Studies of beam propagation characteristics on apertured fractional Fourier transforming systems

Based on the principle that a rectangular function can be expanded into a sum of complex Gaussian functions with finite numbers, propagation characteristics of a Gaussian beam or a plane wave passing through apertured fractional Fourier transforming systems are analyzed and corresponding analytical formulae are obtained. Analytical formulae in different fractional orders are numerically simulated and compared with the diffraction integral formulae, the applicable range and exactness of analytical formulae are confirmed.It is shown that the calculating speed of using the obtained approximate analytical formulae, is several hundred times faster than that of using diffraction integral directly. Meanwhile, by using analytical formulae the effect of different aperture sizes on Gaussian beam propagation characteristics is numerically simulated, it is shown that the diffraction effect can be neglected when the aperture size is 5 times larger than the beam waist size.     

Data-Based Optimal Bandwidth for Kernel Density Estimation of Statistical Samples

It is a common practice to evaluate probability density function or matter spatial density function from statistical samples. Kernel density estimation is a frequently used method, but to select an optimal bandwidth of kernel estimation, which is completely based on data samples, is a long-term issue that has not been well settled so far. There exist analytic formulae of optimal kernel bandwidth, but they cannot be applied directly to data samples, since they depend on the unknown underlying density functions from which the samples are drawn. In this work, we devise an approach to pick out the totally data-based optimal bandwidth. First, we derive correction formulae for the analytic formulae of optimal bandwidth to compute the roughness of the sample's density function. Then substitute the correction formulae into the analytic formulae for optimal bandwidth, and through iteration we obtain the sample's optimal bandwidth. Compared with analytic formulae, our approach gives very good results, with relative differences from the analytic formulae being only 2%~3% for sample size larger than 10⁴. This approach can also be generalized easily to cases of variable kernel estimations.     

有限点方法研究

在二维散乱离散点集上研究一类无网格方法——有限点方法(Finite Point Method,简称FPM),建立方法的基础.采用方向微商和方向差商讨论有限点方法,建立各阶各方向微商间的关系式.利用这些关系式,根据被逼近点的邻点数目差异,分别建立数值方向微商的五点公式及少点(两点、三点、四点)公式;研究五点公式的可解性条件与可允许邻点集;获得典型微分算子的数值方向微商公式等.理论分析和数值试验表明,随着邻点数目的增加,相应数值公式的逼近精度随之提高.这类近似公式不仅为在散乱离散点集上构造各类偏微分方程的格式奠定了基础,同时,也可应用于偏微分方程非结构网格计算方法,提高方法的精度.     

Johann单色器能量分辨率分析

Johann单色器是一种广泛应用的X光单色器,对其能量分辨率已有若干工作进行过分析.对此问题进行了进一步的分析,改进了已有的工作.将计算结果和已有公式的计算结果同时与SHADOW追迹结果进行比较,理论计算与追迹结果有更好地吻合,同时适用范围更大.     

轻离子反射系数的标度公式

最近发现，轻离子输运行为具有显著的标度特性，可用一个称为标度输运截面的参数来标定。本文利用这个重要的性质，给出了以标度输运截面为自变量的轻离子粒子数反射系数和能量反射系数的普适拟合公式。此公式具有非常好的单值对应关系，相比之下，经常应用的约化能ε作为标度参数的反射公式，因其分散度大而相形见绌。 关键词：     

On the accuracy of the approximate formulae for the photon counting statistics of mixed coherent and chaotic radiation

We compare the photon counting statistics for mixed coherent and chaotic radiation calculated with the help of approximate formulae and exact recurrence formulae recently obtained. We investigate Lorentzian, Gaussian and rectangular spectrum of chaotic radiation and show that the accuracy of the approximate formulae, which are relatively simple, is better than 1% if the signal-to-noise ratio is greater than 4 for arbitrary values of the other parameters. If the signal-to-noise ratio is less than one the agreement of the approximate and exact values is only fair. The simple closed-form approximate formulae may be useful particularly for systems employing laser radiation.     

失调光学系统光线追迹公式

本文借助坐标变换，对现有的光线追迹公式进行改进，推导出一套失调光学系统的光线追迹公式。利用该套公式可以计算出光学系统存在各种失调情况下的像差值。     

Improvements to formulae for the ensemble relative variance of random noise in a reverberation room

In this paper possible improvements to the formulae previously developed by the author for predicting the ensemble relative variance of the intensity of a frequency band of random noise which has been passed through a reverberation room are investigated. First the experimentally deduced values of the factor in the formulae which depends on the modal spatial functions are considered. It is shown that these values do not always agree well with theoretically deduced values. Better empirical values for the factor are suggested. Then the assumptions behind the different formulae in the literature for the ensemble relative variance of random noise are considered. These different formulae are then applied to the case when random noise is passed through a reverberation room. In Appendix A the different definitions of filter bandwidth are discussed, while in Appendix B the derivations of the different ensemble variance formulae are presented.     

The vectorial angular-spectrum representation and Rayleigh–Sommerfeld diffraction formulae

The equivalence of the vectorial angular-spectrum representation and Rayleigh–Sommerfeld (RS) diffraction formulae is studied. Based on the angular-spectrum representation and the Weyl representation of a spherical wave, the vectorial RS diffraction formulae of the first and second kinds are derived in a simple way. Numercial results of diffracted divergent spherical waves are given to illustrate the application of the two vectorial RS diffraction formulae.     

光谱仪谱线和谱带弯曲现象的精确表述方式

在矢量衍射理论基础上给出了任意倾斜入射下的光栅方程一般形式以及衍射极角和衍射方位角的完整解析表达式,由此首次导出了平面光栅光谱仪谱线和谱带弯曲量的精确计算公式和对应于衍射极角和衍射方位角的两类角色散公式,比较了谱线弯曲精确公式和近似公式的计算结果,进而指出和弥补了近似公式的不足,并数值考察了入射狭缝高度对光谱仪色散能力的影响程度。由于在以上各式的推导过程中未作任何近似,且涉及到了导致谱线和谱带弯曲的所有可能因素,因而它要比以往所用近似公式更为全面和可靠,可作为实际光谱仪器设计、测试、装调和使用的理论依据。     

Fractional Fourier transform for a hollow Gaussian beam

The fractional Fourier transform (FRT) for a hollow Gaussian beam (HGB) is investigated. Based on the definition of FRT in the cylindrical coordinate system, analytical formulae are derived for the FRT for a HGB. By using the derived formulae, the properties of a HGB in the FRT plane are illustrated numerically. The derived formulae provide a convenient way for analyzing and calculating the FRT of a HGB.     

晶体器件中的光线追迹法

本文以Wollaston棱镜为例将作者原先提出的晶体中光线轨迹公式用于具有多个折射面的晶体器件,阐明公式中各物理量的计算方法,把它与坐标转换关系相结合,提出了一种较简单的光线追迹法,用以研究晶体器件的象差.为晶体器件设计提供了一种新方法.     

Extension of aberration theory of grating systems from one- to two-dimension extended source

The aberration theory of grating systems is extended from one-dimension to two-dimension extended source. In deriving the aberration formulae, the concept that pupil in the meridional plane is separate from the one in the sagittal plane is applied and the principal ray is determined by a linear equation. Using the Seidel aberration for reference, the aberrations are classified into seven types according to the geometrical characteristics. The numerical simulation of imaging of one model optical systems is made with the aberration formulae and exact ray tracing to examine the accuracy of the aberration formulae.     

Forced vibrations of a non-uniform thickness rectangular plate with two free sides

A forced vibration of a rectangular plate with thickness varying linearly in one direction is studied. The plate is simply supported along two opposite sides, and is free along the other sides. Approximate formulae are proposed for estimating the maximum deflection and surface stresses of a rectangular plate subject to a uniformly distributed harmonic lateral load. The accuracy of the formulae is discussed. The approximate formulae show the possibility that one can obtain solutions for the forced vibration of a plate with non-uniform thickness from those for the static bending of the same plate when the plate is subjected to a uniformly distributed load.     

任意截面光波导的模式计算

提出了一种计算近理想形状任意截面光波导模式特性的方法。首先运用微扰理论计算出光波导的电磁场分布,然后从麦克斯韦方程出发推导出一个计算光波导传播常数的公式并由此得出其色散曲线,该方法没有繁杂的计算过程。     

Diffraction of light by adjacent fundamental and second or fourth harmonic ultrasound beams: Comparison of exact and simplified formulae with experiment

Theoretical expressions and their approximate formulae for the intensities of light diffracted by two adjacent ultrasonic beams are discussed and compared with experimental data for the case when the second beam is an even harmonic of the first (fundamental). In the special case of the fundamental and its second or fourth harmonic, relatively simple formulae are found for the first and second diffraction orders describing the variation of light intensity with phase shift and intensity ratio.The approximate formulae describe the experiments satisfactorily only for small values of ζ₁ and α_n. For higher values the exact formulae give better agreement. Only the Raman-Nath region is considered.     

彩色数字图像色差计算方法的研究

用5幅ISO SCID图像在EIZO CG19显示器上分别进行了两组图像色差等级评价实验。两组实验分别在室内自然照明环境和标准灯箱中进行,用目视评价实验数据检验并优化彩色图像色差的计算公式。用CIELAB、CIEDE2000、CIE94和CMC色差公式对第一组实验数据计算结果表明,现有色差公式计算的图像明度差和彩度差与实际色差感觉有明显的系统性差异,图像内容对计算色差与色差感觉的关系有一定影响,说明需要对原始色差公式进行改进。提出了用实验数据拟合直线斜率比值优化色差公式的方法,得到CIELAB(1.50:1)、CIEDE2000(2.29:1)、CIE94(3.04:1)和CMC(3.38:1)计算的色差比原始公式有明显的改善,其中CIEDE2000(2.29:1)计算结果受图像内容的影响最小。用第二组实验数据检验色差公式的优化效果也证明,这种优化方法准确,简单,具有普遍适用性。     

Coherent State Representation for a Kind of Operator Disentangling Formulae

We derive a kind of operator disentangling formulae and their coherent state represen tation by virtue of the correspondence be tween symplectic transformations and their quantum mechanical images. The application of the formulae in calculating Feynman transformation matrix elements is discussed.     

Defect interactions in intrinsic silver chloride

Approximate formulae are developed for the activity coefficients of vacancies and interstitials in intrinsic silver chloride. The numerical values are compared with the Debye-Hückel approximation and the useful range of the formulae is discussed.     

Improved analytical formulae for correcting elastic-scattering effects in X-ray photoelectron spectroscopy

We present new predictive formulae for the correction parameters Q_x and β_eff that are used to account for the effects of elastic scattering in quantitative X-ray photoelectron spectroscopy. These formulae were derived from an analysis of published calculations of Q_x and β_eff from extensive Monte Carlo simulations for a group of elemental solids. Two formulae are given for each parameter. One formula is a function of the single-scattering albedo and the photoelectron emission angle, and is useful for emission angles between 0° and 80°. The other formula is a function only of the single-scattering albedo and is useful for emission angles between 0° and 50°. The single-scattering albedo is in turn a simple function of two material parameters, the inelastic mean free path and the transport mean free path. The latter parameters can be determined readily from available predictive formulae for any material or from databases. The root-mean-square and mean deviations of the Monte Carlo values of Q_x and β_eff from those found with the new formulae are comparable to or smaller than those found with formulae published by Seah and Gilmore.