利用Zernike多项式对湍流波前进行波前重构

分析了利用Ｚｅｒｎｉｋｅ多项式进行波前重构时模式耦合与混淆产生的原因，以及模式耦合与混淆对受大气湍流影响的光学波前进行波前重构的精度和稳定性的影响。取除去平移的前６５项Ｚｅｒｎｉｋｅ多项式构造湍流波前。针对６１单元自适应光学系统的波前传感器子孔径布局进行仿真，通过比较Ｎｓ不同取值时的湍流波前重构精度和稳定性，得出了该条件下Ｎｓ的最佳取值。     

受大气湍流影响的光学波前模拟

大气湍流对光学波前的影响可以用Ｚｅｒｎｉｋｅ模式分析，由于Ｚｅｒｎｉｋｅ模式并非统计独立，需要转而寻找一种称为Ｋａｒｈｕｎｅｎ－Ｌｏｅｖｅ（简称Ｋ－Ｌ）函数的系数，它们统计独立，而且可以展开为Ｚｅｒｎｉｋｅ多项式的形式。本文介绍的就是用Ｋ－Ｌ函数构造随机波前的原理和仿真构造尝试。     

热风式大气湍流模拟装置的哈特曼测量

大气湍流是由于大气温度、压强的随机变化引起大气折射率随机改变而产生的。遵循这一原理,热风式湍流模拟装置对传输光束周围的空气加热,用风扇抽吸热空气产生湍流,通过改变加热温度和风扇的转速,模拟不同强弱的湍流。用哈特曼传感器对该装置进行测量并采集数据,通过波前斜率处理复原波前,在此基础上从时间域和空间域等方面分析所产生湍流的特点,并与理论进行比较。结果表明,该装置能产生基本符合大气湍流统计理论的湍流。     

大气湍流畸变相位屏的数值模拟方法研究

利用功率谱反演法和Zernike多项式展开法对符合Kolmogonov统计规律的大气湍流畸变波前相位屏进行了数值模拟研究。通过对比模拟相位屏的相位结构函数与理论结果的差异分析模拟相位屏的准确性。结果表明,功率谱反演法产生的相位屏在高空间频率部分与理论相符,在低空间频率部分明显偏离理论值,通过次谐波补偿有效改善低频不足,次谐波级数达到4级足够;Zernike多项式展开法产生的相位屏的低空间频率成分与理论相符,而高空间频率成分不足随着所用Zernike阶数的增加而有所改善,但同时也带来计算量增大的问题。     

大气相位屏的协方差冲激函数产生法

提出正交基为冲激函数的协方差法,在Matlab中仿真实现符合大气湍流统计分布(vonKarman谱和Kolmogorov谱)的大气相位屏。协方差冲激函数产生法对连续波前相位进行冲激函数正交分解,根据正交分解系数的协方差模拟平行光通过大气后的瞬时波前相位。计算多帧相位屏的波前结构函数,与理论值比对,验证产生大气相位屏的正确性。根据比对结果得,该方法产生的相位屏精度高,对于Kolmogorov谱,该方法产生的大气相位屏的波前结构函数误差约为理论波前结构函数均值的1%。     

弱光61单元自适应光学系统的控制优化

在自适应光学系统中,波前校正残误差主要由未完全补偿湍流所引起的误差和系统闭环噪声组成。基于一阶比例－积分控制器分析了弱光６１单元适应光学系统的控制特性。在此基础上风云地非Ｋｏｌｍｏｇｏｒｏｖ湍流情况,提出一种根据实际测量的大气湍流波前扰动功率谱来确定系统,针对非Ｋｏｌｍｏｇｏｒｏｖ湍流情况,提出一根据实际测量的大气湍流波前扰动功率谱来确定系统最优控制带宽的新方法。应用这种方法对弱光６１单元自适应光     

曲率波前传感器波前重构算法的研究

运用衍射理论计算出两个离焦面上的光强分布,从而获得波前曲率和孔径边缘的波前径向倾斜。根据Zernike多项式的微分特性,采用两种不同的方法分别处理曲率和倾斜,使用积分的方法解决了在边界上δ函数很难处理的问题。由探测器的几何分布预先算出控制矩阵,用Zernike多项式曲线拟合的方法重构出波前。计算机仿真表明,波前残差小于5％,验证了此波前重构算法的可行性。     

实验模态分析中一种改进的傅氏域离散正交多项式

针对在借助有理分式多项式构建结构频响函数的分析模型的过程中由于负频率引入的虚拟测点将导致方程式在多项式阶次较高时出现病态的问题,对实域离散点列上的正交多项式进行了推广,得到傅氏域离散点列上的正交多项式.该多项式不仅可避免由负频率引入的冗余计算,而且亦使方程式得到解耦,从而使该方法更为高效.     

天文学自适应光学成像望远镜的模拟

为分析天文学自适应光学(AO)望远镜中AO系统的校正性能,利用Matlab仿真其成像过程。采用正交基为Zernike多项式的自相关法产生符合大气统计特性的大气相位屏,仿真平行光通过大气后的瞬时畸变波前相位;采用快速傅里叶变换仿真哈特曼-夏克波前传感器的成像光斑,根据实际成像与参考平面波成像的质心坐标之差,计算波前传感器子孔径内的平均波前斜率。模拟比较了1.2m望远镜两种AO系统布局的校正性能,结果表明,子孔径为正六边形AO系统的校正性能略优于子孔径为正方形AO系统的校正性能,两种AO系统的SR比(StrechlRatio)分别为0.872和0.859。     

矩形孔径上折射率均匀性数据拟合研究

泽尼克圆多项式在圆形光瞳的正交性和能够代表经典像差而被广泛应用到波前分析中,用泽尼克圆多项式作为矩形光瞳基底函数,通过推导得到在矩形光瞳上正交的多项式.这个在矩形光瞳上正交的多项式不仅是唯一的,而且也能够表示经典像差,就像泽尼克圆多项式在表示圆形光瞳时具有这样的特性一样.矩形光瞳上正交多项式像泽尼克圆多项式一样即可以用...     

Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials

A general wavefront fitting procedure with Zernike annular polynomials for circular and annular pupils is proposed. For interferometric data of typical annular wavefronts with smaller and larger obscuration ratios, the results fitted with Zernike annular polynomials are compared with those of Zernike circle polynomials. Data are provided demonstrating that the annular wavefront expressed with Zernike annular polynomials is more accurate and meaningful for the decomposition of aberrations, the calculation of Seidel aberrations, and the removal of misalignments in interferometry. The primary limitations of current interferogram reduction software with Zernike circle polynomials in analyzing wavefronts of annular pupils are further illustrated, and some reasonable explanations are provided. It is suggested that the use of orthogonal basis functions on the pupils of the wavefronts analyzed is more appropriate.     

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.     

Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms

Four modal methods of reconstructing a wavefront from its difference fronts based on Zernike polynomials in lateral shearing interferometry are currently available, namely the Rimmer-Wyant method, elliptical orthogonal transformation, numerical orthogonal transformation, and difference Zernike polynomial fitting. The present study compared these four methods by theoretical analysis and numerical experiments. The results show that the difference Zernike polynomial fitting method is superior to the three other methods due to its high accuracy, easy implementation, easy extension to any high order, and applicability to the reconstruction of a wavefront on an aperture of arbitrary shape. Thus, this method is recommended for use in lateral shearing interferometry for wavefront reconstruction.     

General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes

Zernike polynomials have been widely used to describe the aberrations in wavefront sensing of the eye. The Zernike coefficients are often computed under different aperture sizes. For the sake of comparison, the same aperture diameter is required. Since no standard aperture size is available for reporting the results, it is important to develop a technique for converting the Zernike coefficients obtained from one aperture size to another size. By investigating the properties of Zernike polynomials, we propose a general method for establishing the relationship between two sets of Zernike coefficients computed with different aperture sizes.     

Mathematical construction and perturbation analysis of Zernike discrete orthogonal points

Zernike functions are orthogonal within the unit circle, but they are not over the discrete points such as CCD arrays or finite element grids. This will result in reconstruction errors for loss of orthogonality. By using roots of Legendre polynomials, a set of points within the unit circle can be constructed so that Zernike functions over the set are discretely orthogonal. Besides that, the location tolerances of the points are studied by perturbation analysis, and the requirements of the positioning precision are not very strict. Computer simulations show that this approach provides a very accurate wavefront reconstruction with the proposed sampling set.     

扇形子孔径布局H-S传感器对圆域的探测性能研究

通过Zernike模式波前重构算法进行计算机仿真计算,对扇形子孔径布局的Hartmann-Shack波前传感器探测性能进行仿真研究,对比分析了方形子孔径布局Hartmann-Shack波前传感器的探测性能.结果表明,扇形子孔径布局是一种有效的Hartmann-Shack波前传感器的布局方式,在低子孔径密度且径数目差异不多的条件下,比方形孔径布局探测精度更高.     

人眼像差哈特曼测量仪的重复性测试

人眼像差哈特曼测量仪利用 Hartmann-Shack 波前传感器探测波前,并用 Zernike 多项式实现波前的重建。测试结果的重复性对该仪器来说是至关重要的。此仪器对模拟人眼静态像差和活体人眼动态像差分别进行单帧采集和连续采集,所得的模拟人眼静态像差 Zernike 系数的标准偏差的最大值,PV 与 RMS 依次为 0.016?,0.061?和 0.006?;活体人眼动态像差 Zernike 系数的标准偏差的最大值,PV 与 RMS 依次为 0.045?,0.167? 和 0.030?,这些指标都小于 ?/5,表明该仪器具有较高的重复性。     

非球面非零位横向剪切干涉检测与波前重建

本文分析了非球面非零位检测中3种典型的非球面度及其变化率、调整误差、回程误差以及干涉仪系统误差,提出了非球面度变化率是影响检测误差的主要因素,并给出了非球面非零位检测面形加工误差的确定方法.对基于Zernike多项式拟合的最小二乘法和系数转换法进行了对比研究,分析了剪切量和多项式拟合阶数对波前重建精度的影响,并给出了数值模拟结果.为了验证以上方法的可行性及检测精度,针对一单点金刚石车削的抛物面反射镜,利用顶点球作为测试波前进行非零位横向剪切干涉检测,并用最小二乘法进行波前重建.实验结果表明,在非球面度变化率最大的反射镜边缘处面形误差最大,达到0.203μm,研究结果为横向剪切干涉仪用于非球面加工过程中在位检测提供了技术支撑.