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在实际应用中,使用泽尼克(Zernike)多项式等圆域连续基底,不仅会出现非圆域正交性退化的问题,同时在离散取样时,其表征精度会出现相应的下降。在此选取傅里叶级数对波前进行表达,同时,由于傅里叶变换可以使用快速算法,其计算速度也可以大幅提高。
设最高的拟合阶数为N,将波前Φ(x, y)表示为离散傅里叶级数,如公式(1)所示:
$$\varPhi (x,\;y) = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\tilde \varPhi \left( {n,m} \right)} } {{\rm{e}}^{j\frac{{nx + my}}{N}}}$$ (1) 式中:n、m为整数;x、y为离散空间坐标;
$\tilde \varPhi $ (n, m)为离散傅里叶级数系数。将公式(1)变形可得公式(2):$$\tilde \varPhi (m,\;n) = \sum\limits_{x = 1}^N {\sum\limits_{y = 1}^N {\varPhi \left( {x,\;y} \right)} } {{\rm{e}}^{ - j\frac{{nx + my}}{N}}}$$ (2) 将公式(1)两端进行梯度算子:
$${\nabla ^2}\varPhi (x,\;y) = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{n^2} + {m^2}}}{{{N^2}}}\tilde \varPhi \left( {n,\;m} \right)} } {{\rm{e}}^{j\frac{{nx + my}}{N}}}$$ (3) 曲率传感器由Roddier于1988年提出[16],其基本原理是光瞳处波前局部的曲率变化,所对应的焦内像与焦外像的光强分布会发生对应的变化。根据近场电磁波的传输方程可以解算出波前信息,曲率传感因具有非干涉(无需参考光)、结构简单、环境适应性好、解算稳定(无需迭代)、孔径遮拦影响小以及动态范围大(无需相位解缠)等诸多独特优势,具有非常好的应用价值。
根据曲率传感基本原理,波前相位的曲率与光强沿光轴方向的差分成正比,如公式(4)所示:
$$\frac{{{I_1}(x,\;y) - {I_2}(x,\;y)}}{{{I_0} \cdot 2\Delta {\textit{z}}}} \approx - {\nabla ^2}\varPhi (x,\;y)$$ (4) 式中:I1(x, y)、I2(x, y)为焦前焦后的能量分布;I0为光强总值;Δz为离焦量。结合公式(3)~(4)可得:
$$\begin{array}{l} \dfrac{{{I_1}(x,\;y) - {I_2}(x,\;y)}}{{{I_0} \cdot 2\Delta {\textit{z}}}} = - \displaystyle\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\dfrac{{{n^2} + {m^2}}}{{{N^2}}}\tilde \varPhi \left( {n,m} \right)} } {{\rm{e}}^{j\frac{{nx + my}}{N}}} \end{array} $$ (5) 对公式(5)两端进行变形可得公式(6):
$$ \begin{split} & \displaystyle\sum\limits_{x = 1}^N {\sum\limits_{y = 1}^N {\dfrac{{{I_1}(x,\;y) - {I_2}(x,\;y)}}{{{I_0} \cdot 2\Delta {\textit{z}}}}{{\rm{e}}^{ - j\frac{{nx + my}}{N}}}} } = \\ & -\displaystyle\sum\limits_{x = 1}^N {\sum\limits_{y = 1}^N {\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{n^2} + {m^2}}}{{{N^2}}}\tilde \varPhi \left( {n,\;m} \right)} } {\rm{e}}^{j\frac{{nx + my}}{N}}} {{\rm{e}}^{ - j\frac{{nx + my}}{N}}}} \end{split} $$ (6) 进而可得公式(7),通过公式(1)可得波前Φ(x, y):
$$\tilde \varPhi \left( {n,\;m} \right){\rm{ = }}\sum\limits_{x = 1}^N {\sum\limits_{y = 1}^N {\frac{{{I_2}(x,\;y) - {I_1}(x,\;y)}}{{\left( {{n^2} + {m^2}} \right){I_0} \cdot 2\Delta {\textit{z}}}}{{\rm{e}}^{ - j\frac{{nx + my}}{N}}}} } $$ (7) 类似地,可以建立
$\tilde \varPhi $ (n, m)与斜率S(x, y)的关系,如公式(8)所示:$$\tilde \varPhi \left( {n,\;m} \right){\rm{ = }}\sum\limits_{x = 1}^N {\sum\limits_{y = 1}^N {\frac{{S(x,\;y)}}{{n + m}}{{\rm{e}}^{ - j\frac{{nx + my}}{N}}}} } $$ (8) -
大口径光学元件系统性能评价和误差分配是一个复杂的系统工程,其涉及面广、环节众多、与实际应用需求联系极为紧密。误差分析、分配准则的研究本质上是建立大口径望远镜科学目标与望远镜实际性能指标之间的关系。不仅如此,性能评价和误差分配实际上是相互关联的,即误差分配本身是一个平衡过程:在既定的性能评价标准下,需要综合考虑系统误差的敏感性和实际加工、生产能力以及成本等因素,进而实现误差的分配。标准化点源敏感性(Point Source Sensitivity,PSSn)是点扩散函数在成像区域的平均[17-20]。PSSn的定义与计算方法如公式(9)所示:
$$PSSn = \dfrac{{\displaystyle\iint\limits_{A(x,\;y)} {{{\left| {PS{F_e}} \right|}^2}}{{\left| {PS{F_{t + a}}} \right|}^2}}}{{\displaystyle\iint\limits_{A(x,\;y)} {{{\left| {PS{F_{t + a}}} \right|}^2}}}} = \dfrac{{\displaystyle\iint\limits_{A(fx,\;fy)} {{{\left| {OT{F_e}} \right|}^2}}{{\left| {OT{F_{t + a}}} \right|}^2}}}{{\displaystyle\iint\limits_{A\left( {x,\;y} \right)} {{{\left| {OT{F_{t + a}}} \right|}^2}}}} $$ (9) 式中:PSFe、PSFt+a为望远镜误差的点扩散函数以及理想望远镜在视宁影响下的点扩散函数;OTFe、OTFt+a为望远镜误差的光学传递函数以及理想望远镜在视宁影响下的光学传递函数;A(x, y)与A(fx, fy)为空间域与频域的积分区域。根据公式(1),可选择合适的方式计算PSSn。
Seo在2009年提出PSSn之后,TMT全频段的误差分配均使用PSSn进行表征。Angeli在2016年对LSST进行了系统级建模,利用PSSn分析了各个频段的误差[21]。Angeli在2018年利用PSSn对GMT自然视宁与地表自适应光学两种工作模式行了误差分配,引入进入大于1的PSSn,用于模拟自适应光学中频误差校正环节[22]。
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对流方向模拟实际使用过程中的不同方向采样,并通过对自准直仪示数的直接分析定性地获得湍流强度,指导系统进行初步的温控与湍流调整,同时利用公式(5)也可以对镜面视宁进行初步计算,如图1所示。
为了兼顾测量动态范围以及测量精度,在此采用量级测量的策略,即采用动态范围较大的斜率测量对强湍流情况进行估计,之后采用曲率传感测策略对镜面视宁度进行更加精细与准确的测量。
在进行斜率探测后,通过自准直仪后端光路的调焦将系统进行一定量的离焦(两台自准直仪的离焦方向需要相反),并通过波前沿光轴方向的微分获得波前的估计。获得数据后,基于PSSn这一频域指标对结果进行表征与分析,以期更全面地理解镜面视宁度作用的内在物理机制与统计学特性,最终平衡镜面视宁度、光学误差以及加工成本之间的关系。由于斜率测量采用大像元尺寸(提高读出速度与读出噪声),当其测量值接近其测角精度时,说明热控系统已经抑制部分湍流,在此情况下可切换曲率传感。
具体来说,采用系统测量分为两级,分别是大范围的斜率测量以及小量程高精度的曲率测量。由于斜率探测仅需要几个像元,动态范围较大,因此作为初始阶段的镜面视宁度检测手段。
在镜面视宁度精确测量中,曲率传感因具有非干涉(无需参考光)、结构简单、环境适应性好、解算稳定(无需迭代)、孔径遮拦影响小以及动态范围大(无需相位解缠)等诸多独特优势,具有非常好的应用价值。但高精度的离焦器件系统限制了其应用的场景,在此采用分体球面镜实现固定的离焦量,并最终实现单次曝光测量的小型化、静态化的镜面视宁度测量仪。
Detection method of mirror seeing based on curvature/ slope hybrid sensing
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摘要: 随着光学元件的口径增加,无论是在加工检测还是在站址观测过程中,镜面视宁度的影响已经越来越难以忽略。在此情况下,研究可以定量测量镜面视宁度的方法与设备就显得十分重要。基于曲率/斜率混合传感策略,结合冷冻湍流假设实现镜面视宁度评价在时-空域上的转化,实现对镜面视宁度大动态范围的检测。首先,对曲率/斜率混合传感的基本方法进行了理论推导,并以标准化点源敏感性作为评价指标进一步分析了检测过程中的镜面视宁度变化。实验结果显示,在热扰动流场较为均匀时,镜面视宁度的影响较流场反复变化时小,由于湍流不稳定所引入点源敏感性(PSSn)为0.971 8。实验结果证明基于曲率/斜率混合传感的方法可以定量分析大口径光学元件镜面视宁度,对于后续开展的大口径系统设计优化与检测加工具有很好的指导意义。Abstract: With the increase of aperture of optical elements, the influence of mirror seeing has become more and more difficult to ignore in the process of processing inspection and observation. For strong turbulence, slope detection is used to qualitatively analyze low order components. After the airflow is stable, curvature sensor can be used for fine measurement. After the slope detection, the back-end optical path of the autocollimator can be focused, and the system can be defocused to a certain amount (the defocusing direction of the two autocollimators is opposite), that is, the estimation of wavefront curvature can be obtained by the differential of wavefront along the optical axis. After obtaining the data, the results can be statistically analyzed based on point source sensitivity (PSSn) theory. Firstly, the basic method of curvature/slope hybrid sensing was deduced theoretically. The normalized PSSn was proposed by the US 30-meter telescope (TMT) to evaluate the overall performance, shape and distribution of systematic error and the change of mirror tranquility in the detection process were further analyzed with the PSSn as the evaluation index, the PSSn introduced by turbulence is 0.9718. The experimental results show that when the thermal disturbance flow field is relatively uniform, the influence of specular visibility on the flow field changes little with repeated changes. The work can quantitatively analyze the specular visibility of large-aperture optical elements, which is of great guiding significance for the optimization of large-aperture system design and detection processing.
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Key words:
- large optics system /
- mirror seeing /
- curvature sensing /
- wavefront slope
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