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1.
The theory of orthogonal polynomial (Zernike) expansions of functions on a disk, as used in the diffraction theory of optical aberrations, is applied to obtain (semi-) analytical expressions for the spatial impulse responses arising from a non-uniformly moving, baffled, circular piston. These expressions are in terms of the expansion coefficients of the non-uniformity and the responses of the orthogonal expansion functions. The latter impulse responses have a closed form as finite series involving the Legendre functions and the sinc function. The method is compared with a similar method, proposed by P. R. Stepanishen [J. Acoust. Soc. Am. 70, 1176-1181 (1981)] where zeroth order orthogonal Bessel functions, rather than Zernike polynomials, are used as expansion functions. 相似文献
2.
Aaron N. Fletcher 《Applied Physics A: Materials Science & Processing》1977,13(4):327-332
The Zernike polynomials are orthogonal functions defined on the unit circle, which have been used primarily in the diffraction
theory of optical aberrations. A summary of their principal properties is given. It is shown that the polynomials, which are
closely related to the general spherical harmonics, are especially useful in numerical calculations. In particular, by using
the polynomials as a basis to represent the commonly encountered functions of optical theory, it is often possible to avoid
numerical quadrature and computations are reduced to the simple manipulation of expansion coefficients. 相似文献
3.
Dai GM 《Optics letters》2006,31(4):501-503
The set of Fourier series is discussed following some discussion of Zernike polynomials. Fourier transforms of Zernike polynomials are derived that allow for relating Fourier series expansion coefficients to Zernike polynomial expansion coefficients. With iterative Fourier reconstruction, Zernike representations of wavefront aberrations can easily be obtained from wavefront derivative measurements. 相似文献
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运用泽尼克多项式进行物面波前数据拟合 总被引:8,自引:0,他引:8
本文提出了一种运用泽尼克(Zernike)多项式的波前数据拟合方法.给出了离散点上正交多项式的构造并描述了具体计算步骤.可对待测物面低调制度点、散斑点、噪音及灰尘点区域进行波前数据拟合,通过一定阈值的设定,将误差点用掩模滤除并予以拟合.进一步提高了物体测量精度. 相似文献
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Sergey N. Bezdidko 《Optical Review》2014,21(5):632-638
The author introduces a complete set of polynomials that are orthogonal in the three-dimensional region (generalized Zernike polynomials). These polynomials make it possible to obtain orthogonal expansion of the wave aberration in the three-dimensional region of field-pupil 0 ≤ r ≤ 1, 0 ≤ ρ ≤ 1, 0 ≤ φ ≤ 2π. This permits us to determine the orthogonal system of individual aberrations and introduce a classification of individual aberrations depending on the degree of field r and pupil ρ, φ variables. The author shows that orthogonal aberrations have a number of unique properties. The developed approach, describing the aberration properties of optical systems by means of orthogonal aberrations, and its use in the construction of new methods and techniques for the design of optical systems form a new section of lens design, namely, “Theory of orthogonal aberrations and its applications in the design of optical systems”. 相似文献
6.
针对非圆域波面拟合中Zernike多项式失去正交特性、拟合系数交叉耦合的问题,提出非圆域Zernike正交基底函数构造方法。以圆Zernike为基底,采用Gram-Schimdt正交组构造方法,线性表出单位正交基底。通过构造不同遮光比环形光阑下的正交基底与环Zernike多项式进行比较,验证了此方法的正确性。然后采用圆Zernike多项式和构造的新基底对矩形光阑下的波面进行了拟合,从拟合残余误差、各项基底系数的稳定性、传递矩阵的条件数等分析,结果表明针对特定的非圆域构造的新基底可靠性和抗扰动能力优于圆Zernike多项式。此方法不需要具体求出基底的解析表达式,不同非圆域仅是正交化系数矩阵发生改变,为非圆域正交基底构造提供了一种新途径。 相似文献
7.
An aberrometer was used to measure the monochromatic aberration of 219 eyes in a population of 113 myopic subjects. The Zernike expansion was used to describe the wavefront aberration of the human eyes. From the distribution of Zernike coefficients, the means of almost all Zernike coefficients are approximately zero, and the mean absolute values of Zernike coefficients convey the level of aberration for our study population. Based on the distribution of Zernike coefficients, the averaged optical quality was computed in different methods. The mean MTF and PSF of individual eyes represent the optical quality for a typical myopic eye. Through the calculation of the visual benefits expected from correcting Zernike modes, a large expected visual benefit can be achieved across all spatial frequencies after correcting 5 Zernike modes. Furthermore, the larger improvement of optical quality can be achieved with correcting more Zernike modes at a higher spatial frequency. 相似文献
8.
A D Alhaidari 《理论物理通讯》2020,72(1):15104-76
We present a formulation of quantum mechanics based on the theory of orthogonal polynomials.The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems(both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation(see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases(the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials. 相似文献
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1 Introduction Weoftendescribethestaticordynamicwavefrontaberrationsascombinationofdifferentmodes,suchaspiston ,tilt,defocus,coma,spheralandsoon .ThesemodesaresimilarassomelowerordersofZernikepolynomials.TheZernike polynomialsarenormalizedorthogonalincir… 相似文献
11.
The problem of determining the orthonormal polynomials for hexagonal pupils by the Gram-Schmidt orthogonalization of Zernike circle polynomials is revisited, and closed-form expressions for the hexagonal polynomials are given. We show how the orthonormal coefficients are related to the corresponding Zernike coefficients for a hexagonal pupil and emphasize that it is the former that should be used for any quantitative wavefront analysis for such a pupil. 相似文献
12.
《Current Applied Physics》2015,15(6):739-747
The linear Hilbert phase mask or transform has found applications in image processing and spectroscopy. An optical version of the fractional Hilbert mask is considered here, comprising an imaging system with a circular, unobscured pupil in which a variable phase delay is introduced into one half of the pupil, split bilaterally.The radial Hilbert phase mask is also used in image processing and to produce optical vortices which have applications in optical tweezers and the detection of exoplanets.We subjected the fractional and radial Hilbert phase masks to Zernike function expansion in order to compute the image plane electromagnetic field distribution using Nijboer-Zernike theory. The Zernike functions form an orthogonal basis on the unit circle. The complex-valued Zernike expansion coefficients for these two phase masks were derived for use in the context of the Extended Nijboer-Zernike (ENZ) theory of image formation. The ENZ approach is of interest in that it allows a greater range of defocus to be dealt with, provides a simple means of taking a finite source size into account and has been adapted to high Numerical Aperture (NA) imaging applications.Our image plane results for the fractional Hilbert mask were verified against a numerical model implemented in the commercial optical design and analysis code, Zemax®. It was found that the Nijboer-Zernike result converged to the Zemax® result from below as the number of Zernike terms in the expansion was increased. 相似文献
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中心遮拦干涉图的圆泽尼克拟合对计算赛德尔像差的影响分析 总被引:5,自引:0,他引:5
从波像差的幂级数和圆泽尼克多项式展开理论入手,介绍了圆泽尼克多项式和赛德尔多项式之间的联系,理论上分析了圆泽尼克多项式在环域的相关性,着重讨论了以中心遮拦干涉图的圆泽尼克多项式拟合系数计算赛德尔像差系数的影响。对理论分析进行了实验验证,其结果与理论分析具有良好的一致性,并提出了一种简单直观的误差容限设定方法。研究表明,随着遮拦比的增加,赛德尔系数误差增加,其变化规律和被测元件的像差类型和大小有关。当遮拦比达到某一特定的阈值时,误差曲线将产生较大的变化,为了获得较准确的赛德尔系数,圆泽尼克拟合时应选择适当的阶数;当遮拦比继续增加时,为了计算出准确的赛德尔系数,拟合时应选择环泽尼克多项式。此外,遮拦比对赛德尔系数中畸变、像散的影响较弱,对彗差、场曲、球差的影响较强。 相似文献
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A.B. Utkin R. Vilar A.J. Smirnov 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2001,17(2):145-148
The consideration is carried out in its general formulation: the wave aberration function is represented in terms of classical
aberrations (the Zernike polynomials), the phase transfer function (argument of the complex optical transfer function) is
defined by a chain of transformations originating from the generalized pupil function. Quasi-analytical quadrature formulas
are derived that link the optical transfer function and the phase transfer function with the aberration terms. It is shown
that the phase transfer function contains information on the odd-order aberrations, which can be retrieved from coefficients
to the Taylor expansion of the derived quadrature relation.
Received 16 June 2001 相似文献
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自适应光学系统的模式法数值模拟 总被引:7,自引:2,他引:5
建立了利用模式法笃自适应光学系统进行数值模拟的理论模型,编制了计算程序,并与激光大气传输计算程序衔接起来,进行了大量数值模拟计算。首次发现:存在泽尼特多项式展开的最佳项数。大于一定项数的展开式的效果迅速变坏,竖排和斜排经特面式展开有类似的结果。文献中认为可以采用的15项经特多项式展开的效果不好,最佳项数随着横向风速的增加而减小,在风速较大时最佳项数下的模范地结果稍好于直接斜率控制法的结果。 相似文献
18.
Adaptive optics systems mitigate the atmospheric turbulence-induced distortion of a propagating light wavefront. The use of adaptive optics entails the design of a feedback controller, which requires the development of a model of the plant to be controlled. In adaptive optics, the plant consists of the atmosphere through which light is traveling. Moreover, a distinct feature of the adaptive optics control application is the presence of random signals in the plant. In optics, Zernike orthonormal polynomials are commonly used as a basis set for the expansion of wavefront phase distortions. Due to the atmospheric turbulence-induced random nature of the underlying physical process, the spatial-temporal correlation functions of the Zernike polynomial phase distortion expansion coefficients must be evaluated if a proper stochastic model of the plant is to be developed and adaptive optics is to be employed. In Part 1 of this paper, these correlation functions are developed using a layered atmospheric model and calculations for the first few low-order Zernike modes are performed. Using these correlation functions, an underlying stochastic linear dynamical system, which is adequate for control design, is synthesized. This system models the plant and, in turn, provides the basis for the employment of advanced model-based control and estimation concepts in an adaptive optics system for an airborne platform application. 相似文献
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本文就专著和文献中关于Zernike多项式拟合干涉波面几乎都建设采用Gram-Schimdt正交化方法,而不采用比较简单的传统经典的最小二乘法问题进行了深入研究,从理论和实践上严格地证明了两种方法的等价性。实践中发现,用最小二乘法求解Zernike多项式拟合系数的速度比用Gram-Schimdt正交化方法提高了三倍之多。由于在精密光测技术中,Zernike多项式已被广泛采用,因此,“等价性”的证明具有重要意义,并对于其它类似问题也有着普遍的参考价值。 相似文献