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聚苯乙烯悬浊液由聚苯乙烯原液与去离子水混合而成,原液的微粒粒径分为1 μm,原液的粒径标准差<5%,配置浓度分别为1.04、2.08、4.17、8.33、12.50、16.67 μg/cm3。配置后的聚苯乙烯悬浊液罐装于125 cm3的立方体石英玻璃容器中。入射光为532 nm波长的完全45°线偏振光或完全水平线偏振光,光在容器中的传输距离为5 cm。
为抑制外界和大角度前向散射光进入接收区域,对整个测试系统进行了相关设置。测试环境为黑布遮盖的密闭空间,测试时间为夜晚,避免了环境杂散光的干扰;聚苯乙烯悬浊液装配容器,除了前后方向之外,其余面均采用黑胶布覆盖,避免了一部分大角度前向散射光的外溢;实验开展于超净实验室中,环境经过除尘和除静电处理,确保环境干净,避免了空气中的杂质引起的散射;系统采用大口径物镜,而超出物镜接收范围的前向散射光被外围黑色遮布吸收,不会在环境中发生漫散射。
测试装置参照图3。532 nm激光通过偏振片P1转化为完全线偏振光Sin,线偏振光Sin通过聚苯乙烯悬浊液输出为Sout,之后由物镜收集,经由偏振片P2后获得Smea,最终由AvanSpec-ULS2048LTEC型光谱仪接收。
设聚苯乙烯悬浊液的穆勒矩阵为MP,Sout和Smea分别求解为:
$$ \boldsymbol{S}_{\rm{out}}=\boldsymbol{M}_{\rm{P}} \boldsymbol{S}_{\rm {in }} $$ (1) $$ \boldsymbol{S}_{\rm{mea}}={\boldsymbol P}_{2} \boldsymbol{M}_{\rm{P}} \boldsymbol{S}_{\rm{in}} $$ (2) 式中:P2代表米勒矩阵。公式(2)转化为:
$$ \left[\begin{array}{l} S_{0 \rm{mea}} \\ S_{1 \rm{mea}} \\ S_{2 \rm{mea}} \\ S_{3 \rm{mea}} \end{array} \right]=\left[\begin{array}{cccc} 1 \;\;\cos 2 {\rm {\theta}} \;\;\sin 2 {\rm {\theta}} \;\;0 \\ \cos 2 {\rm {\theta}} \;\;\cos ^{2} 2 {\rm {\theta}} \;\;\sin 2 {\rm {\theta}} \cos 2 {\rm {\theta}} \;\;0 \\ \sin 2 {\rm {\theta}} \;\;\sin 2 {\rm {\theta}} \cos 2 {\rm {\theta}} \;\;\sin ^{2} 2 {\rm {\theta}} \;\;0 \\ 0 \;\;0 \;\;0 \;\;0 \end{array}\right] \\ \cdot \frac{1}{2}\left[\begin{array}{c} S_{0 \rm { out }} \\ S_{1 \rm { out }} \\ S_{2 \rm { out }} \\ S_{3 \rm { out }} \end{array}\right] $$ (3) 将公式(3)展开得:
$$ \begin{split} 2 S_{0 \rm { mea }}= &S_{0 \rm { out }}+S_{\rm {1out }} \cos 2 {\rm {\theta}}+S_{2 \rm { out }} \sin 2 {\rm {\theta}} \\ 2 S_{1 \rm { mea }}= &S_{0 \rm { out }} \cos 2 {\rm {\theta}}+S_{1 \rm { out }} \cos ^{2} 2 {\rm {\theta}}+ \\ &S_{2 \rm { out }} \sin 2 {\rm {\theta}} \cos 2 {\rm {\theta}} \\ 2 S_{2 \rm { mea }} =&S_{0 \rm { out }} \sin 2 {\rm {\theta}}+S_{\rm {1out }} \sin 2 {\rm {\theta}} \cos 2 {\rm {\theta}} +\\ &S_{2 \rm { out }} \sin ^{2} 2 {\rm {\theta}} \end{split} $$ (4) 公式(4)中有 3 个待求参数,通过测量偏振片旋转至不同角度下的 3 组光强值即可求解,求解后得到 Sout。
为了求解聚苯乙烯悬浊液的穆勒矩阵,需选择4种不同的偏振光作为输入光,入射光分别设置为 Sin−1、Sin−2、Sin−3、Sin−4,求解输出光分别为Sout−1、Sout−2、Sout−3、Sout−4。分别建立秩为 4 的输入矩阵和输出矩阵,输入矩阵和输出矩阵之间关系为:
$$ \left[\boldsymbol{S}_{\rm {out }-1}, \boldsymbol{S}_{\rm {out- } 2}, \boldsymbol{S}_{\rm {out }-3}, \boldsymbol{S}_{\rm {out- } 4}\right] \\ =\boldsymbol{M}_{\rm{p}}\left[\boldsymbol{S}_{\rm{in}-1}, \boldsymbol{S}_{\rm{in}-2}, \boldsymbol{S}_{\rm{in}-3}, \boldsymbol{S}_{\rm{in}-4}\right] $$ (5) 求解聚苯乙烯悬浊液的穆勒矩阵 Mp为:
$$ \boldsymbol{M}_{\rm{p}}=\left[\boldsymbol{S}_{\rm {out- } 1}, \boldsymbol{S}_{\rm {out- } 2}, \boldsymbol{S}_{\rm {out- } 3}, \boldsymbol{S}_{\rm {out-4 }}\right] \\ ·\left[\boldsymbol{S}_{\rm{in}-1}, \boldsymbol{S}_{\rm{in}-2}, \boldsymbol{S}_{\rm{in}-3}, \boldsymbol{S}_{\rm{in}-4}\right]^{-1} $$ (6) -
由于偏振度计算过程不仅引入了初始入射光的正交分量光强,而且掺杂着具有其他偏振态光波的光强,评估时缺乏一定的精度。偏振状态保持率(Retention rate of polarization state,RoPS)[15-16]用于表征前向散射光中与初始入射光具有相同偏振态的光波光强占前向散射光总光强的比例,它不仅能够规避正交分量光强度差值计算所引入的误差,而且规避了其他偏振态光强的影响,因此,采用RoPS作为评估指标。其中,RoPS可由穆勒矩阵元素求得,具体求解过程参考文献[16]。
当 45°线偏振光前向传输时,RoPS 表达为:
$$ R o P S_{\rm {light-45 }}=\frac{P_{45 \rm {-out-forward }}}{P_{0 \rm {-out-forward }}} $$ (7) 式中:P0-out-forward表示前向散射光的总光强度;P45-out-forward代表前向散射光中45°线偏振光的光强度。
当水平线偏振光前向传输时,RoPS表达为:
$$ R o P S_{\rm {light-H }}=\frac{P_{\rm {H-out-forward }}}{P_{0 \rm {-out-forward }}} $$ (8) 式中:P0-out-forward表示前向散射光的总光强度;PH-out-forward代表前向散射光中水平线偏振光的光强度。
图4显示了 45°线偏振光在聚苯乙烯悬浊液中前向传输的计算和测试结果,重复模拟计算和实验测试 5 次以确保结果的准确性。结果显示:随着浓度的增加,偏差范围基本呈现逐渐增加的趋势;且在低浓度条件下,改进算法与原算法计算差异较小,随着浓度的增加,改进算法较原算法更贴近实测结果。结果表明:在较高浓度的聚苯乙烯悬浊液中,改进算法的计算结果相比偏振子午面蒙特卡洛法更接近测量值。
图 4 45°线偏振光在聚苯乙烯悬浊液中前向传输的改进算法、原算法与实测结果
Figure 4. Improved algorithm, original algorithm and measured results of 45° linearly polarized light propagating in polystyrene suspension
图5显示了水平线偏振光在聚苯乙烯悬浊液中前向传输的计算和测试结果,重复模拟计算和实验测试5次以确保结果的准确性。结果显示:随着浓度的增加,偏差范围基本呈现逐渐增加的趋势;且在低浓度条件下,改进算法与原算法计算差异较小,随着浓度的增加,改进算法较原算法更贴近实测结果。结果表明:在较高浓度的聚苯乙烯悬浊液中,改进算法的计算结果相比偏振子午面蒙特卡洛法更接近测量值。
图 5 水平线偏振光在聚苯乙烯悬浊液中前向传输的改进算法、原算法与测试结果
Figure 5. Improved algorithm, original algorithm and measured results of horizontally linearly polarized light propagating in polystyrene suspension
统观45°线偏振光和水平圆偏振光在聚苯乙烯悬浊液中前向传输的计算结果和测试结果,得出结论:相比偏振子午面蒙特卡洛法,改进算法的计算结果与测试结果更为匹配。
Optimized algorithm to limit receiving range of polarized light forward propagation into scattering media
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摘要: 为解决偏振子午面蒙特卡洛法的计算结果与测试结果存在出入的问题,文中提出了一种限定接收范围的改进算法。偏振子午面蒙特卡洛法计算过程中采集整个截面的前向散射光,但实际接收装置通常接收部分前向散射光,这是引起偏振子午面蒙特卡罗法的计算结果与实验结果存在出入乃至失配的重要因素之一。为此,改进算法限定了前向散射光的接收范围,限定的接收范围与实际接收范围一致。采用原算法和改进算法分别模拟 45°线偏振光和水平线偏振光在聚苯乙烯悬浊液中传输。选用粒径 1 μm 的聚苯乙烯悬浊液开展验证实验,通过对照测试结果,改进算法相比原算法能够更好地匹配实测结果。通过计算单个微粒的偏振状态保持情况,阐明改进算法提高计算精度的作用机理,即前向散射角愈外延,散射光的偏振态变化愈剧烈,而改进算法规避掉了绝大部分前向散射角外延程度较大的散射光。Abstract: To solve the problem of discrepancies between calculation results and test results of the polarization meridian Monte Carlo method, an improved algorithm which can limit receiving range was proposed. All forward scattered light was received generally in real receiving devices in the process of calculation, but part of forward scattered light was collected. This is one of the important factors causing discrepancies and mismatch between calculation results and test results of the polarization meridian Monte Carlo method. For this reason, an improved algorithm was proposed to limit receiving range of forward scattered light. Receiving range of the improved algorithm was consistent with real receiving range. Original algorithm and improved algorithm were used to simulate the propagation of 45°- and horizontally- linearly polarized light in polystyrene suspension, respectively. Then, the polystyrene suspension with particle size of 1 μm were selected to carry out verification experiments. Compared with the original algorithm, simulation results of the improved algorithm were closer to measured values. Finally, the polarization state distribution of a single particle was calculated to clarify mechanism of improved accuracy. Retention of polarization state(RoPS) changes drastically as forward scattered angle extends. Improved algorithm avoids most of forward scattered light with larger forward scattered angles.
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