曲率波前传感器探测高斯光束时的信号误差

曲率波前传感器已被用于天文自适应光学和光学度量等领域。在这些领域使用时都假设入射波前光强均匀,但这种假设与曲率传感技术的基本原理不一致。利用傅里叶光学理论,给出了光强不均匀情况下曲率波前传感器的曲率信号解析式,并利用光强均匀和不均匀情况下的信号表达式对探测高斯光束时的信号误差进行了数值分析。结果表明：曲率波前传感器探测高斯光束时存在一定误差,相位分布为4阶Zernike多项式时,误差最大,且阶数越高,误差越小;分区平均曲率信号误差较小,一般在10％以下。     

Spatial impulse responses from a flexible baffled circular piston

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.     

Calculation of resonance Raman excitation profiles

The present work describes a novel mathematical scheme for the evaluation of multi-dimensional time domain integrals that arise in the calculation of resonance Raman excitation profiles and the optical absorption spectra when the electronic transitions take place between displaced, distorted, and rotated harmonic potential energy surfaces. A recipe is provided for formulating the time domain integrals in terms of a tractable summation of products involving hermite polynomials. Expressions for the special cases (displaced and/or distorted oscillators) are written in terms of mathematical special functions which are computationally easier to deal with. Relations with the so-called transform theory are discussed. Resonance Raman excitation profiles and the optical absorption spectrum for a 7-mode system are calculated as an illustration of our expressions.     

Scaling pseudo-Zernike expansion coefficients to different pupil sizes

Orthogonal polynomials are routinely used to represent complex surfaces over a specified domain. In optics, Zernike polynomials have found wide application in optical testing, wavefront sensing, and aberration theory. This set is orthogonal over the continuous unit circle matching the typical shape of optical components and pupils. A variety of techniques has been developed to scale Zernike expansion coefficients to concentric circular subregions to mimic, for example, stopping down the aperture size of an optical system. Here, similar techniques are used to rescale the expansion coefficients to new pupil sizes for a related orthogonal set: the pseudo-Zernike polynomials.     

基于泽尼克多项式进行面形误差拟合的频域分析

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。     

Application of jacobi polynomial methods to the one-speed transport equation

The transport equations associated with radiation damage studies are often solved using expansions in Legendre polynomials. The radiation damage distribution functions which satisfy these equations may be sharply peaked in the forward direction, while the Legendre polynomials, as a set, are isotropic. This situation requires the use of many terms in the Legendre expansion in order to adequately represent the distribution functions. The Jacobi polynomials, on the other hand, can have strong peaking built into their associated weight function. To test the usefulness of the Jacobi polynomials we use them to solve the simple, one-speed, neutron transport equation. The results are then compared to the exact theory and to the results of applying Legendre methods to the same problem. This sample calculation demonstrates the advantage of the Jacobi polynomials in strongly non-isotropic situations.     

Numerical study of paraxial beam formation

A method for numerical study of different integral problems with sufficiently strongly localized solutions is considered. Approximations of the solution with different degrees of accuracy were constructed and, using the existing approximation, an approach to an increase of the solution accuracy is proposed. Numerical solutions are obtained for the problem of free oscillations of laser cavities the optical surface of which was approximated by polynomials of the second and fourth degree inclusive. The obtained numerical solutions were stable.     

Theory of a FIR optical digital differentiator

An optical realization of a proposed family of the finite impulse response (FIR) qth-order mth-derivative digital differentiators (under coherent operation), whose transfer functions are derived from the well-known numerical differentiation scheme (namely, the backward Taylor series expansion), using the amplified fiber optic transversal filters is presented. Several input pulse shapes, such as polynomials of various orders and the Gaussian and exponential pulses, are chosen as examples for illustration of the accuracy of the proposed optically realized differentiators. A mathematical model describing the dynamic (or transient) response of any optical discrete-time systems using the z-transform technique is proposed. The real-time performances of the proposed differentiators have been analyzed using both the steady-state and the dynamic analyses, Although the analysis is directed at fiber optic systems, the methodology and the results are applicable to other physical systems.     

Perturbation expansion of quantum resonance energies

We present new results concerning the theoretical treatment of quantum resonances based on the use of optical potentials and perturbation theory. A new criterion is presented for selecting an optical potential. As a result, a wide class of optical potentials which lead to efficient asymptotic series is available. A numerical illustration for two shape resonances show that accurate resonance energies and lifetimes can be obtained from a very small number of real square-integrable functions.     

基于衍射光栅曲率波前传感器的实时波前重构研究

自适应光学系统要求波前传感器能实现动态实时测量,曲率波前传感技术符合这一发展要求。一种新型的基于扭曲衍射光栅的曲率波前传感器在探测装置的实现方法方面具有较大优势,其波前重构已应用于光学度量。根据衍射光学理论,对其探测信号进行数值模拟,并利用Neumann边界条件的Green函数法对其波前重构进行数值模拟。结果表明：Green函数法归结为2矩阵相乘,计算速度快,达到实时重构要求; Green函数法对阶数不高的Zernike多项式重构效果较好;影响重构误差的主要因素是光强梯度的边界噪声。     

光学干涉波面数字化处理方法与应用

全面论述了激光数字平面检测系统中的核心技术——干涉波面数字化分析理论和处理方法,对光学干涉波面的拟合函数系的选择、拟合算法以及系统像差和被测表面平面误差类型判定方法进行了深入研究,提出了在"Zernike像差空间"确定激光数字平面检测系统精度的理论和方法.     

Quantum theory of super-resolution for optical systems with circular apertures

We present the two-dimensional quantum theory of super-resolution, applicable for a large variety of optical systems with circular pupils. Our theory is formulated in terms of circular prolate spheroidal functions which form the eigen basis of two-dimensional imaging system with circular pupils. We provide, in particular, analytical and numerical results for the point-spread function characterizing reconstruction of optical objects with super-resolution from diffraction-limited images. We evaluate the super-resolution factor as a function of the signal-to-noise ratio in the input object for coherent light and multimode squeezed light.     

Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármán's nonlinear strain-displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.     

Correlations of RMT characteristic polynomials and integrability: Hermitean matrices

Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of τ functions, we (i) identify a zoo of hierarchical relations satisfied by τ functions in an abstract infinite-dimensional space and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasis is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.     

数字平面检测系统误差和精度评价方法的研究

通过对光学平面玻璃样品表面检测结果的分析，对激光数字平面检测系统中，应用泽尼克多项式进行平面分析的系统理论误差以及被测表面形状类型的判别方法进行了深入的研究，并应用误差分析理论，提出了一个在“泽尼克像差空间”确定激光数字平面检测系统精度的理论和方法。     

Theory of a FIR optical digital differentiator

Abstract

An optical realization of a proposed family of the finite impulse response (FIR) qth-order mth-derivative digital differentiators (under coherent operation), whose transfer functions are derived from the well-known numerical differentiation scheme (namely, the backward Taylor series expansion), using the amplified fiber optic transversal filters is presented. Several input pulse shapes, such as polynomials of various orders and the Gaussian and exponential pulses, are chosen as examples for illustration of the accuracy of the proposed optically realized differentiators. A mathematical model describing the dynamic (or transient) response of any optical discrete-time systems using the z-transform technique is proposed. The real-time performances of the proposed differentiators have been analyzed using both the steady-state and the dynamic analyses, Although the analysis is directed at fiber optic systems, the methodology and the results are applicable to other physical systems.     

Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations

A numerical method for solving nonlinear initial-value problems is proposed. The Lane-Emden type equations which have many applications in mathematical physics are then considered. The method is based upon hybrid function approximations. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials are presented and are utilized to reduce the computation of nonlinear initial-value problems to a system of non-algebraic equations. The method is easy to implement and yields very accurate results.     

Biorthogonal ensembles

One object of interest in random matrix theory is a family of point ensembles (ramdom point configurations) related to various systems of classical orthogonal polynomials. The paper deals with a one-parametric deformation of these ensembles, which is defined in terms of the biorthogonal polynomials of Jacobi, Laguerre and Hermite type.Our main result is a series of explicit expressions for the correlation functions in the scaling limit (as the number of points goes to infinity). As in the classical case, the correlation functions have determinatal form. They are given by certain new kernels which are described in terms of Wright's generalized Bessel function and can be viewed as a generalization of the well-known sine and Bessel kernels.In contrast to the conventional kernels, the new kernels are non-symmetric. However, they possess other, rather surprising, symmetry properties.Our approach to finding the limit kernel also differs from the conventional one, because of lack of a simple explicit Christoffel-Darboux formula for the biorthogonal polynomials.     

Axisymmetric vibration of prestressed non-uniform cantilever cylindrical shells

The extended Galerkin method has been used in the investigation of axially loaded clamped-free homogeneous, isotropic and elastic cylindrical shells. Both mass and stiffnesses are considered to vary along the longitudinal direction. Legendre polynomials have been used as shape functions which lead to a simple and systematic procedure in determining the natural frequencies and mode shapes. Some numerical results are presented.     

VIBRATION OF TWISTED AND CURVED CYLINDRICAL PANELS WITH VARIABLE THICKNESS

A numerical procedure, with an exact strain-displacement relationship of twisted and curved cylindrical panels having variable thickness derived by considering the Green strain tensor on general shell theory, is presented using the principle of virtual work and the Rayleigh-Ritz method with algebraic polynomials as in-plane and transverse displacement functions. The accuracy and applicability of the procedure are verified by comparing the present results with previous experimental and theoretical results for several panels. The effects of variation ratio of thickness in chordwise and lengthwise directions, twist, and curvature both in two directions aforementioned on vibrations of cylindrical panels are studied in detail, and typical vibration mode shapes are plotted to demonstrate the effects.