两种正交多项式模拟湍流波前的比较

针对Zemike多项式仅在连续单位圆上正交,用于在离散点上构造光学波前必然会引起误差的问题,本文提出用能够在离散点上正交的多项式来模拟经过大气湍流的光学波前.该方法根据湍流的统计理论,采用Gram-Schmidt正交化方法,构造了Malacara多项式表示的湍流波前,并进行了数值模拟.将模拟结果与直接用Zemike多项式模拟的结果进行了比较分析,结果表明:在相同的条件下,该方法的模拟结果更接近统计理论值.     

用干涉法测量标准镜头的传递函数

在对标准镜头的干涉图进行数字化后,利用Zernike圆正交多项式进行波面拟合,得到光瞳函数。最后用自相关积分计算标准镜头的光学传递函数。对一组(3个)标准镜头所作的实测表明,这种测量方法的不确定度小于0.03。     

具有附加弹性支承的圆形薄板结构振动分析全文替换

采用瑞利-里兹法对附加弹性铰支承的圆形薄板结构进行振动分析,研究不同约束边界条件下,支承的刚度、数量、位置对圆板结构振动特性（频率、振型）的影响。运用正交梁多项式作为圆板的径向试函数,用傅里叶级数作为圆板的周向试函数,保证了振动特征参数计算结果的完备性和准确性。通过一些经典算例分析附加支承的数量、刚度和位置等对圆板结构固有频率的影响规律,揭示圆形薄板附加支承对称布局设计的重要性。     

基于离轴四反的空间引力波探测激光发射望远镜设计

用于空间引力波探测的星载望远镜在航天器间进行激光的传输以支持精密干涉测量系统,因此望远镜的光程稳定性已经成为一项关键的技术指标。在此系统中,光瞳像差与传统的像平面像差相比在了解系统光程稳定性需求、评价望远镜成像质量以及抑制抖动光程耦合噪声等方面可提供更深刻的见解。本文基于传统像平面像差理论和光瞳像差理论,建立了望远镜的初始结构,然后利用光学软件Zemax的宏编程实现了光瞳像差和像平面像差的自动校正,从而实现了高性能星载望远镜的设计,仿真结果显示满足天琴任务的需求。     

泽尼克像差组合对人眼光学质量的影响

基于人眼光学质量客观评价标准区域调制传递函数（AreaMTF）、斯特列尔比（SRX）和同心相对瞳孔平面（PFWc）,分析了波前像差RMS为0．25μm时泽尼克像差项组合对人眼光学质量的影响。C4（离焦）与C12（球差）等组合后,像差比例在1．33到4．90的范围内AreaMTF和SRX有极大值;C6（三叶草）与C7（彗差）等组合后,像差比例在0．20到1．00的范围内它们有极小值。二阶像差（离焦和像散）组合时,C4（离焦）起着主导作用。三阶像差（彗差和三叶草）组合时,SRX和PFWc值的变化范围均较大,分别为0．066到0．228和0．048到0．656。     

空间相干光通信外差效率及天线像差的影响

空间相干光通信中接收天线像差会使光外差效率下降.对本振光为高斯分布,信号光为爱里斑分布的光外差效率进行了研究,给出了无像差时外差效率的解析表达式.当焦平面上爱里斑半径与本振高斯光束光腰半径之比为1.71时,有最大外差效率81.45%.然后以本振光为理想的高斯光束,信号光受像差的影响,研究了倾斜、离焦、球差、彗差、像散等像差引起的光外差效率损失,给出了存在像差时外差效率的一维积分表达式.研究表明即使在采用离焦校正后,一个波长的球差引起的附加外差效率损失仍可达0.9 dB.因此对于爱里斑位于光轴上的接收天线,在设计时需仔细处理球差的影响.     

弹性边界约束矩形板的振动特性分析：理论、有限元和实验

提出一种有效的理论方法研究弹性边界约束矩形板的振动特性,并设计实验测试不同边界矩形板的固有频率。矩形板的弹性边界约束采用一系列的均布线性弹簧模拟,用特征正交多项式来表示矩形板的位移容许函数,并采用瑞利-里茨法获得弹性边界约束矩形板的固有频率和固有振型。通过改变边界弹簧的刚度即可模拟矩形板不同的边界条件,提高计算效率。基于理论方法计算获得结构固有频率并和有限元及实验结果进行对比,验证所提理论方法的正确性。此外,通过实验测试的方法分析弹性-简支、弹性-固支等不同边界组合条件下矩形板的振动特性,分析调整不同边界弹簧刚度对矩形板振动特性的影响。     

空间光学合成孔径成像系统原理与关键技术分析

根据傅里叶成像理论,分析了光学合成孔径系统成像原理,指出光瞳函数的离散化,提高了光学合成孔径成像系统的分辨本领,但降低了系统传函中频性能,并使系统在信息获取时具有方向选择性。运用遗传算法对光瞳构形进行优化,光瞳优化后,四孔径系统的优化参数从310.8降到13.7,减小了23倍。在合成孔径成像系统单个子孔径存在像差的情况下,Piston误差对系统的影响最大,其次是离焦误差、倾斜误差、球差、彗差和像散,对于四孔径系统,Piston误差对系统的影响几乎是离焦误差的2倍,像散的4倍。系统曝光时间与填充比的平方或立方成反比。     

带分划板的连续变倍双筒望远镜设计

介绍一种用两组元透镜来实现倒象和连续变倍的双筒望远镜设计。该设计在物镜焦面位置安装带刻线的分划板 ,可对目标进行瞄准和测量。特别是在系统不另设可变光栏的情况下 ,利用光学系统透镜边框作孔径光栏 ,使系统在整个变倍过程中不仅出瞳直径和出瞳距离均变化不大 ,而且系统与定倍望远镜一样具有较大的象方视场、出瞳直径和出瞳距离 ,从而使系统无论昼夜使用都具有真正的变倍作用和良好的观察效果。     

离轴反射式光学系统设计

提出通过光瞳和视场离轴,实现无中心遮拦的离轴反射式光学系统设计方法。在同轴三反射光学系统基础上,将光瞳和视场适当离轴,实现镜间遮拦的消除。分主镜或次镜为系统孔径光阑两种情况,导出同轴三反射光学系统初级像差公式和初始结构参数计算公式。由三反射系统成像性质,进一步总结无焦光路条件。根据设计理论计算离轴三反射系统初始结构,利用Zemax优化得到无中心遮拦的离轴三反射空间观测望远镜。入瞳320mm,视场(±0.3°)×(±0.6°),焦距1800mm。     

Orthonormal polynomials in wavefront analysis: analytical solution

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. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.     

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.     

Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.     

Zernike annular polynomials and optical aberrations of systems with annular pupils

Zernike annular polynomials that represent orthogonal andbalanced aberrations suitable for systems with annular pupilsare described. Their numbering scheme is the same asfor Zernike circle polynomials. Expressions for standard deviationof primary and balanced primary aberrations are given.     

Zernike circle polynomials and optical aberrations of systems with circular pupils

Zernike circle polynomials, their numbering scheme, and relationship to balanced optical aberrations of systems with circular pupils are discussed.     

Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.     

Zernike-gauss polynomials and optical aberrations of systems with gaussian pupils

In the first two Notes of this series,(l,2) we discussed Zernike circle and annular polynomials that represent optimally balanced classical aberrations of systems with uniform circular or annular pupils, respectively. Here we discuss Zernike-Gauss polynomials which are the corresponding polynomials for systems with Gaussian circular or annular pupils.(3-5) Such pupils, called apodized pupils, are used in optical imaging to reduce the secondary rings of the pointspread functions of uniform pupils.(6) Propagation of Gaussian laser beams also involves such pupils.     

Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts

The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.     

Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils

Zernike polynomials and their associated coefficients are commonly used to quantify the wavefront aberrations of the eye. When the aberrations of different eyes, pupil sizes, or corrections are compared or averaged, it is important that the Zernike coefficients have been calculated for the correct size, position, orientation, and shape of the pupil. We present the first complete theory to transform Zernike coefficients analytically with regard to concentric scaling, translation of pupil center, and rotation. The transformations are described both for circular and elliptical pupils. The algorithm has been implemented in MATLAB, for which the code is given in an appendix.     

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.