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同步辐射镜外形多为狭长的长方体,长度一般在100 mm到1 m之间。因此,除非采用掠入射方式,基本上很难使用常见的100 mm或150 mm口径的菲索干涉仪直接测量整个镜面。此外,在测量曲面镜时,为减少由机械震动和相移不准引起的相位测量误差和回程误差,文中尽量采用近零位检测的方式,于是,待测镜面的局部曲率也限制着有效的视场范围。为了能在视场受限的状态下,测量整个有效光学区域的面形,子孔径拼接干涉技术是一种较为合理的测量解决方案。
简单地讲,拼接的主要目的是通过计算所有子孔径面形数据的相对几何关系来重构整个镜面面形。如图1所示,子孔径面形数据之间的几何关系主要包括x方向的倾斜量anx,和y方向的倾斜量bny以及z方向的平移量cn。图1和公式(1)中的这些几何参数an,bn和cn可以通过重叠区域内的冗余测量数据估计出来[24, 25],或是通过其他的距离或角度测量来确定[14-17, 19, 21, 26]。一般地,通过拼接调整后的第n块高度数据
${z_n}\left( {x,y} \right)$ 可表达为$${z_n}\left( {x,y} \right) = {m_n}\left( {x,y} \right) + {a_n}x + {b_n}y + {c_n},\;n = 1,2, \cdots ,N$$ (1) 式中:
${m_n}\left( {x,y} \right)$ 是总共N块子孔径测量数据中的第n块高度数据。公式中的几何参数an, bn和cn分别是第n块子孔径数据的x方向斜率,y方向斜率和z方向平移的调整量。这些几何参数一旦确定,拼接的主要工作就基本完成了。考虑到在测量X光反射镜时,通常并不关心最终拼接结果的总体倾斜量和z方向平移量,所以,为拼接N块子孔径数据,总共需要确定$3\left( {N - 1} \right)$ 个几何参数。图 1 拼接的根本目的是为了确定子孔径数据之间的几何关系
Figure 1. The essential purpose of stitching is to determine the geometric relation for the subset data
当所有子孔径数据的几何位置按照所得的几何参数全部调整好后,属于不同子孔径的高度数据还要以一定的权重函数融合为一整块最终的镜面面形结果。笔者通常使用Hann窗作为的权重函数来进行数据融合。
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在纯软件拼接模式下,拼接的基本思路是利用最小二乘法从重叠区域的测量数据中估计出每一块子孔径数据的几何参数。在重叠区域内,两块相邻子孔径的高度差异理论上应为一个倾斜的平面。拟合相邻的第i和第j个子孔径的在重叠区域内的差异斜面(共K个数据点),由
$$\underbrace {\left[ {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1\\ \vdots & \vdots & \vdots \\ {{x_K}}&{{y_K}}&1 \end{array}} \right]}_{{{{C}}_{{{ij}}}}}\underbrace {\left[ {\begin{array}{*{20}{c}} {{\alpha _{ij}}}\\ {{\beta _{ij}}}\\ {{\gamma _{ij}}} \end{array}} \right]}_{{{{r}}_{ij}}} \buildrel \Delta \over = \underbrace {\left[ {\begin{array}{*{20}{c}} { - {m_i}\left( {{x_1},\;{y_1}} \right) + {m_j}\left( {{x_1},\;{y_1}} \right)}\\ { - {m_i}\left( {{x_2},\;{y_2}} \right) + {m_j}\left( {{x_2},\;{y_2}} \right)}\\ \vdots \\ { - {m_i}\left( {{x_K},\;{y_K}} \right) + {m_j}\left( {{x_K},\;{y_K}} \right)} \end{array}} \right]}_{{{{m}}_{ij}}},$$ (2) 式中:“
$ \buildrel \Delta \over = $ ”表示其两端的表达式在最小二乘的意义下相等。这两个子孔径数据的相对几何关系参数${{r}_{ij}} = {\left[ {{\alpha _{ij}},{\beta _{ij}},{\gamma _{ij}}} \right]^ \top }$ 可以通过线性最小二乘法确定为$${{{r}}_{ij}} = {\left( {{{{C}}_{ij}}^ \top {{{C}}_{ij}}} \right)^{ - 1}}{{{C}}_{ij}}^ \top {{{m}}_{ij}}$$ (3) 每一组测得的关系参数rij被认为是相邻子孔径相对几何关系的“测量值”来对待,进一步建立新的几何关系方程以求解每个子孔径的几何位置参数p。
$$\begin{split} & \underbrace {\left[ {\begin{array}{*{20}{c}} 1&0&0& \cdots &0& \cdots &0& \cdots &0&0\\ { - 1}&1&0& \cdots &0& \cdots &0& \cdots &0&0\\ { - 1}&0&1& \cdots &0& \cdots &0& \cdots &0&0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0&0&0& \cdots &{ - 1}& \cdots &1& \cdots &0&0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0&0&0& \cdots &0& \cdots &0& \cdots &{ - 1}&1 \end{array}} \right]}_{{D}} \\ & \underbrace {\left[ {\begin{array}{*{20}{c}} {{{{p}}_1}}\\ {{{{p}}_2}}\\ {{{{p}}_3}}\\ \vdots \\ {{{{p}}_i}}\\ \vdots \\ {{{{p}}_j}}\\ \vdots \\ {{{{p}}_{N - 1}}}\\ {{{{p}}_N}} \end{array}} \right]}_{{p}} \buildrel \Delta \over = \underbrace {\left[ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ 0\\ \end{array}} \right]}\\ {{{{r}}_{12}}}\\ {{{{r}}_{13}}}\\ \vdots \\ {{{{r}}_{ij}}}\\ \vdots \\ {{{{r}}_{N - 1,N}}} \end{array}} \right]}_{{r}}, \end{split}$$ (4) 式中:向量
$1 = \left[ {\begin{array}{*{20}{c}}1&1&1\end{array}} \right]$ ,$0 = \left[ {\begin{array}{*{20}{c}}0&0&0\end{array}} \right]$ ,第一块子孔径p1的几何参数被约束为${{{p}}_1} = {\left[ {\begin{array}{*{20}{c}}0&0&0\end{array}} \right]^ \top }$ ,该方程的最小二乘解为$${{p}} = {\left( {{{{D}}^ \top }{{D}}} \right)^{ - 1}}{{{D}}^ \top }{{r}}$$ (5) 至此,已经从重叠区域的数据中,计算出每个子孔径的几何参数以拼接出整体面形。
更深入地,在数据采集时,笔者等获得了大量的高重叠率的冗余数据,完全可以从这些冗余数据中提取出额外信息来更好的了解测量设备中的加性高阶系统误差。关于这方面的内容,请详见笔者等近期的另一工作[27]。
Study on stitching interferometry for synchrotron mirror metrology
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摘要: 随着同步辐射光源和自由电子激光器相关技术的发展和光束质量的提升,对用于转递和聚焦光束能量的X光反射镜的指标要求也逐渐提高。为避免引入额外的波前误差,反射镜面形高度误差均方根值的要求已逼近至亚纳米量级。如此苛刻的面形要求对X光反射镜的测量工作带来了极大的困难和挑战。除了在各国同步辐射光源得到广泛使用的长程轮廓仪等基于角度测量的轮廓扫描仪器之外,基于激光干涉仪的拼接干涉技术也发展为测量同步辐射镜的一种有效手段。文中主要介绍了近期笔者等为测量X光反射镜而开发的拼接干涉平台。利用这一测量平台,研究了在不同的拼接参数下的多种拼接模式。着重讲述了其中纯软件拼接模式的基本原理和实际测量。用实测结果与不同测量仪器和不同研究机构的结果进行比对,验证了拼接干涉测量用于检测同步辐射镜的有效性,并展示了此拼接平台的测量表现。根据所得的测量数据看来,使用纯软件拼接模式来测量X光平面反射镜时,测量重复性的均方差值可以达到0.1 nm左右;而测量X光双曲柱面镜时(曲率半径的变化范围为50~130 m),测量重复性的均方差值为0.2~0.3 nm。此结果基本满足平面和接近平面(曲率半径大于50 m)的同步辐射镜常规检测和为确定性加工提供面形反馈的需要。Abstract: With the quick development and progress of the synchrotron radiation and free electron laser facilities, the figure error requirement for X-ray delivering and focusing mirrors is getting higher. To preserve the wave-front without introducing additional wave-front error, the mirror surface figure error is typically specified approaching the sub-nm root mean square level. This kind of ultimate specification challenges the X-ray mirror metrology and inspection. In addition to the profile scanners based on the angular measurement, such as the long trace profilometer, which have been widely used in the synchrotron light source in various countries, stitching interferometry has also been developed as an effective method for synchrotron mirror metrology. In this work, the dedicated stitching metrology platform for X-ray mirror metrology was developed. Several stitching methods with varying stitching parameters were investigated at the proposed stitching interferometric system. In this paper, we focused on the principle and measurements of using the software stitching mode. The measurement results were compared with those from several different instruments in different research institutions to verify the effectiveness of the stitching interferometry for the synchrotron mirror inspection and to demonstrate the performance of the stitching platform. According to the measurement data, the repeatability of measuring an X-ray flat mirror in the software stitching mode is at 0.1 nm RMS level. When measuring a curved mirror with its radius of curvature changing from 50 m to 130 m, the repeatability is around 0.2-0.3 nm RMS. Basically, it meets the requirement for the routine inspection and fabrication feedback of flat and near-flat (radius of curvature larger than 50 m) X-ray mirrors.
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图 4 纯软件拼接模式下,比较了采用不同大小的子孔径(黄色高亮区域)来测量X光平面镜的拼接结果:(a)600 pixel
×150 pixel;(b)256 pixel×64 pixel Figure 4. Different sub-apertures (highlighted in a yellow rectangle) with (a) 600 pixel×150 pixel and (b) 256 pixel×64 pixel are used to measure this X-ray flat mirror in software stitching mode for comparison
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