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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
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A general theoretical approach has been developed for the determination of orthonormal polynomials over any integrable domain, such as a hexagon. This approach is better than the classical Gram-Schmidt orthogonalization process because it is nonrecursvie and can be performed rapidly with matrix transformations. The determination of the orthonormal hexagonal polynomials is demonstrated as an example of the matrix approach. 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
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基于DDE接口技术的计算机辅助装调方法 总被引:1,自引:0,他引:1
根据波前拟合和计算机辅助装调的算法原理,基于动态数据交换(DDE)接口技术和泽尼克多项式拟合技术,采用阻尼最小二乘法,运用MATLAB和ZEMAX联合计算实现了计算机辅助装调。利用DDE接口在ZEMAX和MATLAB之间进行通讯,并用MATLAB编写了能够生成各阶泽尼克圆多项式和环多项式的基底矩阵函数的可视化的计算机辅助装调程序,实现了程序的通用性及易用性。使用该程序对大遮拦比光学系统进行模拟装调,验证了程序的正确性。为了确认泽尼克圆多项式在环域上的相关性对计算机辅助装调结果的影响,分别采用泽尼克圆多项式和环多项式进行模拟装调。模拟结果表明:计算机辅助装调使用这两种泽尼克多项式均可行。 相似文献
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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
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基于泽尼克多项式进行面形误差拟合的频域分析 总被引:3,自引:3,他引:0
获得泽尼克多项式的频谱信息是正确利用该多项式进行误差拟合的关键。推导出了泽尼克多项式的傅里叶变换公式,在频域中分析了不同阶数该多项式的径向频谱信息和幅角频谱信息,得到了有限项泽尼克多项式能够有效表达面形误差的最大径向空间频率和角频率。基于频域分析理论,利用泽尼克多项式对不同口径局部误差进行了拟合,并利用齐戈(Zygo)干涉仪对带有不同面形误差的光学元件进行了试验分析。结果表明,当误差的径向空间频率或角频率超出泽尼克多项式所能表达的频谱范围时,拟合误差迅速变大。 相似文献
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We compare seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk. Separation of variables for the Laplace operator yields an analytic solution as a Fourier–Bessel series, but this usually converges at an algebraic (sub-spectral) rate. The cylindrical Robert functions converge geometrically but are horribly ill-conditioned. The Zernike and Logan–Shepp polynomials span the same space, that of Cartesian polynomials of a given total degree, but the former allows partial factorization whereas the latter basis facilitates an efficient algorithm for solving the Poisson equation. The Zernike polynomials were independently rediscovered several times as the product of one-sided Jacobi polynomials in radius with a Fourier series in θ. Generically, the Zernike basis requires only half as many degrees of freedom to represent a complicated function on the disk as does a Chebyshev–Fourier basis, but the latter has the great advantage of being summed and interpolated entirely by the Fast Fourier Transform instead of the slower matrix multiplication transforms needed in radius by the Zernike basis. Conformally mapping a square to the disk and employing a bivariate Chebyshev expansion on the square is spectrally accurate, but clustering of grid points near the four singularities of the mapping makes this method less efficient than the rest, meritorious only as a quick-and-dirty way to adapt a solver-for-the-square to the disk. Radial basis functions can match the best other spectral methods in accuracy, but require slow non-tensor interpolation and summation methods. There is no single “best” basis for the disk, but we have laid out the merits and flaws of each spectral option. 相似文献
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1 Introduction Weoftendescribethestaticordynamicwavefrontaberrationsascombinationofdifferentmodes,suchaspiston ,tilt,defocus,coma,spheralandsoon .ThesemodesaresimilarassomelowerordersofZernikepolynomials.TheZernike polynomialsarenormalizedorthogonalincir… 相似文献
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The deduction of Zernike coefficients is usually influenced by the finite number of sampling dots on interferogram and their inherited measurement errors. In this paper, a simplified Gram-Schmidt method for solving the Zernike polynomial with the higher fitting precision is presented and used to analyze the wave front aberrations for the circle interference fringe of the fine polished aluminum disk surface captured by a Twyman-Green interferometer system. We find the stability of the Zernike coefficients changes with changing the Zernike term, which has lead to the wrong expression for the wave front aberration. By analyzing the condition number of the coefficients matrix and the fitting precision of the method, it is indicated that the instability can be avoided when the Zernike term is lower than 14. Such an analysis will be valuable in solving the Zernike polynomial for the wave front aberration analysis in optical testing. 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
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《中国光学快报(英文版)》2020,(6)
Polarization aberration caused by material birefringence can be partially compensated by lens clocking. In this Letter, we propose a fast and efficient clocking optimization method. First, the material birefringence distribution is fitted by the orientation Zernike polynomials. On this basis, the birefringence sensitivity matrix of each lens element can be calculated. Then we derive the rotation matrix of the orientation Zernike polynomials and establish a mathematical model for clocking optimization. Finally, an optimization example is given to illustrate the efficiency of the new method. The result shows that the maximum RMS of retardation is reduced by 64% using only 48.99 s. 相似文献
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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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用Zernike多项式实现光机分析的技术方法 总被引:2,自引:1,他引:2
由于光学软件不能直接利用有限元分析的结果,而Zernike多项式的各项与光学像差有对应关系,因此常用Zernike多项式作为光机接口。针对目前常用轴向位移作为拟合量描述拟合面形的不足,给出了几种常用的表面位移校正方法并说明了其优缺点。用具体实例比较各校正位移,并对其进行Zernike多项式拟合,从拟合系数的差异可以看出,曲率比较大的表面必须采用校正位移进行拟合。最后指出:在不知道初始表面方程的情况下,轴向和法向校正位移均采用从初始表面出发的方法,如果已知初始表面方程,则轴向校正位移采用从变形表面出发的方法,法向校正位移仍采用从初始表面点出发进行计算。 相似文献
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Zernike polynomials have been widely used for wave-front analysis because of their orthogonality over a uniform circular pupil. However, the pupil is not uniform but weighted by the backpropagated fiber mode in analyzing fiber coupling efficiency. Zernike polynomials are not appropriate for a weighted pupil due to their lack of orthogonality over such pupil. We emphasize the advantages of using orthonormal polynomials in fiber coupling systems. The orthonormal polynomials over weighted pupil are derived by matrix approach. The effects of primary aberrations are investigated based on the orthonormal polynomials. The accuracy of the Strehl ratio approximation for primary aberrations is evaluated. 相似文献
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P. Di Francesco 《Communications in Mathematical Physics》1998,191(3):543-583
We prove a determinantal formula for quantities related to the problem of enumeration of (semi-) meanders, namely the topologically
inequivalent planar configurations of non-self-intersecting loops crossing a given (half-) line through a given number of
points. This is done by the explicit Gram-Schmidt orthogonalization of certain bases of subspaces of the Temperley-Lieb algebra.
Received: 13 December 1996 / Accepted: 21 May 1997 相似文献