Matrix formulation of axis-symmetric laser beams through a paraxial ABCD system containing hard-edged apertures: A comparative study

A matrix formulation is presented, which enables us to study the propagation of axis-symmetric beams through a paraxial optical ABCD system containing hard-edged aperture. Numerical calculation results of super-Gaussian beams passing through a multi-aperture-lens system are given to illustrate the advantage of the method. A comparison of the matrix formulation, complex Gaussian expansion and direct numerical integration of the Collins formula is made, where the propagation of apertured Laguerre-Gaussian beams is chosen as an illustrative example. It is shown that the matrix formulation provides satisfactory results in both Fraunhofer and Fresnel regions, and reduces the computational time greatly in comparison with the direct integration. However, this method is suited only to axis-symmetric optical beams and systems. By using the complex Gaussian expansion discrepancies exist in the near zone closer to the aperture, but usually its computational efficiency is higher than the matrix formulation.     

On the Second-Order Correlation Function of the Characteristic Polynomial of a Hermitian Wigner Matrix

We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit formula for the exponential generating function of the second-order correlation function of the characteristic polynomial. Furthermore, we show that the second-order correlation function of the characteristic polynomial is closely related to that of the permanental polynomial. Supported by CRC 701 “Spectral Structures and Topological Methods in Mathematics”.     

On a Graph Theoretic Formula of Gammelgaard for Berezin–Toeplitz Quantization

We give a proof of (a slightly refined version of) a graph theoretic formula due to Gammelgaard, Karabegov and Schlichenmaier for Berezin–Toeplitz quantization on Kähler manifolds. We obtain the formula by inverting the Berezin transform using a composition formula for the ring of differential operators encoded by linear combinations of strongly connected graphs. The same method is also used to identify the dual Karabegov–Bordemann–Waldmann star product. Our proof has the merit of giving more insight into Karabegov–Schlichenmaier’s identification theorem (Karabegov in J Reine Angew Math 540:49–76, 2001) that the Karabegov classifying form of the Berezin and Berezin–Toeplitz star products are, respectively, obtained by deforming the Kähler metric along the Ricci curvature and the logarithm of the Bergman kernel.     

高斯光束通过含失调窄缝光阑的失调光学系统的传输特性

 利用硬边窄缝光阑的近似展开式和适用于失调光学系统的广义衍射公式,得出了高斯光束经含失调窄缝光阑的失调光学系统传输的近似解析式。模拟结果表明输出光束场分布与光束参量、光阑尺寸、ABCD矩阵元、光阑失调量和光学系统失调量有关。针对给定的光学系统和高斯光束定量分析了各失调量对输出光束场分布的影响,结果表明:光阑横向位移、光学系统横向位移和角位移均引起垂直于z轴截面内明显的光强非轴对称分布。当光阑半宽度为1 mm时,光阑的衍射作用使腰斑半径为0.5 mm的高斯光束产生-1.586π～1.465π范围的相对相移,且光阑横向位移、光学系统横向位移和角位移均引起焦平面前后相对相移的迅速变化。随光阑宽度变小,各失调量对输出光束特性的影响越明显。     

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derived an approximate analytical formula for a partially coherent twisted anisotropic Gaussian Schell-model (AGSM) beam propagating through an apertured paraxial general astigmatic (GA) optical system by use of a tensor method. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. Our formulas avoid time-consuming numerical integration and provide a convenient and effective way for studying the propagation and transformation of a partially coherent twisted AGSM beam through an apertured paraxial GA optical system.     

Correlations Between Zeros and Supersymmetry

In our previous work [BSZ2], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions. Received: 13 November 2000 / Accepted: 23 February 2001     

Measures for Fermionic Integration

The formal Berezin integral for anticommuting variables is modified to a functional which can be applied directly to an infinite dimensional Grassmann algebra. Then a Gaussian measure is constructed on an extended Grassmann algebra and the functional integral is realized as integral on this measure space.     

一种新空心光束的理论及实验研究

提出了一种用来描述空心光束的新理论模型——空心高斯光束模型，通过控制光束参数可以方便地调制空心光束黑斑的大小.导出了空心高斯光束经傍轴光学系统的解析传输变换公式.利用导出的公式计算分析了空心高斯光束在自由空间的传输特性.在实验上利用矩阵本征值方法设计了能输出空心高斯光束的光学谐振腔，对实验上得到的空心高斯光束进行了传输特性的研究.结果表明，理论和实验相吻合.该空心高斯光束产生的光势阱为控制冷原子提供了一条新途径. 关键词： 空心高斯光束 光学谐振腔 传输特性     

核能谱单能峰快速高精度高斯函数拟合算法研究

在核能谱分析中,高斯函数最小二乘拟合法是计算单能峰净峰面积常用的方法,该方法精度较高,但噪声敏感性较强,导致拟合出的高斯函数在峰位附近的残差向量较大。针对该问题,对高斯函数最小二乘拟合法进行了详细推导,分析了峰位附近残差向量较大的原因,提出了一种基于高斯函数最小二乘拟合法的高斯函数加权最小二乘拟合法,即在高斯函数最小二乘拟合法的基础上,引入了权重因子。该权重因子与取对数后数据权重削弱趋势相反或与数据本身趋势相符,以减小噪声敏感性。由于在求解高斯函数参数的过程中涉及到求逆矩阵运算,计算量较大,耗时较长,为了提高实时性,将求逆矩阵的运算过程转换为了简单的方程组运算,并给出了高斯函数的幅值、中心及方差参数的快速求解公式。将这两种方法用于55Fe的特征X射线单能峰的实际拟合中,结果表明,高斯函数加权最小二乘拟合法效果均较好,这说明该方法降低了噪声敏感性,减小了高斯函数在峰位附近的残差向量,进一步提高了拟合精度。另外,使用快速求解公式,也减小了运算量,增强了实时性,为在便携式设备中的有效使用提供了可能。     

A Factor Analysis Perspective on Linear Regression in the ‘More Predictors than Samples’ Case

Linear regression (LR) is a core model in supervised machine learning performing a regression task. One can fit this model using either an analytic/closed-form formula or an iterative algorithm. Fitting it via the analytic formula becomes a problem when the number of predictors is greater than the number of samples because the closed-form solution contains a matrix inverse that is not defined when having more predictors than samples. The standard approach to solve this issue is using the Moore–Penrose inverse or the L2 regularization. We propose another solution starting from a machine learning model that, this time, is used in unsupervised learning performing a dimensionality reduction task or just a density estimation one—factor analysis (FA)—with one-dimensional latent space. The density estimation task represents our focus since, in this case, it can fit a Gaussian distribution even if the dimensionality of the data is greater than the number of samples; hence, we obtain this advantage when creating the supervised counterpart of factor analysis, which is linked to linear regression. We also create its semisupervised counterpart and then extend it to be usable with missing data. We prove an equivalence to linear regression and create experiments for each extension of the factor analysis model. The resulting algorithms are either a closed-form solution or an expectation–maximization (EM) algorithm. The latter is linked to information theory by optimizing a function containing a Kullback–Leibler (KL) divergence or the entropy of a random variable.     

复变几何光学

本文在复平面上描述高斯光束的特性，并利用复数分式线性变换及其几何反演点对研究高斯光束的传输与变换，推导出几何光学的高斯光束类牛顿公式和类高斯公式，同时阐明了高斯光束菲涅耳数与焦移概念的本性．     

*-Products on some Kähler manifolds

Starting from work of F. A. Berezin, in this Letter we define an invariant *-product on every nonexceptional Kähler symmetric space. We then obtain a recursion formula to calculate the corresponding invariant Hochschild 2-cochains for types I and IV spaces.     

BEPCII横向束流反馈系统前端电子学

选择了一台商业化的前端电子学RFTF（radio frequency front-end for transverse feedback system）用于BEPCII横向束流反馈系统, 这是跟进世界先进加速器技术的具体实施。在BEPC同步运行模式下对RFTF的性能进行了测试实验，结果表明使用RFTF可以得到理想的波形，能够满足BEPCII横向反馈系统的需要，但是，当储存环出现不稳性时，系统无法正常工作，这就要求BEPCII纵向不稳定很小或者用纵向反馈系统来抑制存在的纵向不稳定。     

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential

In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.     

Visualization of the Barkhausen effect by magnetic force microscopy

By visualization of the Barkhausen effect using magnetic force microscopy we are able to provide detailed information about the physical principles that govern the magnetization reversal of a granular ferromagnetic thin film with perpendicular anisotropy. Individual Barkhausen volumes are localized and distinguished as either newly nucleated or grown by domain wall propagation. The Gaussian size distribution of nucleated Barkhausen volumes indicates an uncorrelated random process, while grown Barkhausen volumes exhibit an inverse power law distribution, which points towards a critical behavior during domain wall motion.     

Analytical study of a flattened Gaussian beam through multi-apertured optical imaging systems of B = 0

By using the methods of the matrix decomposition and expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, the recurrence propagation expressions for a flattened Gaussian beam (FGB) through multi-apertured optical imaging systems of B = 0 are derived and illustrated with numerical examples. Comparisons with the straightforward numerical integration of the Collins formula and with the previous work are made. It is shown that the main advantages of our methods and results are the more accuracy and great reduction of computer time.     

Correlation functions and correlation times for models with multiplicative white noise

Correlation functions and correlation times for the Stratonovich and Verhulst model are investigated. By transforming the Fourier transform of the corresponding Fokker-Planck equation into a tridiagonal vector recurrence relation, the Fourier transform of the correlation function and the correlation time are expressed in terms of matrix continued fractions or by similar iterations and are thus obtained numerically. By using the inverse Fourier transform, the correlation function itself is calculated. Furthermore an analytic expression in terms of an integral is obtained for the correlation time, which is evaluated exactly in the Verhulst model and asymptotically for large and weak noise strength in the Stratonovich model. A Padé expansion approximating the correlation time for all noise strength is also given.     

二维高斯分布光斑中心快速提取算法研究

在深入分析二维高斯分布公式的基础上,通过将光斑中心整像素坐标和亚像素坐标进行分离,推导出一种无需求解广义逆矩阵的高斯曲面解析算法,该方法综合利用窗口内的所有像素灰度信息,通过解析表达式直接计算高斯分布光斑的亚像素中心位置;并且对传统高斯曲面拟合法求解过程进行了优化,提出一种更加高效的定参高斯拟合法。与传统高斯曲面拟合法相比,提出的两种方法具有基本相同的稳定性和定位精度,但运行效率分别提高了278倍和78倍以上。     

双曲余弦高斯光束通过有光阑限制的ABCD光学系统的传输

对双曲余弦高斯光束通过有硬边光阑的一阶ABCD光学系统的传输进行了研究,采用将矩形域函数表示为复高斯函数叠加的技巧,推导出了解析的传输公式,在特殊情况下,该公式简化为在无光阑情况下的传输公式,对双曲余弦高斯光束通过有光阑限制的薄膜镜聚焦进行了数值计算,计算结果与直接由柯林斯（Collins)公式所得结果一致,且此解析方法便于进行物理分析,可节约大量机时。