Closed cartesian representation of the Zernike polynomials

A general closed representation is presented of Zernike polynomials in cartesian coordinates. This form substantially increases the precision of numerical calculations when x-y coordinates are used to sample an interferogram, by avoiding any numerical conversions from cartesian to polar coordinates.     

基于Zernike多项式进行波面拟合研究

基于Zernike多项式利用Householder变换法对不同空间频率径向误差和不同口径局部误差进行了拟合,得到了拟合误差的均方根(RMS)值,分析了Zernike多项式的局限性。结果表明,Zernike在拟合径向误差时,受到最大空间频率的限制;在拟合局部误差时要受到局部误差口径大小的限制,并且会引起额外的波动。     

On the application of symmetrization to the transmission of electromagnetic waves through small convex apertures of arbitrary shape

The transmission of an electromagnetic wave through a small aperture in a perfectly conducting screen is examined from the viewpoint of symmetrization.     

Some comments on the use of the Zernike polynomials in optics

The advantages of a modified application of the Zernike polynomials are demonstrated.     

用Zernike多项式实现光机分析的技术方法

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。     

基于Zernike多项式表示色域边界的色域映射

在彩色图像的跨媒体复制中,由于不同媒体有不同的色域,所以在大多数情况下色域映射是不可避免的。不论采用何种映射算法,色域映射的第一步是确定有关媒体的色域边界。目前大多数的色域边界都是用一组测量的或用其它方法得到的离散数据来表示的。然而,由于这种表示方式是利用一组离散数据来实施色域映射的两个基本处理(即确定任一指定的等色调面平面中色域的边缘线以及确定色域边界与一条映射线的交点),比较复杂和费时,因此希望找到一种色域边界的解析表达式来解决这一问题。简述了不久前提出的一种解析函数色域边界表示方法,并用实例说明了如何利用这种解析表示(Zernike多项式)来实现上述的边缘线确定和交点确定。最后给出了几种映射算法的色域映射完整过程及其结果。     

利用泽尼克系数求取衍射光栅的分辨本领

光栅分辨本领检测设备的焦距通常达几米甚至十几米,采用直接测量法难度大、成本高,利用衍射波前间接求取光栅分辨本领是解决该问题的有效途径之一。在光栅光谱成像傅里叶变换理论基础上,建立了利用泽尼克多项式拟合系数求解衍射光栅分辨本领的归一化模型,揭示了光栅衍射波前与分辨本领的求取关系,提出了依据泽尼克多项式拟合系数求取衍射光栅分辨本领的新方法。根据该方法实测了一块衍射光栅的分辨本领,并与直接测量法进行对比测试。结果表明该方法误差小于4.42%,降低了分辨本领的测试难度,是衍射光栅分辨本领求取的有效手段,应用于ZYGO干涉仪等仪器中,通过简单的波前测试即可得到定量的衍射光栅分辨本领指标。     

环形大口径平面镜圆Zernike多项式拟合精度分析

采用瑞奇-康芒实验装置和利用圆Zernike多项式分别对不同遮拦比环域进行了拟合。结果表明,在遮拦比小于0.4的环域上,用圆Zernike多项式拟合,残差PV和残差RMS值不是很大;在遮拦比大于0.4的环域上,需用环Zernike多项式拟合才能保证一定的拟合精度。     

Generating functions for Hermite polynomials of arbitrary order

We propose a systematic method for the construction of generating functions for Hermite polynomials of arbitrary order. The procedure is based on a suitable formula for the Hermite polynomials and our results contain ones obtained earlier by Nieto and Truax [Phys. Lett. A 208 (1995) 8] as particular cases.     

Error evaluation for Zernike polynomials fitting based phase compensation of digital holographic microscopy

Combining the experimental research with the simulation calculation, the error evaluation for Zernike polynomials fitting (ZPF) based phase compensation of digital holographic microscopy (DHM) is performed. The obtained results show that the reconstructed phase with high precision can be obtained by ZPF phase compensation algorithm. Moreover, the phase error for ZPF based phase compensation algorithm increases with both the variation of object height and object transverse area, the larger variation of object height, the larger of phase error, and the larger of object transverse area, the faster increase of RMS phase error. To decrease the error of ZPF phase compensation algorithm, it is required to ensure one of the variations of object height and object transverse area to be a small value. Importantly, the proposed method supplies a useful tool for the error evaluation of phase compensation algorithm.     

Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers

A novel method of reconstructing wavefront aberrations by use of Zernike polynomials for radial shearing interferometers is discussed. This method uses matrix formalism to calculate the Zernike coefficients of a wavefront under test and shows the validity of reconstructing an arbitrary wavefront aberration from an interferogram taken by a radial shearing interferometer. We also propose a new interferometer setup to determine the shape and the direction (concave or convex) of wavefront aberration in a single measurement.     

Similarity of light fluxes diffracted in the far field by apertures of complicated shape

Translation of Preprint No. 189, Lebedev Physics Institute, Academy of Sciences of the USSR, Moscow, 1990.     

Free vibration of composite membranes with arbitrary shape

This paper is concerned with a method for solving problems of a composite membrane with an arbitrarily shaped outer boundary and an arbitrarily shaped inner boundary. The boundary conditions and the conditions of continuity are satisfied directly by using a Fourier expansion collocation method which has been given by Nagaya. The general equation for finding the natural frequencies of the composite membranes has been presented. As examples, numerical calculations have been carried out for composite polygonal membranes, composite elliptical membranes and composite circular membranes with eccentric circular boundaries.     

Two-dimensional thermal illusion device with arbitrary shape based on complementary media

On the basis of transformation thermodynamics and compensation medium theory, we develop a method to design a two-dimensional thermal illusion device with arbitrary shape, and the general expression of thermal conductivity in the each region is obtained. Simulation results show that when an object is covered with the thermal illusion device, it will accurately perform the same temperature distribution signature as another object we have predetermined. Owing to the property of deceiving and interfering with the observer, the thermal illusion device can achieve generalized thermal stealth by using thermal metamaterials, which may have a potential application in military field.     

Two-step demodulation based on the Gram-Schmidt orthonormalization method

This Letter presents an efficient, fast, and straightforward two-step demodulating method based on a Gram-Schmidt (GS) orthonormalization approach. The phase-shift value has not to be known and can take any value inside the range (0,2π), excluding the singular case, where it corresponds to π. The proposed method is based on determining an orthonormalized interferogram basis from the two supplied interferograms using the GS method. We have applied the proposed method to simulated and experimental interferograms, obtaining satisfactory results. A complete MATLAB software package is provided at http://goo.gl/IZKF3.     

Accurate and fast simulation of Kolmogorov phase screen by combining spectral method with Zernike polynomials method

A new method is proposed to simulate the Kolmogorov phase screen. A hybrid phase screen is achieved by linearly combining two phase screens simulated by both spectral and Zernike polynomials methods. Unlike the imperfection of phase structure function existing at some spatial frequencies in each of the two methods, the mean structure function of the hybrid phase screens coincides remarkably well with theory in the whole spatial frequency domain. Apart from advantage on accuracy, the proposed method also saves considerable computational time. For a phase screen with certain accuracy, the new method is ten times and seven times faster compared with the spectral method and Zernike polynomials method, respectively.     

Emission of a linear current pulse of an arbitrary shape

A general formula is obtained, which makes it possible in some cases to easily calculate the electromagnetic field of a linear current in the wave band. The emission of a linear current pulse of triangular form is considered as an application of the formula.Polytechnical University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 14–18, June, 1992.