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.     

Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces

We propose a more accurate and efficient reconstruction method used in testing large aspheric surfaces with annular subaperture interferometry. By the introduction of the Zernike annular polynomials that are orthogonal over the annular region, the method proposed here eliminates the coupling problem in the earlier reconstruction algorithm based on Zernike circle polynomials. Because of the complexity of recurrence definition of Zernike annular polynomials, a general symbol representation of that in a computing program is established. The program implementation for the method is provided in detail. The performance of the reconstruction algorithm is evaluated in some pertinent cases, such as different random noise levels, different subaperture configurations, and misalignments.     

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.     

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.     

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.     

横向剪切干涉测量中一种获得无耦合Zernike系数的模式复原方法

模式耦合误差常见于横向剪切干涉测量中基于波前梯度数据的模式复原法,其原因是用于表示波前的基函数——Zernike圆多项式的导数不正交。使用一种含有Gram矩阵的矩阵方程进行复原,直接利用Zernike圆多项式m≠0模式角向导数对于权重函数w(ρ) = ρ (极坐标下)的正交性,以及Zernike圆多项式m = 0模式径向导数对于权重函数w(ρ) = ρ(1-ρ2)(极坐标下)的正交性进行复原。该方法无需构造辅助的向量函数,并可得到无耦合的Zernike系数,复原结果表明,模式耦合得到了避免。该方法可推广到环上,得到无耦合的Zernike环多项式系数。     

对称结构模态振型的Zernike矩描述方法

讨论了利用Zernike矩描述对称结构模态振型的方法,通过对结构的模态振型数据进行Zernike矩变换,将其分解成一系列Zernike矩的线性组合,而每一个Zernike矩反映模态振型的一部分形状特征.不同特征矩的线性组合,可以代表各阶模态的振型.在此基础上进一步提出了确定Zernike多项式最高阶数的方法,并讨论了Zernike矩描述对称结构模态振型的方法及其去噪声的能力.通过对简单圆盘结构的仿真实例研究,验证了应用Zernike矩描述对称结构模态振型的优越性.结果表明;利用Zernike矩描述对称结构的模态振型可以更有效地描述对称结构的模态,包括重模态,同时还能有效地消除测试数据中噪声的影响,对进一步实现对称结构有限元模型修正和模型确认具有重要的应用价值.     

Aberrations and spherocylindrical powers within subapertures of freeform surfaces

A method is described for the derivation of refractive properties and aberration structure of subapertures of freeform surfaces. Surface shapes are described in terms of Zernike polynomials. The method utilizes matrices to transform between Zernike and Taylor coefficients. Expression as a Taylor series facilitates the translation and size rescaling of subapertures of the surface. An example operation using a progressive addition lens surface illustrates the method.     

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.     

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.     

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.     

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...     

Corneal surface reconstruction algorithm using Zernike polynomial representation: improvements

Recently Sicam et al. [J. Opt. Soc. Am. A21, 1300 (2004)] presented a new corneal reconstruction algorithm for estimating corneal sag by Zernike polynomials. An equivalent but simpler derivation of the model equations is presented. The algorithm is tested on a sphere, a conic, and a toric. These tests reveal significant height errors that accrue with distance from the corneal apex. Additional postprocessing steps are introduced to circumvent these errors. A consistent and significant reduction in height errors is observed across the test surfaces. Finally, Sicam used the conic p-value p as a measure of algorithm efficacy. Further investigation shows that the finite Zernike representation affected the reported results. The p-value should therefore be used with caution as an efficacy measure.     

Zernike coefficients of a scaled pupil

By expressing a scaled Zernike radial polynomial as a linear combination of the unscaled radial polynomials, we give a simple derivation for determining the Zernike coefficients of an aberration function of a scaled pupil in terms of their values for a corresponding unscaled pupil.     

Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.     

Variable aberration generators using rotated Zernike plates

The rotational properties of Zernike polynomials allow for an easy generation of variable amounts of aberration using two rotated phase plates, each one encoding one or several Zernike modes. This effect may be used to build variable aberration generators useful for calibrating different kinds of aberrometer.     

Nonnull interferometer simulation for aspheric testing based on ray tracing

The nonnull interferometric method that employs a partial compensation system to compensate for the longitude aberration of the aspheric under test and a reverse optimization procedure to correct retrace errors is a useful technique for general aspheric testing. However, accurate system modeling and simulation are required to correct retrace errors and reconstruct fabrication error of the aspheric. Here, we propose a ray-tracing-based method to simulate the nonnull interferometer, which calculates the optical path difference by tracing rays through the reference path and the test path. To model a nonrotationally symmetric fabrication error, we mathematically represent it with a set of Zernike polynomials (i.e., Zernike deformation) and derive ray-tracing formulas for the deformed surface, which can also deal with misalignment situations (i.e., a surface with tilts and/or decenters) and thus facilitates system modeling extremely. Simulation results of systems with (relatively) large and small Zernike deformations and their comparisons with the lens design program Zemax have demonstrated the correctness and effectiveness of the method.     

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.     

Absolute measurement of planarity with Fritz's method: uncertainty evaluation

Fritz's method [Opt. Eng. 23, 379 (1984)] of using Zernike polynomials to assess the absolute planarity of test plates is revisited. A refinement is described that takes into account the data decorrelation that appears in experiments. An uncertainty balance is defined by propagation of error contributions through the steps of the method. The resultant measuring procedure is demonstrated on a data set from experiments, and a nanometer level of uncertainty is achieved.     

AUTOMATED COMPACT PARAMETRIC REPRESENTATION OF IMPACT CRATERS

We have developed a procedure for the measurement and automated parametric representation of 3-D impact craters. The 3-D stucture of the crater is measured using a lowcoherence optical interference technique and the parametric representation achieved in a two-step procedure. Sobel edge detection and morphological operations are used to define rigorously the impact region. The parameter set is defined with respect to a coordinate system whose origin and orientation are uniquely determined from the data. Subjective, arbitrary choices for this position and angle are thus eliminated and the coefficients of the Zernike polynomials are calculated by direct integration over this region to produce a parameter set. This set allows the representation of the three-dimensional geometry by a small set of numbers. Results are presented for impact craters generated at DERA Fort Halstead by the penetration of shaped charge jet particles. It is shown that the agreement between the crater profile generated by the Zernike parameter set and the real crater is very good.