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基于光学偏折技术的子孔径拼接测量系统原理图如图1所示,系统组成部分包括LCD投影屏、CCD相机、被测元件以及子孔径扫描机构。在图1所示的条纹投影偏折测量系统中,利用LCD投影屏将编码正弦条纹投射到被测反射元件表面,并通过CCD相机采集经被测元件表面反射后的变形条纹。利用五轴调整架作为子孔径扫描机构,通过调节被测元件位置来实现各个测量子孔径区域的调整,由此实现被测元件全口径覆盖。在各个子孔径区域测量中,LCD投影屏上的发光像素点、被测区域投影位置与CCD相机上亮点像素之间存在一一对应关系,可据此确定各入射光线与对应的反射光线,以此计算得到被测面形的投影位置在x和y方向的斜率,进而实现被测元件的面形测量。
图 1 基于光学偏折技术的子孔径拼接测量系统原理图
Figure 1. Schematic diagram of optical deflectometric system for sub-aperture stitching measurement
按照实际测量系统的结构参数可建立对应的系统光线追迹模型,由此获得接收像面上的理想光斑分布(xmodel, ymodel);在实际测量中,利用条纹相移技术测得被测面形在投影屏上的实际光斑分布(xtest, ytest)。被测面相对其理想面的斜率偏差(∆wx, ∆wy)可通过理想光斑分布(xmodel, ymodel)与实际光斑分布(xtest, ytest)的坐标偏差(∆xspot, ∆yspot)计算得到,即
$$\left\{ \begin{gathered} \Delta {w_x} = \frac{{\partial W\left( {x,y} \right)}}{{\partial x}} = \frac{{{x_{{\rm{test}}}} - {x_{{\rm{model}}}}}}{{2{d_{{\rm{s2s}}}}}} = \frac{{\Delta {x_{{\rm{spot}}}}}}{{2{d_{{\rm{s2s}}}}}} \\ \Delta {w_y} = \frac{{\partial W\left( {x,y} \right)}}{{\partial y}} = \frac{{{y_{{\rm{test}}}} - {y_{{\rm{model}}}}}}{{2{d_{{\rm{s2s}}}}}} = \frac{{\Delta {y_{{\rm{spot}}}}}}{{2{d_{{\rm{s2s}}}}}} \\ \end{gathered} \right.$$ (1) 式中:ds2s为被测元件与投影屏的距离;(x, y)为测量系统出瞳面上的坐标;W(x, y)为被测面引入的波前像差。根据公式(1)可求得实际面形与其理想面形的斜率偏差(∆wx, ∆wy),通过对斜率偏差(∆wx, ∆wy)进行表面积分,即可获得被测元件的面形信息。
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图1所示的光学偏折测量系统最大可测的斜率范围∆L与LCD投影屏的尺寸、系统孔径有关,并且有
$$\Delta L = {L_{\rm screen}}/2{d_{\rm s2s}}$$ (2) 式中:Lscreen为LCD投影屏的实际有效尺寸。因此可以通过增大LCD投影屏的有效尺寸以及调整被测元件与LCD投影屏、CCD相机之间相对距离的方式,实现对更大口径及更高斜率动态范围光学元件表面的测量。但是当被测元件的口径及斜率动态范围过大而超出系统可测范围时,依旧存在无法实现全口径测量的问题。此时可对被测元件各部分区域(子孔径)进行测量,再将其拼接得到全口径面形。
在CCD相机的视场范围中,通过调整五轴调整架来改变被测元件的位姿,使被测面形部分区域的变形条纹保持在合理的密集度。在当前位姿状态下,可同时利用五轴调整架对被测元件进行子孔径扫描,再通过CCD相机逐次采集各子孔径数据,直至覆盖全口径面形。在子孔径划分设计中应遵循子孔径间重叠区域大于子孔径区域的1/4,同时子孔径的数目尽量少的原则[12]。
实际测量中,由于被测元件的移动会导致不同子孔径之间产生倾斜、平移和离焦误差。因此需要计算各子孔径之间对应的倾斜、平移与离焦系数,将各个子孔径数据变换到统一坐标下进行拼接处理。对于任意两个重叠区域的子孔径,其面形之间的关系可表示为:
$$ {w_m}(x,y) - {w_n}(x,y) {\rm{ = }}{A_{m{\rm{,}}n}}x + {B_{m{\rm{,}}n}}y + {C_{m{\rm{,}}n}}\left( {{x^{\rm{2}}}{\rm{ + }}{y^{\rm{2}}}} \right) + {D_{m{\rm{,}}n}} $$ (3) 式中:wm(x, y)与wn(x, y)分别表示第m、n个子孔径的面形数据;Am, n和Bm, n分别表示第m个子孔径相对于第n个子孔径在x、y方向的倾斜系数;Cm, n和Dm, n分别为其对应的离焦系数和平移系数。根据最小二乘拟合法,在子孔径间的重叠区域面形平方和最小条件下,可求取各子孔径间的倾斜、平移与离焦系数,即
$$\begin{split} {\rm{min}} =& \sum\limits_m {\sum\limits_n^{m \cap n} {\left\{ {{w_m}(x,y) - {w_n}(x,y) - } \right.} } \\& {A_{m{\rm{,}}n}}x - {B_{m{\rm{,}}n}}y - {C_{m{\rm{,}}n}}({x^{\rm{2}}}{\rm{ + }}{y^{\rm{2}}}) - {\left. {{D_{m{\rm{,}}n}}} \right\}^2} \end{split} $$ (4) 在重叠区域数据的计算处理中,利用传统均值计算法获取的拼接面形难免会产生“拼接痕迹”。为实现拼接面形的平滑过渡,可采用基于子孔径重叠区域位姿的加权融合算法。以图2所示的任意两个具有重叠区域的子孔径为例,以该重叠区域的质点P与距质点最远点Q两点间连线
$PQ$ 和x轴正半轴夹角的绝对值α作为任一子孔径位姿方位角的表征值。根据子孔径相对于x轴的位姿分布特征,可设定方位临界值α0(如α0 = π/4),当方位角α > α0时,取在重叠区域任一横坐标区间[x1, x2]内渐变系数为kx;当方位角α ≤ α0时,取在重叠区域任一纵坐标区间[y1, y2]内渐变系数为ky。其中kx、ky的取值范围均为[0, 1],并且有
$$\left\{ \begin{gathered} {k_x} = 0.5 - 0.5\cos \left( {\frac{{x - {x_1}}}{{{x_2} - {x_1}}}} \right)\pi \\ {k_y} = 0.5 - 0.5\cos \left( {\frac{{y - {y_{\rm{1}}}}}{{{y_2} - {y_1}}}} \right)\pi \\ \end{gathered} \right.$$ (5) 因此,两子孔径间重叠区域的面形w可表示为:
$$ w=\left\{ {\begin{array}{*{20}{c}} {\left(1-{k}_{x}\right){w}_{1}+{k}_{x}{w}_{\rm{2}},\;\;{\rm{if}}\;\;\alpha \leqslant {\alpha }_{0}}\\ {\left(1-{k}_{y}\right){w}_{1}+{k}_{y}{w}_{2},\;\;{\rm{if}}\;\;\alpha >{\alpha }_{0}} \end{array}} \right.$$ (6) 式中:w1、w2分别为任意两个子孔径在重叠区域内的面形数据。由公式(5)和公式(6)可得,任一重叠区域内的渐变系数变化趋势均为平滑上升的曲线[13],因而能够有效避免重叠区域边缘点突变导致的“拼接痕迹”。当渐变系数为kx时,由kx的变化趋势可得,越靠近重叠区域任一区间端点x1时造成“拼接痕迹”的面形数据w1的权重越低,越靠近重叠区域任一区间端点x2时造成“拼接痕迹”的面形数据w2的权重越低;当渐变系数为ky时,同理如上。利用上述处理方法可以改善“拼接痕迹”,实现拼接面形的平滑过渡。
Sub-aperture stitching deflectometric testing technology for optical surfaces
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摘要: 针对大口径的高斜率动态范围光学元件的测量需求,提出了基于光学偏折技术的子孔径拼接测量方法。利用所搭建的条纹投影光学偏折测量系统,结合子孔径划分拼接方法,对各子孔径分别进行测量,并根据实际测量结果与测量系统模型光线追迹结果的偏差,高精度测得各个子孔径的面形数据,由此对各子孔径进行拼接来实现全口径面形测量。光学偏折测量技术相对干涉法具有很大的测量动态范围和视场,可极大降低所需的子孔径数量,由此大大提高了检测效率。同时提出了针对重叠区域的加权融合算法来实现拼接面形的平滑过渡。为验证所提出方案的可行性,分别进行了仿真分析以及实验验证。对一高斜率反光灯罩进行拼接测量实验,并将拼接测量与全口径测量结果进行对比。结果表明,利用所提出测量方法获得的拼接面形连续光滑,且与全口径测量面形RMS值偏差为0.0957 µm,优于微米量级。该测量具有较高的测量精度和大动态测量范围,并且系统结构简单,为各类复杂光学反射元件提供了一种有效可行的检测方法。Abstract: Aiming at the measurement needs of large-aperture optical elements with ultra-large dynamic range, a sub-aperture stitching testing method based on optical deflectometry was proposed. According to the surface feature, the sub-apertures were divided and sequentially measured with the proposed fringe-illumination deflectometric testing system. Based on the slope data measured with actual testing system and ray-tracing result in the system model, the tested surface in each sub-aperture could be reconstructed with high accuracy and be stitched for full-field testing. Compared with the interferometric testing method, the optical deflectometry testing was larger in dynamic range and field of view, which could greatly reduce the number of subapertures required, thus greatly improving the measurement efficiency. Additionally, a weighted-fusion algorithm based on overlapped regions was proposed to obtain the smooth stitching result. To demonstrate the feasibility of the proposed method, both the numerical analysis and experimental verification were carried out. The high accuracy and large dynamic range were validated in the reflective lampshade testing. The result shows that the stitched surface obtained by the proposed method is consistent and smooth, and its surface deviation RMS compared with the full-aperture measurement result is 0.0957 µm, which is smaller than microns. The proposed method is high in measurement accuracy, large in dynamic range and also simple in system configuration, providing an effective and feasible testing method for various optical elements with complex reflective surfaces.
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图 4 仿真中拼接面形与残差分布。(a) 基于加权融合拼接法获取的拼接面形;(b) 真实面形;(c) 图(a)与图(b)的残差分布;(d) 均值计算法获取的拼接面形与真实面形残差分布
Figure 4. Stitched surfaces and residual errors in simulation. (a) Stitched surface with weighted-fusion method; (b) Nominal surface; (c) Residual error between (a) and (b); (d) Residual error between stitched surface based on mean calculation method and nominal surface
图 8 实验中反光灯罩全口径测量结果与残差分布。(a) 基于加权融合拼接法的测量结果;(b) 激光扫描仪测量结果;(c) 图(a)与图(b)的残差分布;(d) 均值计算拼接法的测量结果与扫描仪测量结果残差分布
Figure 8. Full-aperture measurement results and residual errors for a reflective lampshade in experiment. (a) Measurement result with weighted-fusion method; (b) Measurement result with a laser scanner; (c) Residual error between (a) and (b); (d) Residual error between measurement result stitched with mean calculation method and that with laser scanner
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