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湿度与颗粒半径,颗粒折射率有很大的关系,半径与折射率的变化最终会影响散射光的偏振特性,干洁环境中颗粒半径与湿度关系如下:
$$\frac{{{r_h}}}{{{r_0}}} = {\left( {1 - f} \right)^{ - 1/u}}$$ (1) 式中:
${r_h}$ 为受湿度影响变化的粒子半径;${r_0}$ 为干洁环境中的颗粒半径;$f$ 为湿度;u为常数,对于雾天环境而言,气溶胶吸湿性强,u=3.9;${r_0}$ 为初始半径。湿度导致了粒径的变化,而粒径变化又会导致复折射率变化:
$${m_{re}} = {m_{rw}} + \left( {{m_{r0}} - {m_{rw}}} \right){\left[ {\frac{{{r_h}}}{{{r_0}}}} \right]^{ - 3}}$$ (2) $$\frac{{{m_{ie}}}}{{m_{re}^2 + 2}} = \dfrac{{{m_{iw}}}}{{m_{rw}^2}} + \left( {\frac{{{m_{i0}}}}{{m_{r0}^2 + 2}} - \frac{{{m_{iw}}}}{{m_{rw}^2}}} \right){\left[ {\frac{{{r_h}}}{{{r_0}}}} \right]^{ - 3}}$$ (3) ${m_{re}}$ 为复折射实部;${m_{ie}}$ 为虚部。r,i,e,0和w分别表示实部、虚部吸湿后气溶胶,干气溶胶粒子和水。湿度影响下的气溶胶折射率为:$${m_e} = {m_{re}} + {m_{ie}}$$ (4) 水雾环境中粒子尺度较大,通常都在0.1~10 μm之间,水雾粒子等效半径0.5 μm,折射率为1.33+1.96E-4i,通过公式(1)、(4)对不同湿度的粒子物理参数进行了计算,如表1所示。
表 1 水雾环境粒子湿度与折射率的关系
Table 1. Relationship between humidity and refractive index of particles in water mist environment
Relative
humidityParticle
radius /μmIndex of
real partIndex of
imaginary part30% 0.548 1.369 1.0298E-4 40% 0.570 1.365 1.3907E-4 50% 0.597 1.361 1.7635E-4 60% 0.632 1.356 2.1518E-4 70% 0.681 1.352 2.5608E-4 80% 0.755 1.347 3.0002E-4 90% 0.902 1.341 3.4926E-4 从表1可以看出:水雾粒子折射率实部随着湿度的增加而变小,水雾环境折射率实部值基本不变。由于水雾粒子吸湿性强,所以粒子折射率改变值小。
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为了得到湿度与偏振光传输特性的关系,定义参考平面如图1所示。首先,确定光子传播方向,夹角θ为散射角,x轴与z轴和散射平面分别构成了散射前后的参考平面,散射前后的斯托克斯分量由垂直和平行于这两个平面的E∥、E⊥分量定义,每次散射发生时,都必须对斯托克斯分量进行调整以使参考平面为新的参考面。
参考平面定义以后,设定激光波长、粒子直径、粒子复折射率等,散射系数
${\mu _s}$ ,吸收系数${\mu _a}$ ,大气的消光系数${\mu _t} = {\mu _s} + {\mu _a}$ ,不同偏振态的光的斯托克斯参量初始值等。光子沿z轴正方向入射,初始位置${u_0}$ (0,0,0),初始方向余弦${D_0}$ (0,0,1),偏振参考平面为X轴与Z轴构成的平面,定义光子初始斯托克斯参量,并且先将散射后各斯托克斯参量都设为0。粒子传输的过程的步长就是光子的自由程, 其中
$\xi $ 为关于0.5对称分布的随机数,消光系数与光子自由程${S_1}$ 关系为:$${S_1} = \ln \left( \xi \right){\mu _t}$$ (5) 一旦光子传输的自由程确定后,光子将发生移动,当前位置
$\left( {x,y,z} \right)$ 和传输方向$\left( {{\mu _x},{\mu _y},{\mu _z}} \right)$ 确定光子到达的下一个散射点的坐标$\left( {x',y',z'} \right)$ 为:$$\left\{ {\begin{aligned} & {x' = x + {\mu _x}{S_1}} \\ & {y' = y + {\mu _y}{S_1}} \\ & {z' = z + {\mu _z}{S_1}} \end{aligned}} \right.$$ (6) 并且根据坐标位置,判断光子是否射出参考平面所确定的边界,如果射出参考平面确定的边界,则对光子进行斯托克斯参量计算。反之,必须先对斯托克斯分量进行调整以使参考平面为新的子午面,再重新进行粒子半径选择。
光子与粒子碰撞后,散射角
$\alpha $ 和方位角$\beta $ 抽样由联合概率密度函数(PDF)得到,该函数与入射光斯托克斯分量[I0,Q0,U0,V0]T的关系为:$$\rho \left( {\alpha ,\beta } \right) = {m_{11}}\left( \alpha \right) + {m_{12}}\left( \alpha \right)\left[ {{Q_0}\cos \left( {2\beta } \right) + {U_0}\sin \left( {2\beta } \right)} \right]/{I_0}$$ (7) ${m_{11}}\left( \alpha \right)$ 、${m_{12}}\left( \alpha \right)$ 分别为球形粒子Mueller矩阵$M\left( \alpha \right)$ 中相应元素。$$M\left( \alpha \right) = \left[ {\begin{array}{*{20}{c}} {{m_{11}}\left( \alpha \right)}&{{m_{12}}\left( \alpha \right)}&0&0 \\ {{m_{12}}\left( \alpha \right)}&{{m_{11}}\left( \alpha \right)}&0&0 \\ 0&0&{{m_{33}}\left( \alpha \right)}&{{m_{34}}\left( \alpha \right)} \\ 0&0&{ - {m_{34}}\left( \alpha \right)}&{{m_{33}}\left( \alpha \right)} \end{array}} \right]$$ (8) ${m_{11}}\left( \alpha \right)$ 、${m_{12}}\left( \alpha \right)$ 、${m_{33}}\left( \alpha \right)$ 、${m_{34}}\left( \alpha \right)$ 与散射幅度值${S_1}$ ,${S_2}$ 有如下关系:$$\begin{split} & {m_{11}}\left( \alpha \right) = \frac{1}{2}\left( {{{\left| {{S_1}} \right|}^2} + {{\left| {{S_2}} \right|}^2}} \right),\;\;{{m_{12}}\left( \alpha \right) = \frac{1}{2}\left( {{{\left| {{S_1}} \right|}^2} - {{\left| {{S_2}} \right|}^2}} \right)} \\ & {m_{33}}\left( \alpha \right) = \frac{1}{2}\left( {{S_1}S_2^ * - S_{_1}^ * {S_2}} \right),\;\;{{m_{44}}\left( \alpha \right) = \frac{i}{2}\left( {{S_1}S_2^ * - S_2^ * {S_1}} \right)} \end{split}$$ (9) 由于散射幅度值
${S_1}$ ,${S_2}$ 表达式为:$$\left\{ {\begin{aligned} & {{S_1}(\theta ) = \sum\limits_{n = 1}^\infty {\frac{{2 n + 1}}{{n(n + 1)}}} \left[ {{a_n}\frac{{P_n^1(\cos \theta )}}{{\sin \theta }} + {b_n}\frac{{P_n^1(\cos \theta )}}{{d\theta }}} \right]}\\ & {{S_2}(\theta ) = \sum\limits_{n = 1}^\infty {\frac{{2 n + 1}}{{n(n + 1)}}} \left[ {{a_n}\frac{{P_n^1(\cos \theta )}}{{d\theta }} + {b_n}\frac{{P_n^1(\cos \theta )}}{{\sin \theta }}} \right]} \end{aligned}} \right.$$ (10) 式中:an,bn为Mie散射系数,它们与球形颗粒的大小,折射率有很大关系,其表达式为:
$$\left\{ {\begin{aligned} & {{a_n} = \frac{{\psi _n^\prime (mx){\psi _n}(x) - m{\psi _n}(mx)\psi _n^\prime (x)}}{{\psi _n^\prime (mx){\xi _n}(x) - m{\psi _n}(mx)\xi _n^\prime (x)}}}\\ & {{b_n} = \frac{{m\psi _n^\prime (mx){\psi _n}(x) - {\psi _n}(mx)\psi _n^\prime (x)}}{{m\psi _n^\prime (mx){\xi _n}(x) - {\psi _n}(mx)\xi _n^\prime (x)}}} \end{aligned}} \right.$$ (11) 式中:
${\psi _n}\left( x \right)$ 和${\xi _n}\left( x \right)$ 为Riccati-Bessel函数,可用第一类Bessel球函数${J_{n + 1/2}}\left( x \right)$ 和半整数阶第二类Hankel函数$H_{n + 1/2}^{\left( 2 \right)}$ 表示:$$\left\{ {\begin{aligned} & {{\psi _n}(x) = \sqrt {\pi x/2} {J_{n + 1/2}}(x)}\\ & {{\xi _n}(x) = \sqrt {\pi x/2} H_{n + 1/2}^{(2)}(x)} \end{aligned}} \right.$$ (12) 所以,相应的散射强度函数为:
$${i_1}(\theta ) = {\left| {{S_1}(\theta )} \right|^2} = {\left| {\sum\limits_{n = 1}^\infty {\frac{{2 n + 1}}{{n(n + 1)}}} \left( {{a_n}{\pi _n} + {b_n}{\tau _n}} \right)} \right|^2}$$ (13) $${i_2}(\theta ) = {\left| {{S_2}(\theta )} \right|^2} = {\left| {\sum\limits_{n = 1}^\infty {\frac{{2 n + 1}}{{n(n + 1)}}} \left( {{a_n}{\pi _n} + {b_n}{\tau _n}} \right)} \right|^2}$$ (14) 光子经过n次散射后的能量权重变为:
$${W_n} = {W_{n - 1}} \cdot {\mu _s}/\left( {{\mu _s} + {\mu _a}} \right)$$ (15) 当光子能量权重低于某一阈值(一般为10−4)或飞离边界时,光子传输终止。光子从多分散系边界射出时,其斯托克斯矢量必须经过最后一次旋转以保证参考平面与探测器所在的平面相同,旋转的角度为:
$$\omega = \pm {\rm{arctan}}\left( {{\mu _{{y}}}/{\mu _{{x}}}} \right)$$ (16) 式中:反射模式取正号,透射取负号。由于散射路径的不同,光子到达探测器的时间不同,对于偏振分量形如 [I(t),Q(t),U(t),V(t)]T的光束,时域偏振度定义为:
$${\mathop{\rm DOP}\nolimits} (t) = \frac{{\sqrt {{Q^2}(t) + {U^2}(t) + {V^2}(t)} }}{{I(t)}}$$ (17) 光束经散射后总的偏振度定义为:DOP=(Q2+U2+V2)1/2/I,其中,I、Q、U、V表示不同时刻到达探测器的光子偏振分量的累加值。
将公式(1)、(2)、(3)代入公式(11),其中f为湿度,得到湿度影响下的Mie散射系数为:
$$\left\{ {\begin{aligned} & {a{{\left( f \right)}_n} = \frac{{\psi _n'(m(f)x(f)){\psi _n}(x(f)) - m(f){\psi _n}(m(f)x(f))\psi _n'(x(f))}}{{\psi _n'(m(f)x(f)){\xi _n}(x(f)) - m(f){\psi _n}(m(f)x(f))\xi _n'(x(f))}}}\\ & {b{{(f)}_n} = \frac{{m(f)\psi _n'(m(f)x(f)){\psi _n}(x(f)) - {\psi _n}(m(f)x(f))\psi _n'(x(f))}}{{m(f)\psi _n'(m(f)x(f)){\xi _n}(x(f)) - {\psi _n}(m(f)x(f))\xi _n'(x(f))}}} \end{aligned}} \right.$$ (18) 再将公式(18)代入公式(10),可以得到湿度影响下的散射振幅表达式:
$$\left\{ {\begin{array}{*{20}{l}} {{S_1}(\theta ) = \displaystyle\sum_{{n} = 1}^\infty {\dfrac{{2{n} + 1}}{{{n}({n} + 1)}}} \left[ {a{{(f)}_{n}}\dfrac{{P_n^1(\cos \theta )}}{{\sin \theta }} + b{{({\rm{f}})}_{n}}\dfrac{{P_n^1(\cos \theta )}}{{{\rm{d}}\theta }}} \right]}\\ {{S_2}(\theta ) = \displaystyle\sum_{{n} = 1}^\infty {\dfrac{{2{n} + 1}}{{{n}({n} + 1)}}} \left[ {a{{(f)}_{n}}\dfrac{{P_n^1(\cos \theta )}}{{d\theta }} + b{{({\rm{f}})}_{n}}\dfrac{{P_n^1(\cos \theta )}}{{\sin \theta }}} \right]} \end{array}} \right.$$ (19) 再将公式(19)代入公式(9),可以计算出Mueller矩阵中相应元素,再乘以入射偏振光,最终通过公式(17)即可得到改进的雾霾粒子湿度与偏振传输特性的关系模型。
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室内测试搭建实验系统如图5所示,系统实物图如图6所示。发射端组成为激光器、衰减片、偏振片、波片、滤光片组成。激光器输出功率为50 mW的固体激光器,根据实验波长来选择相应激光器,偏振片透过波段为400~700 nm,各波长透过率可以达到80%以上。接收端分成两路一路由波片、偏振片、光功率计组成,一路由偏振态测量仪组成。
接收端偏振角度为0°、60°、120°,采集的强度信息通过下式计算出相应的偏振信息,并与偏振态测量仪数据做比对。
$$\left\{\begin{aligned} & {{{I}} = \dfrac{1}{2}\left( {{{{I}}^\prime }\left( {60,45,\dfrac{\pi }{2}} \right) + {{{I}}^\prime }\left( {120,45,\dfrac{\pi }{2}} \right) - 3{{{I}}^\prime }\left( {0,45,\dfrac{\pi }{2}} \right)} \right)}\\ & Q = 2{I^\prime }(0,0,0) - \dfrac{1}{2}{I^\prime }(60,45,0) - \dfrac{1}{2}{I^\prime }\left( {120,45,\dfrac{\pi }{2}} \right) \\ & \quad\quad + \dfrac{3}{2}{I^\prime }\left( {0,45,\dfrac{\pi }{2}} \right)\\ & {U = \dfrac{{2\sqrt 3 }}{3}\left( {{{{I}}^\prime }\left( {60,45,\dfrac{\pi }{2}} \right) - {{{I}}^\prime }\left( {120,45,\dfrac{\pi }{2}} \right)} \right)}\\ & {{{V}} = \dfrac{1}{2}\left( {{{{I}}^\prime }\left( {60,45,\dfrac{\pi }{2}} \right) + {{{I}}^\prime }\left( {120,45,\dfrac{\pi }{2}} \right) - {{{I}}^\prime }\left( {0,45,\dfrac{\pi }{2}} \right)} \right)} \end{aligned}\right.$$ (20) 该测量方法增加了圆偏振分量,实现全偏振度测量,弥补了以往仅针对线偏振分量进行偏振特性测量造成的误差。
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通过对450、532、671 nm的6种偏振态在不同湿度的水雾环境中实验数据分析可知:随着水雾环境湿度的增加,偏振度是不断降低的,而且圆偏振光与线偏振光在低湿度30%−40%情况下较为接近,随着湿度的增加,圆偏振光的偏振度明显高于线偏振光。
图7~图9中每一种偏振态都是随着波长的增大而增大,由此可见对于可见光波段水雾环境,较长波长偏振特性保持良好,即在湿度较大的环境中,应尽量选取波长较长的偏振光成像,以达到较好的成像效果。
图 7 波长450 nm偏振度与湿度关系实验图
Figure 7. Relationship between the polarization degree and humidity of 450 nm polarized light experiment
图 8 波长532 nm偏振度与湿度关系实验图
Figure 8. Relationship between the polarization degree and humidity of 532 nm polarized light experiment
图 9 波长671 nm偏振度与湿度关系实验图
Figure 9. Relationship between the polarization degree and humidity of 671 nm polarized light experiment
为了验证仿真与实验的一致性,依据公式(21)计算了仿真与实测的偏振度的置信度。
$$M = 1 - \left( {\sum\limits_1^n {\frac{{\left| {R - {R_m}} \right|}}{{{R_m}}}} } \right)/n \times 100\% $$ (21) 式中:R和Rm分别为同一条件下的仿真和实测得到的偏振度值,由于对7种湿度进行了仿真与实测实验,所以n=7,分别计算了6种偏振态的偏振度置信度,450 nm的0°、90°、45°、135°、左旋、右旋偏振光的置信度分别为63.9%、64.1%、64.2%、64.1%、75.4%、75.5%;532 nm的置信度分别为73.1%、73.2%、73.1%、73%、82.6%、82.5%。671 nm的置信度分别为74.0%、74.1%、74.4%、74.1%、85.4%、85.3%。仿真模型置信度均大于60%,结果可信。
Laser polarization characteristics of visible light band in different humidity environments
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摘要: 雾霾天气严重干扰了可见光成像效果,利用可见光偏振特性可以有效提升探测效率。雾霾环境会受到气溶胶颗粒湿度的影响,湿度是雾霾环境重要物理参数。为了获得可见光在雾霾环境的偏振特性规律,分析了环境湿度对偏振特性的影响。在非偏振光气溶胶单粒子散射特性基础上,采用改进蒙特卡罗方法建立了偏振传输模型,对不同湿度水雾环境下可见波段偏振光传输特性进行研究,重点分析水雾环境湿度改变对不同可见光波段偏振光偏振特性的影响情况并建立了接近真实水雾环境,通过室内实验对偏振模型进行验证,对不同湿度环境下的 450、532 、671 nm 线偏振光与圆偏振光的偏振度与偏振态改变进行了比较与分析,仿真模型置信度>60%。研究结果表明:偏振光的偏振度随水雾环境湿度是呈下降趋势。随着波长的增大,偏振度变化趋于平缓,出射偏振度随着波长的增加而变大,当激光波长为 450、532、671 nm 时,偏振度下降点的湿度值分别为 50%、70%、90%;圆偏振光入射,圆偏振光的旋性对于偏振度没有影响,且高于线偏振度值。对于水雾这种容易受湿度影响较大的环境而言,在可见光波段,应该尽量选择较长波长的偏振光进行传输探测,因为湿度对较短波长的影响比较长波长的影响要大,所以,在湿度较大的环境中,较长波长的圆偏振光是偏振保持特性最好的。在湿度较大的环境中,应尽量选取波长较长的偏振光成像,以达到较好的成像效果。Abstract: Haze weather interferes with the visible light imaging effect, and the polarization characteristic of visible light can effectively improve the detection efficiency. Haze environment is affected by aerosol particle humidity, which is an important physical parameter of haze environment. In order to obtain the polarization characteristics of visible light in haze environment, the influence of humidity in haze environment on polarization characteristics was analyzed. Based on the single particle scattering characteristics of non-polarized light aerosol, the polarization transmission model was established by improved Monte Carlo method, research on transmission characteristics of polarized light in visible bands under different humidity and water mist was conducted, the influence of humidity change in water mist environment on polarization characteristics of polarized light in different visible bands was analyzed, and a near-real water mist environment was built. The polarization model was verified by laboratory experiments, the change of polarization degree and polarization state of linear polarized light on 450, 532 and 671 nm was compared and analyzed under different humidity conditions, the confidence of simulation model was more than 60%. The results show that the polarization degree of polarized light decreases with the increase of humidity of water-fog environment. With the increase of the wavelength, the polarization tends to be flat, while the exit polarization degree increases with the increase of wavelength. The humidity values of the descending point of polarization degree are 50%, 70% and 90% when the laser wavelengths are 450, 532, 671 nm, respectively. For water mist, which is easily affected by humidity, polarized light with longer wave length should be selected as far as possible for transmission detection in visible band. Because humidity has a greater influence on shorter wavelengths than longer wavelengths, circularly polarized light with longer wavelengths has the best polarization retention characteristics in environments with higher humidity. In the environment with high humidity, polarized light imaging with long wavelength should be selected as far as possible to achieve better imaging effect.
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Key words:
- polarization characteristics /
- ambient humidity /
- polarization degree /
- Monte Carlo
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表 1 水雾环境粒子湿度与折射率的关系
Table 1. Relationship between humidity and refractive index of particles in water mist environment
Relative
humidityParticle
radius /μmIndex of
real partIndex of
imaginary part30% 0.548 1.369 1.0298E-4 40% 0.570 1.365 1.3907E-4 50% 0.597 1.361 1.7635E-4 60% 0.632 1.356 2.1518E-4 70% 0.681 1.352 2.5608E-4 80% 0.755 1.347 3.0002E-4 90% 0.902 1.341 3.4926E-4 -
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