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在激光波长尺度,常见水泥路面可以看作随机粗糙表面,当激光照射到物体表面时会发生散射。根据麦克斯韦方程和矢量格林函数可以得到Stratton-Chu方程,由本地场通过Stratton-Chu方程数值积分得到空间散射场。随机粗糙表面粗糙程度可以用概率统计参量表征,无法确定具体的粗糙面曲面方程。现有散射场计算方法有两种,一种是近似解析法,用入射角θi代替大量随机分布的本地面元入射角θi, locall,计算量小但误差较大,结果不够精确。另一种是使用蒙特卡罗方法生成大量随机粗糙表面,将每一个粗糙面网格化离散为大量微面元组合,计算每个面元的本地场,然后数值积分计算空间散射场分布,计算精度高,同时可以处理遮蔽效应、多重散射与大角度入射等情况。光电传感器对电场的敏感度远高于磁场,因此只计算散射场电场分量,空间散射场计算如下式所示[13]:
$ \begin{array}{*{20}{c}} \boldsymbol{E}(\boldsymbol{r}) \approx \sum \approx \sum \boldsymbol{n} \times\left[\nabla \times \boldsymbol{E}\left(\boldsymbol{r}^{\prime}\right)\right] G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)+\\ \left[\boldsymbol{n} \times \boldsymbol{E}\left(\boldsymbol{r}^{\prime}\right)\right] \times G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)+\\ \left.\left[\boldsymbol{n} \cdot \boldsymbol{E}\left(\boldsymbol{r}^{\prime}\right)\right] \nabla^{\prime} G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)\right\} \mathrm{d} S^{\prime} \end{array} $
(1) 式中,E(r)为空间散射场,G(r, r′)≈$\frac{{\exp \left( {{\rm{i}}kR} \right)}}{{4{\rm{ \mathsf{ π} }}R}}$exp(-ikr·r′)为远场格林函数,r′为源点位置,r′为源点位置矢量,r为观测点位置矢量,R为观测点到源点的距离, $k = \frac{{2{\rm{ \mathsf{ π} }}}}{\lambda }$为入射波波数(λ为入射光波长),n为本地面元单位法向量,E(r′)为本地场,$\nabla = {\mathit{\boldsymbol{e}}_x}\frac{\partial }{{\partial x}} + {\mathit{\boldsymbol{e}}_y}\frac{\partial }{{\partial y}} + {\mathit{\boldsymbol{e}}_z}\frac{\partial }{{\partial z}}$为哈密顿算符(ex,ey,ez代表笛卡尔直角坐标系3个方向的单位矢量),${\nabla '}$代表对源点进行求偏导数运算,dS′为微面元面积。
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水泥路面高度起伏服从高斯分布,当条件lc2>2.76δλ成立时(其中lc为相关长度,δ为高度均方根),物体表面曲率半径远大于入射光波长。此时可以将入射点曲面近似为切平面。设入射场电场分量为Ei=(As, isi+Ap, ipi)exp(ikier′),As, i代表s光振幅,Ap, i代表p光振幅,er′代表粗糙面上的入射点位置矢量,即源点位置单位矢量,ki为入射光波矢量,${\mathit{\boldsymbol{s}}_{\rm{i}}} = \frac{{\mathit{\boldsymbol{z}} \times {\mathit{\boldsymbol{k}}_{\rm{i}}}}}{{\left| {\mathit{\boldsymbol{z}} \times {\mathit{\boldsymbol{k}}_{\rm{i}}}} \right|}}$为垂直于入射面方向单位矢量,${\mathit{\boldsymbol{p}}_{\rm{i}}} = \frac{{{\mathit{\boldsymbol{s}}_{\rm{i}}} \times {\mathit{\boldsymbol{k}}_{\rm{i}}}}}{{\left| {{\mathit{\boldsymbol{s}}_{\rm{i}}} \times {\mathit{\boldsymbol{k}}_{\rm{i}}}} \right|}}$为平行于入射面方向单位矢量,本地面元单位法向量为n=${\frac{{ - {f_x}\mathit{\boldsymbol{x}} - {f_y}\mathit{\boldsymbol{y}} + \mathit{\boldsymbol{z}}}}{{\sqrt {1 + f_x^2 + f_y^2} }}}$,其中f(x, y)为表面高度,${{f_x} = \frac{{\partial f}}{{\partial x}}}$,${{f_y} = \frac{{\partial f}}{{\partial y}}}$。本地入射角${{\theta _{{\rm{i}}, {\rm{ locall }}}} = \arccos \left( { - \frac{{{\mathit{\boldsymbol{k}}_{\rm{i}}} \cdot \mathit{\boldsymbol{n}}}}{{\left| {{\mathit{\boldsymbol{k}}_{\rm{i}}}} \right||\mathit{\boldsymbol{n}}|}}} \right)}$,由菲涅耳公式计算本地反射场Er=(rsAs, isr, locall+rpAp, i×pr, locall)exp(ikier′),${\mathit{\boldsymbol{s}}_{{\rm{r}}, {\rm{ locall }}}} = \frac{{\mathit{\boldsymbol{n}} \times {\mathit{\boldsymbol{k}}_{{\rm{r}}, {\rm{ locall }}}}}}{{\left| {\mathit{\boldsymbol{n}} \times {\mathit{\boldsymbol{k}}_{{\rm{r}}, {\rm{ locall }}}}} \right|}}$为垂直于本地面元反射面方向的单位矢量(kr, locall为本地面元反射波波矢量),${\mathit{\boldsymbol{p}}_{{\rm{r}}, {\rm{ locall }}}} = \frac{{{\mathit{\boldsymbol{s}}_{{\rm{r}}, {\rm{ locall }}}} \times {\mathit{\boldsymbol{k}}_{{\rm{r}}, {\rm{ locall }}}}}}{{\left| {{\mathit{\boldsymbol{s}}_{{\rm{r}}, {\rm{ locall }}}} \times {\mathit{\boldsymbol{k}}_{{\rm{r}}, {\rm{ locall }}}}} \right|}}$为平行于本地面元入射面方向的单位矢量,本地场为E(r′)=Ei+Er,其中s光和p光的菲涅耳反射系数如下:
$ \left\{\begin{array}{l} r_{s}=\frac{\cos \theta_{\mathrm{i}, \text { locall }}-\sqrt{n^{2}-\sin ^{2} \theta_{\mathrm{i}, \text { locall }}}}{\cos \theta_{\mathrm{i}, \text { locall }}+\sqrt{n^{2}-\sin ^{2} \theta_{\mathrm{i}, \text { locall }}}} \\ r_{p}=\frac{n^{2} \cos \theta_{\mathrm{i}, \text { locall }}-\sqrt{n^{2}-\sin ^{2} \theta_{\mathrm{i}, \text { locall }}}}{n^{2} \cos \theta_{\mathrm{i}, \text { locall }}+\sqrt{n^{2}-\sin ^{2} \theta_{\mathrm{i}, \text { locall }}}} \end{array}\right. $
(2) 式中,n为水泥路面折射率。将大量微面元本地场代入(1)式进行数值积分, 可以计算空间散射场。
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设计粗糙度测量系统,使用高精度千分表作为测量元件,精度可达0.001mm,使用步进电机控制精密丝杆旋转,每旋转一周丝杆前进0.5mm,可以获取路面高度起伏与采样间隔数据。计算高度均方根与相关长度的方法如下:
$ \delta^{2}=\frac{1}{N-1}\left[\sum\limits_{i=1}^{N}\left(f_{i}^{2}-N \cdot \bar{f}^{2}\right]\right. $
(3) 式中,N为采样点数,fi为水泥路面高度采样点数值,$\bar f = \frac{1}{N}\sum\limits_{i = 1}^N {{f_i}} $为采样高度平均值。
水泥路面高度采样数值为离散数值,其相关系数的计算公式如下:
$ \rho(l)=\frac{\sum\limits_{i=1}^{N+1-j} f_{i} f_{i+j-1}}{\sum\limits_{i=1}^{N} f_{i}^{2}}, (j=1, 2, \cdots, N) $
(4) 式中,l=(j-1)Δx, Δx为采样间隔,当j的取值使ρ(l)最接近1/e时,此时l的值称为相关长度,记作lc;fi和fi+j-1均为水泥路面高度离散采样值。
取A, B两块水泥路面样块进行测量,A和B长宽均约为100mm。经测量,A样块高度均方根为δA=0.54mm,相关长度为lA=6.12mm,为较光滑样块; B样块高度均方根为δB=1.21mm,相关长度为lB=4.23mm,为较粗糙样块。
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常见水泥路面可以看作随机粗糙面,随机粗糙面可以看作不同频率的谐波叠加而成,谐波的振幅是独立的高斯变量,使用不同的功率谱函数对大量服从高斯分布的随机数进行频域滤波,然后进行逆快速傅里叶变换可以得到粗糙面模型[14-15]。
$ \begin{aligned} f\left(x_{m}, y_{n}\right)=& \frac{1}{L_{x} L_{y}} \sum\limits_{m_{k}=-M / 2+1}^{M / 2} \sum\limits_{n_{k}=-N / 2+1}^{N / 2} F\left(k_{m_{k}}, k_{n_{k}}\right) \times \\ & \exp \left[\mathrm{i}\left(k_{m_{k}} x_{m}+k_{n_{k}} y_{n}\right)\right] \end{aligned} $
(5) 其中,
$ \begin{array}{c} F\left( {{k_{{m_k}}}, {k_{{n_k}}}} \right) = 2{\rm{ \mathit{ π} }}\sqrt {{L_x}{L_y}S\left( {{k_{{m_k}}}, {k_{{n_k}}}} \right)} \times \\ \left\{ {\begin{array}{*{20}{l}} {\frac{{N(0, 1) + {\rm{i}}N(0, 1)}}{{\sqrt 2 }}, \left( {{m_k} \ne 0, \frac{M}{2}{\kern 1pt} {\kern 1pt} {\rm{and}}{\kern 1pt} {\kern 1pt} {n_k} \ne 0, \frac{M}{2}} \right)}\\ {N(0, 1), \left( {{m_k} = 0, \frac{M}{2}{\kern 1pt} {\kern 1pt} {\rm{or}}{\kern 1pt} {\kern 1pt} {n_k} = 0, \frac{M}{2}} \right)} \end{array}} \right. \end{array} $
(6) 式中, mk和nk为离散变量,取值范围分别为$\left[ {\frac{{ - M}}{{2 + 1}}, \frac{M}{2}} \right]$,$\left[ {\frac{{ - N}}{{2 + 1}}, \frac{N}{2}} \right]$之间的整数,S(kmk, knk)=${\delta ^2}\frac{{{l_x}{l_y}}}{{4{\rm{ \mathsf{ π} }}}}\exp \left( { - \frac{{k_{{m_k}}^2l_x^2 + k_{{n_k}}^2l_y^2}}{4}} \right)$为高斯型功率谱密度函数,lx和ly分别为粗糙面x方向和y方向的相关长度,Lx和Ly分别为粗糙面的长度和宽度,M, N为采样点数,xm=mΔx, yn=nΔy为采样点,Δx和Δy为采样间隔,N(0, 1)为服从标准正态分布的随机数,${k_{{m_k}}} = \frac{{2{\rm{ \mathsf{ π} }}{m_k}}}{{{L_x}}}$和${k_{{n_k}}} = \frac{{2{\rm{ \mathsf{ π} }}{n_k}}}{{{L_y}}}$为离散波数。设置粗糙面长度和宽度为Lx=Ly=100mm,采样点数N=1024×1024,生成2维高斯型随机粗糙表面, 如图 1所示。高度均方根δ=0.3mm,相关长度lc=6mm。
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仿真参量设置如下:令粗糙面长度和宽度为Lx=Ly=100mm,采样点数N=1024×1024,入射角θi=30°,入射光波长λ=650nm,水泥样块折射率为n=1.5。入射光偏振状态为典型的s光与p光。s光偏振方向垂直于入射面,p光偏振方向平行于入射面。随机粗糙面高度均方根δ为0.3mm, 0.6mm, 0.9mm, 1.2mm;相关长度为lc=6mm。以上参量均与水泥路面比较样块统计参量在同一数量级,便于研究散射光分布特征。使用频域滤波生成10000个随机粗糙面表面,计算散射场平均值。
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当s光与p光入射水泥路面时,散射光强度分布如图 2所示。s光入射时散射光强度明显大于p光入射,约为其2倍。在高度均方根δ=0.3mm时,在30°镜像方向具有明显的峰值,随着高度均方根的增大,峰值逐渐减小,分布逐渐展宽,均出现峰值位置移动的现象。s光入射时,散射光峰值位置逐渐向散射角增大方向移动,p光入射时向散射角减小方向移动。结果表明, 相关长度lc与高度均方根δ同时影响随机粗糙表面光学粗糙度,相关长度lc越小,高度均方根δ越大,则粗糙面光学粗糙度越大。不同粗糙程度的水泥路面对入射光峰值大小与位置产生调制作用。
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对于单站式激光测量系统,如激光雷达,后向散射特性对于目标探测与识别具有重要意义,s光和p光入射水泥路面其后向散射光强度分布如图 3所示。随着高度均方根δ的增大,后向散射光强度随入射角的增大其衰减速度变慢,对于p光与s光,其散射规律相近。当均方根增大到一定程度,粗糙面后向散射表现出朗伯体的散射特征,原因是粗糙度增大造成的多重散射效应使光强度空间分布较为均匀。
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高度均方根δ增大,本地面元入射角分布将发生改变,每种均方根高度生成10000个粗糙面,统计本地面元入射角的频率分布,如图 4所示。高度均方根δ=0.3mm时,本地面元入射角θi, locall在11°~48°之间,相对集中于镜像附近,随着高度均方根δ的增大,本地面元入射角范围逐步扩大,分布逐渐展宽。本地面元对s光反射率为Rs=rs2,对p光反射率为Rp=rp2。当θi, locall变化时,反射率变化如图 4所示。本地入射角概率密度函数p(θi, locall)和反射率的乘积与散射光强度成正比,使用p(θi, locall)与反射率的乘积拟合散射光强度,如图 5所示。在不同本地入射角条件下,面元对s光的反射率随入射角的增大单调增大,s光入射时水泥路面散射光呈现明显的峰值位置右移。p光在比镜像角度大的方向,面元对p光的反射率在不断减小。随着本地入射角θi, locall分布不断展宽,p光入射时镜像右侧的散射光强度减小,水泥路面出现峰值位置左移的现象。由于反射率Rs>Rp,所以s光入射时, 散射光强度明显大于p光入射情况。
水泥路面激光散射特性研究
Research on laser scattering characteristics of cement road surface
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摘要: 为了通过激光散射特性识别不同粗糙程度水泥路面, 设计了路面粗糙度测量系统。使用千分表测量水泥路面高度分布, 计算高度均方根与相关长度。根据以上参量采用功率谱频域滤波生成随机粗糙表面以模拟水泥路面, 通过切平面近似将粗糙面离散为大量微面元, 由菲涅耳公式计算本地场, 利用蒙特卡罗方法获取不同偏振光入射条件下粗糙面双向与后向散射光强度统计平均值。基于虚拟仪器技术进行高精度自动化激光散射测量, 并根据实验数据对理论模型进行了验证。结果表明, 双向散射小粗糙度水泥路面散射光强度在镜像方向散射角40°附近具有峰值90lx, 在镜像方向两侧逐步递减, 大粗糙度水泥路面光学特性近似为朗伯体, 散射光强度在各散射角方向变化不大; 后向散射在垂直入射时, 小粗糙度水泥路面散射光强度峰值为103lx, 随散射角增大逐渐递减, 大粗糙度水泥路面具有朗伯体散射特征。此研究结果可为车辆自主驾驶方案路面信息感知提供参考。Abstract: In order to identify cement road surface with different roughness by laser scattering characteristic, a road surface roughness measurement system was designed. The height distribution of cement road surface was measured by micrometer, and the root mean square of height and relevant length were calculated. According to the above parameters, the random rough surface was generated by power spectral frequency domain filtering to simulate cement road surface, and the rough surface was dispersed into a large number of surface elements by tangential plane approximation. The local field was calculated by Fresnel formula, and the Monte Carlo method was used to obtain the statistical mean values of bidirectional and the back scattered light intensity on rough surface under different polarized light incident conditions. Based on the virtual instrument technology, the high precision automatic laser scattering measurement was carried out, and then the theoretical model was verified according to the experimental data. The results show that the scattered light intensity of bidirectional scattered cement road surface with small roughness has a peak value of 90lx near the scattering angle of 40° in the mirror direction, and decreases gradually on both sides of the mirror direction. The optical characteristics of cement road surface with large roughness are approximate to Lambert body, and the scattered light intensity has little change in each scattering angle direction. The back scattered light intensity peak of small roughness cement road surface is 103lx when the back scattering is perpendicular to the incidence, which decreases gradually with the increase of scattering angle. The large roughness cement road surface has Lambert scattering characteristics. The above conclusions could provide reference for road information perception of autonomous driving schemes.
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Key words:
- scattering /
- virtual instrument /
- Monte Carlo method /
- cement road surface /
- laser
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