\rm{Im}} \left( {{S_2}S_1^* + {S_4}S_3^*} \right)。 \end{array}$$ 矩阵元素
${S_{11}}$ 反映总的入射场强度在散射发生前后的变化情况;${{{S_{12}}} / {{S_{11}}}}$ 表示平行和垂直于散射平面的线性极化程度;${{{S_{33}}} / {{S_{11}}}}$ 描述线性极化入射光($\pm {45^{\rm o}}$ )相对于线性极化的散射光($\pm {45^{\rm o}}$ )的变化情况;${{{S_{34}}} / {{S_{11}}}}$ 描述圆极化入射光($\pm {90^{\rm o}}$ )相对于圆极化的散射光($\pm {90^{\rm o}}$ )的变化情况。
Scattering properties of non-spherical cluster core-shell structure particle laser
-
摘要: 在冰晶粒子异质核化理论基础上,建立了椭球形、六角平板和六角棱柱3种簇团形核壳结构冰晶粒子模型,利用离散偶极子近似法(DDA)数值计算同一入射波长下,有效尺寸对核壳结构冰晶粒子消光效率、吸收效率和散射效率的影响、中间均匀混合层对核壳结构冰晶粒子散射强度的影响以及Mueller矩阵元素随散射角度的变化情况。数值结果表明:随着有效尺寸的增大,椭球形、六角平板和六角棱柱3种簇团形核壳结构冰晶粒子的消光效率、吸收效率和散射效率随着有效尺寸的增大分别呈现不同的变化趋势;在相等尺寸条件下,散射强度随散射角度的变化情况与粒子形状有密切关系,且相比于椭球形和六角平板两种簇团形核壳结构冰晶粒子,六角棱柱核壳结构冰晶粒子的前向散射强度最大,散射强度随散射角度的变化曲线振荡更明显。根据Mueller矩阵元素随散射角度的分布情况,可以看出六角棱柱簇团形冰晶结构的散射方向最明显,前向散射强度最大,六角平板和六角棱柱簇团形冰晶结构的Mueller矩阵元素相对球形和椭球形在后向散射场区域的偏差更明显。论文的研究结果为进一步分析复杂冰晶粒子的散射特性,开展高空云层中各种复杂几何形状簇团冰晶粒子的散射特性研究和分析提供支持。Abstract: On the basis of heterogeneous nucleation theory of ice crystal particles, three types of ice crystal particle models were established with nucleation-shell structures of ellipsoid, hexagonal flat plate and hexagonal prism. The extinction, absorption and scattering efficiency of these three special cluster-shaped core-shell structures were numerically calculated by discrete dipole approximation (DDA) method. Under the same incident wavelength, the effect of effective size on the extinction efficiency, absorption efficiency and scattering efficiency of core-shell ice crystal particles, the influence of the intermediate uniform mixing layer on the scattering intensity of core-shell ice crystal particles, and the variation of Mueller matrix elements with the scattering angle were calculated. The numerical results show that the extinction coefficient, absorption coefficient and scattering coefficient of ice crystallites with three clusters of ellipsoidal, hexagonal and hexagonal prisms show different trends with the increase of effective size. Under the condition of equal size, the scattering intensity with the change of the scattering angle and particle shape have close relations, and compared with the ellipsoid and hexagonal flat two cluster core-shell structure of ice crystal particles, core-shell structure of ice crystal particles hexagonal prisms forward scattering intensity, the largest scattering intensity curve along with the change of the scattering angle oscillation is more obvious. According to the distribution of the Mueller matrix elements with the scattering angle, it can be seen that the scattering direction of the hexagonal prism cluster ice crystal structure is the most obvious, and the forward scattering intensity is the largest. The Mueller matrix elements of the hexagonal plate and hexagonal prism cluster ice crystal structure are relatively spherical and the deviation of the ellipsoid in the backscattered field area is more obvious. The research results of the thesis provide support for further analysis of the scattering characteristics of complex ice crystal particles, and the research and analysis of the scattering characteristics of various complex geometric clusters of ice crystal particles in high-altitude clouds.
-
-
[1] 贺秀兰, 吴建, 杨春平. 冰晶粒子散射理论模型[J]. 红外与激光工程, 2006, 10(35): 385-389. He Xiulan, Wu Jian, Yang Chunping. Model of scattering theory of ice crystal particles [J]. Infrared and Laser Engineering, 2006, 10(35): 385-389. (in Chinese) [2] Yang P, Liou K N, Bi L, et al. On the radiative properties of ice clouds: light scattering, remote sensing, and radiation parameterization [J]. Advances in Atmospheric Sciences, 2015, 32(1): 32-63. doi: 10.1007/s00376-014-0011-z [3] Mishchenko M I, Lacis A A, Travis L D. Errors induced by the neglect of polarization in radiance calculations for rayleigh-scattering atmospheres [J]. Journal of Quantitative Spectroscopy and Radiative Transfer, 1994, 51(3): 491-510. doi: 10.1016/0022-4073(94)90149-X [4] Raisanen P, Bogdan A, Sassen K, et al. Impact of H<sub>2</sub>SO<sub>4</sub>/H<sub>2</sub>O coating and ice crystal size on radiative properties of sub-visible cirrus [J]. Atmos Chem Phys, 2006, 6(12): 5231-5250. [5] 张晋源, 张成义, 郑改革. 水包冰球包层粒子散射特性的研究[J]. 激光与红外, 2016, 46(5): 597-601. doi: 10.3969/j.issn.1001-5078.2016.05.017 Zhang Jinyuan, Zhang Chengyi, Zheng Gaige. Study on the scattering characteristics of water-encasing ice hockey cladding particles [J]. Laser & Infrared, 2016, 46(5): 597-601. (in Chinese) doi: 10.3969/j.issn.1001-5078.2016.05.017 [6] 陈洪滨, 孙海冰. 冰-水球形粒子在太阳短波段的吸收与衰减[J]. 大气科学, 1999, 23(2): 233-238. doi: 10.3878/j.issn.1006-9895.1999.02.12 Chen Hongbin, Sun Haibin. Absorption and attenuation of ice-water spherical particles in the short-wave segment of the sun [J]. Atmospheric Science, 1999, 23(2): 233-238. (in Chinese) doi: 10.3878/j.issn.1006-9895.1999.02.12 [7] Sassen Kenneth, Paul J DeMott, Joseph M Prospero, et al. Saharan dust storms and indirect aerosol effects on clouds: CRYSTAL-FACE results [J]. Geophysical Research Letters, 2003, 30(12): 1633. [8] 孟立新, 孟令臣, 李小明, 等. 星载激光通信端机可重复锁紧/解锁机构设计[J]. 光学 精密工程, 2019, 27(7): 1544-1551. doi: 10.3788/OPE.20192707.1544 Meng Lixin, Meng Lingchen, Li Xiaoming, et al. Design of a repeatable locking/unlocking mechanism for satellite-borne laser communication terminal [J]. Optics and Precision Engineering, 2019, 27(7): 1544-1551. (in Chinese) doi: 10.3788/OPE.20192707.1544 [9] Purcell E M, Pennypacker C R. Scattering and absorption by non-spherical dielectric grains [J]. The Astrophysical Journal, 1973, 186: 705-714. doi: 10.1086/152538 [10] 保秀娟. 团聚形核壳结构冰晶粒子的光散射特性研究[D]. 西安: 西安理工大学, 2019: 13-14. Bao Xiujuan. Agglomerated core-shell structure of ice crystal particles light scattering characteristics[D]. Xi 'an: Xi 'an institute of Technology University, 2019: 13-14. (in Chinsese) [11] 柯程虎, 张辉, 保秀娟. 团聚形核壳结构冰晶粒子的激光散射特性[J]. 红外与激光工程, 2019, 48(8): 0805008. doi: 0805008 Ke Chenghu, Zhang Hui, Bao Xiujuan. Laser scattering characteristics of ice crystal particles with agglomerated core shell structure [J]. Infrared and Laser Engineering, 2019, 48(8): 0805008. (in Chinese) doi: 0805008 [12] 王明军, 于记华, 刘雁翔, 等. 多激光波长在不同稀薄随机分布冰晶粒子层的散射特性(英文)[J]. 红外与激光工程, 2019, 48(3): 0311002. Wang Mingjun, Yu Jihua, Liu Yanxiang, et al. Scattering characteristics of multiple laser wavelengths in different thin and randomly distributed ice crystal particle layers [J]. Infrared and Laser Engineering, 2019, 48(3): 0311002. (in Chinese) [13] Wendisch M, Yang P. 大气辐射传输原理[M]. 李正强, 李莉, 候伟真, 等, 译. 北京: 高等教育出版社, 2014: 82-88. Wendisch M, Yang P. Principle of Atmospheric Radiation Rransmission[M]. Li zhengqiang, Li Li, Hou Weizhen, et al translated. Beijing: Higher Education Press, 2014: 82-88. [14] Liou K N. 大气辐射导论[M]. 第2版. 郭彩丽, 周诗健, 译. 北京: 气象出版社, 2004: 260-264. Liou K N. Introduction to atmospheric radiation[M]. 2nd ed. Guo Caili, Zhou Shijian, trans. Beijing: Meteorological Press, 2004: 260-264. [15] Jullien R, Botet R. Aggregation and Fractal Aggregates[M]. Singapore: World Scientific, 1987. [16] Xing Zhangfan. Martha S Hanner. Light Scattering by Aggregate Particles [J]. Astron Astrophys, 1997, 324(2): 805-820. [17] 类成新, 吴振森, 冯东太. 随机取向内外混合凝聚粒子辐射特性[J]. 红外与激光工程, 2013, 42(10): 2692-2696. doi: 10.3969/j.issn.1007-2276.2013.10.019 Lei Chengxin, Wu Zhensen, Feng Dongtai. Radiation characteristics of randomly oriented mixed condensed particles [J]. Infrared and Laser Engineering, 2013, 42(10): 2692-2696. (in Chinese) doi: 10.3969/j.issn.1007-2276.2013.10.019 [18] 魏邦海. 气溶胶和冰水两相粒子的散射特性[D]. 南京: 南京信息工程大学, 2015: 62. Wei Banghai. Scattering properties of aerosols and ice water two-phase particle[D]. Nanjing: Nanjing University of Information Science & Technology, 2015: 62. (in Chinese) [19] 王德江, 孙翯, 孙雪倩. 消光比与探测器噪声对基于纳米线栅偏振成像系统偏振精度的影响[J]. 光学 精密工程, 2018, 26(10): 2371-2379. doi: 10.3788/OPE.20182610.2371 Wang Dejiang, Sun He, Sun Xueqian. Extinction ratio and the detector noise based on polarization effect of nanowire grid polarization imaging system [J]. Optics Precision Engineering, 2018, 26(10): 2371-2379. (in Chinese) doi: 10.3788/OPE.20182610.2371 [20] 陈露, 高志山, 袁群, 等. 星载激光测高仪距离参数地面标定方法[J]. 中国光学, 2019, 12(4): 897-905. Chen Lu, Gao Zhishan, Yuan Qun, et al. Ground calibration method for distance parameters of satellite-borne laser altimeter [J]. Chinese Optics, 2019, 12(4): 897-905. (in Chinese) [21] 饶瑞中. 现代大气光学[M]. 北京: 科学出版社, 2012: 30-34. Rao Ruizhong. Modern Atmospheric Optics[M]. Beijing: Science Press, 2012: 30-34. (in Chinese)