Deciphering vibration holograms

In vibration holography one measures the temporal Fourier coefficients of the complex amplitude emanating from the vibrating object. However, the user wants to know the temporal Fourier coefficients of the mechanical vibration itself. The mechanical vibrations are proportional to the phase of the complex light amplitude. Hence, the temporal Fourier coefficients of the mechanical vibrations depend in a complicated manner on the temporal Fourier coefficients of the complex amplitude itself. So far, either narrow approximations or complicated Bessel algorithms were used for deducing the mechanical Fourier coefficients from the optical Fourier coefficients. In other words, deciphering the mechanical data from the optical measurements was not easy.We propose a simple and rigorous algorithm for this deciphering project. In addition we outline a holographic experiment that allows one to reconstruct a space- and time-dependent complex amplitude and also its temporal derivative. This experiment is capable of displaying the mechanical velocity distribution as image brightness. Also the spatial distribution of mechanical acceleration can be displayed.     

光学介质表面波度的计算机模型

任一光学介质表面波度对光散射具有不可忽视的影响.本文对用激光轮廓仪所测物体表面曲线进行研究,运用傅里叶级数和最小二乘法拟合光学介质表面波形形态分布,建立了光学介质表面波度的计算机一维模型.     

Evaluation of left ventricular function with cardiac magnetic resonance imaging using Fourier fitting

The aims of this study were to investigate the applicability of Fourier fitting in the magnetic resonance (MR) evaluation of left ventricular (LV) function and to determine the optimal number of harmonics for fitting. Cine cardiac MR imaging was performed in 10 subjects, and an LV time–volume curve was generated. Fourier fitting was applied to the original curve using 1–10 harmonics, and the qualities of the time–volume curve and first-derivative curve were evaluated. LV functional parameters were calculated from curves generated with and without fitting. The quality of the original time–volume curve was good, and Fourier fitting had no substantial effect on functional parameters obtained directly from the time–volume curve such as ejection fraction. The first-derivative curve generated without fitting showed substantial artificial fluctuation. The application of Fourier fitting depressed the fluctuation and tended to decrease estimates of peak ejection rate and peak filling rate. Five or six harmonics appeared to be appropriate for obtaining a high-quality first-derivative curve. In conclusion, Fourier fitting was indicated to aid in reducing the artificial fluctuation of the first-derivative curve generated from cine cardiac MR imaging and to contribute to the evaluation of functional parameters derived from the first-derivative curve.     

Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation

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.     

几种傅里叶子的深入比较研究

用于模式识别的傅里叶子主要有基于极半径函数、转角函数和曲率函数等三种方法 ,通过比较研究 ,找出一种运算简便且精度高的傅里叶子。对不同的图形用三种方法分别计算出原函数和傅里叶频谱 ,然后从大到小依次选取k个 (k =1,2 ,… ,N)傅里叶系数进行反变换 ,计算出反变换函数与原函数之间存在的误差 ,根据误差随k变化的曲线判断三种方法的优劣。实验结果表明 ,基于极半径函数的傅里叶子最为理想。将它用于孢子的识别和计数 ,正确识别率达 99%。     

轨道板中超声兰姆波传播的有限元计算和实验*

该文针对我国高速铁路轨道板缺陷的非接触动态检测问题,研究了空气耦合超声兰姆波在轨道板中的传播规律。首先,给出了轨道板中超声兰姆波的相速度和群速度频散曲线,结果表明:随着频厚积的增加,频散现象越明显,并且A0相速度收敛于Rayleigh波的波速。然后,建立轨道板中波传播的有限元模型,计算得到兰姆波传播的群速度为2220 m/s,且二维傅里叶变换系数的较大值沿Rayleigh波的频散曲线分布。最后,在沪杭高铁嘉兴南站进行了现场测试,以8.8°倾斜角向轨道板激励产生超声兰姆波,激发产生的兰姆波模态群速度为2325 m/s,且二维傅里叶变换分析其系数的较大值沿Rayleigh波的频散曲线分布。有限元计算结果和实验结果均与理论计算结果一致。该研究为后续轨道板缺陷的非接触动态检测提供了理论依据和实验方法。     

Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element

A curve strip Fourier p-element for free vibration analysis of circular and annular sectorial thin plates is presented. The element transverse displacement is described by a fixed number of polynomial shape functions plus a variable number of trigonometric shape functions. The polynomial shape functions are used to describe the element's nodal displacements and the trigonometric shape functions are used to provide additional freedom to the edges and the interior of the element. With the additional Fourier degrees of freedom (dof) and reduce dimensions, the accuracy of the computed natural frequencies is greatly increased. Results are obtained for a number of circular and annular sectorial thin plates and comparisons are made with exact, the curve strip Fourier p-element, the proposed Fourier p-element and the finite strip element. The results clearly show that the curve strip Fourier p-element produces a much higher accuracy than the proposed Fourier p-element, the finite strip element.     

Asymptotic behaviour of the Fourier coefficients of MAPW wave functions

In contrast to the conventional APW method the MAPW method imposes continuity conditions on the trial wave functions which guarantee that the wave functions and their first derivatives are continuous throughout the whole atomic polyhedron. It is shown that as a consequence of this fact the Fourier coefficients of the MAPW wave functions approach zero as 1/p ⁴ whereas conventional APW functions decay as 1/p ² only. For the MAPW Fourier coefficients asymptotic formulas are derived that may be useful in numerical calculations. Numerical results are given for metallic Li and Cu.     

Fourier analysis for rotating-element ellipsometers

We introduce a Fourier analysis of the waveform of periodic light-irradiance variation to capture Fourier coefficients for multichannel rotating-element ellipsometers. In this analysis, the Fourier coefficients for a sample are obtained using a discrete Fourier transform on the exposures. The analysis gives a generic function that encompasses the discrete Fourier transform or the Hadamard transform, depending on the specific conditions. Unlike the Hadamard transform, a well-known data acquisition method that is used only for conventional multichannel rotating-element ellipsometers with line arrays with specific readout-mode timing, this Fourier analysis is applicable to various line arrays with either nonoverlap or overlap readout-mode timing. To assess the effects of the novel Fourier analysis, the Fourier coefficients for a sample were measured with a custom-built rotating-polarizer ellipsometer, using this Fourier analysis with various numbers of scans, integration times, and rotational speeds of the polarizer.     

Reconstructing the Thermal Green Functions¶at Real Times from Those at Imaginary Times

By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism. The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the “unique Carlsonian analytic interpolations” of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes. Starting from the Fourier coefficients regarded as “data set”, we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations. Received: 17 February 2000 / Accepted: 12 July 2000     

3～5μm角度调谐LiNbO3晶体光参量振荡器的技术研究

用LiNbO₃作参量晶体、调QNd:YAG(1.064μm)激光器做泵源,通过求出相位匹配点对有效非线性系数进行数值计算,找出有效非线性系数的最大值,从而求出该点对应的相位角和方位角,确定出进行角度调谐LiNbO₃晶体的切割角在理论计算过程中,用计算机模拟,得出了OPO的角度调谐曲线、有效非线性系数曲线和参量增益曲线实验研究中实现了369μm单点参量光输出,实验结果与理论计算结果相吻合.     

Analytical expression for the nth Fourier coefficient of a modulated Lorentzian dispersion lineshape function

Modulated spectroscopic detection techniques that rely on dispersion, e.g. Faraday modulation/rotation spectroscopy and wavelength-modulated noise-immune cavity-enhanced optical heterodyne molecular spectroscopy, are often described in terms of Fourier coefficients of a modulated dispersion lineshape function. This work derives a non-complex analytical expression for the nth Fourier coefficient of a modulated Lorentzian dispersion lineshape function. The expression is easier to implement, and orders of magnitude faster to execute, than previous approaches involving numerical calculations of integrals. The first six Fourier coefficients are explicitly given and illustrated for their optimum modulation amplitudes, which are also given.     

Soluble Boltzmann equations for internal state and Maxwell models

We consider a class of scalar nonlinear Boltzmann equations describing the evolution of a microcanonical ensemble in which sub-systems exchange internal energy ‘randomly’ in binary interactions.In the continuous variable version these models can equally be interpreted as Boltzmann equations for Maxwell type molecules in arbitrary dimensionality.We construct general solutions in the form of a Fourier series; the expansion coefficients (Sonine or Meixner moments) satisfy the same recursive system of coupled equations as the ordinary moments, and can be solved sequentially.     

Electronic bandstructure of disordered superlattices

The disordering of semiconductor superlattices is analysed in the spatial Fourier domain. It is demonstrated that disordering is equivalent to suppressing the higher-order Fourier coefficients of the alloy profile. Two superlattice bandstructure algorithms are developed in the Fourier domain which allow the electronic envelope function and energy to be determined. The first employs the effective mass approximation. The second is a 14-bandk·pwhich includes nonparabolicity, anisotropy and noncentrosymmetry which are particularly relevant in the determination of nonlinear optical coefficients.     

The physical interaction of gases with crystalline solids: I. Gas-solid energies and properties of isolated adsorbed atoms

It is shown that the potential energy of a gas atom interacting with a solid having a surface made up of single type of exposed lattice plane can be expressed as a Fourier series in the position variables in the plane parallel to the surface. Assuming that the total interaction is a pair-wise sum of gas atom-solid atom terms, an analytic expression for the Fourier coefficients is obtained for inverse power law interactions that is a generalization of an earlier result of Hove and Krumhansl. Gas-solid potentials calculated for several surface lattices by direct summation are compared with those given by the truncated Fourier series, and it is concluded that a satisfactory representation of the energy is obtained even when the number of terms included is small. The truncated series is then utilized to calculate Henry's Law adsorption constants and average energies for an isolated adsorbed atom. These results are compared with calculations based on completely localized and perfectly mobile adsorption models.     

ATLAS potential for elliptic flow measurements in Pb+Pb collisions at LHC

Study of particle azimuthal anisotropies, quantified by coefficients of the Fourier decomposition of the azimuthal angle distributions, has emerged as an important tool to probe the early stages of nucleusnucleus collisions. In this paper the ATLAS capability of measuring elliptic flow, the second Fourier coefficient, is presented. It is shown that various analysis techniques can be applied for different combinations of the detector sub-systems. A systematic comparison of various analysis methods is also shown.     

Regeneration of Form by Fourier Technique and its Application in Powder Characterization

Conventional shape parameters have serious limitations because of their inability to capture finer surface morphology as well as to regenerate the particle form. This problem has been successfully circumvented by applying Fourier analysis technique to the projected image of the particle profile to obtain Fourier coefficients which have been shown to be size inveriant and insensitive to starting point for digitization. Synthesis technique to regenerate the particle profile from Fourier coefficients has revealed that gradual addition of higher order harmonics brings the regenerated form very close to its original counterpart. The regeneration of the form of particles of different shapes as well as application of few strategy to harmonic amplitudes have amply demonstrated the capability of the technique as a potential tool for characterization of powders.     

The Rate of Convergence of Fourier Coefficients for Entire Functions of Infinite Order with Application to the Weideman-Cloot Sinh-Mapping for Pseudospectral Computations on an Infinite Interval

We analytically compute the asymptotic Fourier coefficients for several classes of functions to answer two questions. The numerical question is to explain the success of the Weideman-Cloot algorithm for solving differential equations on an infinite interval. Their method combines Fourier expansion with a change-of-coordinate using the hyperbolic sine function. The sinh-mapping transforms a simple function like exp(-z²) into an entire function of infinite order. This raises the second, analytical question: What is the Fourier rate of convergence for entire functions of an infinite order? The answer is: Sometimes even slower than a geometric series. In this case, the Fourier series converge only on the real axis even when the function u (z) being expanded is free of singularities except at infinity. Earlier analysis ignored stationary point contributions to the asymptotic Fourier coefficients when u(z) had singularities off the real z-axis, but we show that sometimes these stationary point terms are more important than residues at the poles of u(z).