Electromagnetic wave scattering by small bodies

A reduction of the Maxwell's system to a Fredholm second-kind integral equation with weakly singular kernel is given for electromagnetic (EM) wave scattering by one and many small bodies. This equation is solved asymptotically as the characteristic size of the bodies tends to zero. The technique developed is used for solving the many-body EM wave scattering problem by rigorously reducing it to solving linear algebraic systems, completely bypassing the usage of integral equations. An equation is derived for the effective field in the medium, in which many small particles are embedded. A method for creating a desired refraction coefficient is outlined.     

Application of the singularity-subtraction technique to isotropic scattering in a planar layer

A numerical procedure for solving a singular Fredholm integral equation, which describes radiative transfer in an absorbing and isotropically scattering layer exposed to collimated radiation, is studied. The integral equation is solved by subtracting out the singularity and then approximating the integral term by a Gaussian quadrature. Bidirectional and hemispherical properties are found from the source function. Any arbitrary directional distribution of incident radiation can be handled by superimposing the collimated radiation case. In particular, the case of isotropic incident radiation is presented. The method gives accurate results, even for the extreme conditions of large optical thickness and large angle of incidence.     

The density of states in the mobility gap of amorphous-silicon films computed by a new analytical method

Summary The field effect technique has been used to measure the density of states in the mobility gap of silicon films. We used a new method to compute the density of states distribution. The method is based on an analytical solution of the Fredholm integral equation, which relates the space charge density to the density of states. To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

水下光场的迭代求解

水体对光线的散射是水下图像质量劣化的重要因素,为了定量分析在特定光源照射下水体散射的影响,建立了光线水下传输的散射模型,以此为基础推导出求解水下光场分布的Fredholm积分方程.在水中光线能量随距离的增大呈指数规律衰减,基于此,在水体体散射函数为常数的情况下,给出了有边界条件时该积分方程的数值迭代求解方法,得到高精度...     

The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix

It is shown that the cross-spectral density matrix of a planar, secondary, stochastic electromagnetic source is a correlation matrix. Unlike a previously published proof of this result, our derivation is based on the solution of two uncoupled, rather than two coupled, integral equations.     

环形等离子体平衡时维持场电流的计算

本文介绍一个计算轴对称任意截面环形等离子体平衡问题的方法。先选定等离子体边界和环电流分布,用有限元方法解平衡方程的边值问题。借助于虚壳原理,得到用平衡方程解表示的能够产生平衡时所需要的维持场的虚壳电流。计算虚壳电流在等离子体区的维持场,以它为根据,采用积分方程开拓,求等离子体区外某位形上的维持场电流分布。解决这个问题的主要困难是当磁场向外开拓时遇到了不适定问题。我们用奇异值分解方法解不适定的Fredholm第一类积分方程。这个方法能容易地找到稳定解,对解决这个平衡问题是简单有效的。我们以七种等离子体截面形状,三种电流分布为具体模型,在三种维持场电流分布位形上给出了维持场电流分布。还给出了维持场形态,维持场总电流与等离子体总电流的比较,并简单讨论了维持场对等离子体整体稳定性的影响。 关键词：     

Sources with factorized cross-spectral densities

Model sources that are sometimes employed in optical coherence theory are characterized by cross-spectral density functions of a certain factorized form. One of the factors represents the source intensity and the other the degree of coherence of the light across the source. We show that the intensity must necessarily satisfy a certain functional equation, which is found to imply that the intensity distribution must be an exponential function of position. We also show that when one does not impose any requirement as to the significance of the two factors, or when one demands that the factorization holds only approximately, such factorized cross-spectral densities represent a much broader class of sources, including quasi-homogeneous ones.     

Shielding of a low-frequency electric field by a multilayer circular disk

We have solved the boundary-value problem with nonclassical boundary conditions, which describes the penetration of a low-frequency field through a multilayer thin circular disk. The solution of the problem is reduced to solving the Fredholm integral equation of the second kind. The influence of some parameters of the problem on the shielding factor is investigated numerically.     

光全散射法颗粒测量中的一种自适应非独立模式算法

光全散射颗粒测量法中需求解第一类Ｆｒｅｄｈｏｌｍ积分方程。在不知道被测颗粒的尺寸范围的情况下如何确定积分方程的上下限、即测量范围，迄今仍是个问题。本文针对此问题提出了一种自适应非独立模式算法。该算法可根据双参数颗粒分布函数的分布参数Ｋ自动确定积分的上下限．数值模拟和实验都证实了此算法。     

The reconstruction of density and Young’s modulus of an inhomogeneous rod

The problems of reconstructing the Young modulus and the density of an inhomogeneous elastic rod by analyzing its longitudinal and bending vibrations are considered. For solving these problems, an iteration algorithm based on the apparatus of the Fredholm integral equations is proposed. Specific examples of reconstruction are presented.     

Determination of the Parameters of the Equivalent Two-Pole Circuit for an Electric Discharge Induced by Voltage Pulses

An algorithm (method) of determining the parameters of the two-pole equivalent electric circuit for an electric discharge is described in the present paper. The algorithm (method) is based on a solution of the ill-posed inverse problem of determining a kernel of the Duhamel integral for the first-order Fredholm integral equation. The fast Fourier transform (FFT) algorithm used to solve this problem reduces significantly the execution time of the algorithm. Examples of solving model problems for first- and second-order electric equivalent circuits are presented which demonstrate the universality and efficiency of the suggested algorithm.     

Chapman-Kolmogorov equation and path integrals for discrete chaos in presence of noise

We derive an equation of the Chapman-Kolmogorov type for discrete multi-dimensional mappings under the action of additive and multiplicative noise with arbitrary distribution function. The resulting equation is reduced to a Fredholm integral equation. By iteration of the Chapman-Kolmogorov equation as usual, a path integral solution is found. Specializing the distribution function of the noise to a Gaussian distribution and taking the Fourier transform contant can be made with the path integral formulation used by Shraiman, Wayne and Martin.     

Reduction of radiative transfer problems in semi-infinite media to linear Fredholm integral equations

Linear Fredholm integral equations are derived for the Stokes vector of polarized radiation, emergent from a scattering plane parallel semi-infinite medium, by means of the full range orthogonality and completeness properties of Case's eigensolutions. A renormalization concerning the eigenmode with the greatest discrete eigenvalue is applied, which permits us to obtain a new integral equation for the zeroth Fourier component of the radiation field. The kernel of the integral equations is given in terms of Case's eigenfunctions or of the Green's function matrix for an infinite medium. For isotropic scattering, it is shown that the integral equation can be solved by means of a very rapidly convergent Neumann series. Physical arguments lead to the conclusion that the renormalized Fredholm integral equations are well suited also for arbitrary phase matrices.     

Is the Van Cittert-Zernike Theorem Applicable to Spatially Incoherent Broadband Spectral Source?

This paper deals with the propagation law of mutual intensity and cross-spectral density of the light emitted from an extended broadband spectral source that is spatially incoherent. The mutual intensity of light does not obey the same mathematical formalism as the Van Cittert-Zernike theorem applicable to quasi-monochromatic light, although the cross-spectral density of light does in the propagation zone of the Fresnel diffraction region.     

Generalised Error Functions from the Kerr Metric

Motivated by the effort to understand the mathematical structure underlying the Teukolsky equations in a Kerr metric background,a homogeneous integral equation related to the prolate spheroidal function is studied.From the consideration of the Fredholm determinant of the integral equation,a family of generalized error function is defined,with which the Fredholm determinant of the sinc kernel is also evaluated.An analytic solution of a special case of the fifth Painleve transcendent is then worked out explicitly.     

基于压缩感知的目标频空电磁散射特性快速分析

矩量法是求解目标电磁散射问题的一种常用数值方法,因其精度较高而被广泛应用.应用矩量法求解目标频空电磁散射特性时,随着入射波的角度和频率的变化,需要间隔很小的角度和频率步长反复求解矩量法生成的矩阵方程,运算量极大.为解决此类问题,本文结合压缩感知理论和渐近波形估计形成一种新的有效计算方法.首先,基于压缩感知理论引入一种富含空间信息的新型入射源,其次,在该入射源照射下应用渐近波形估计技术求解,从而快速实现目标频空电磁散射特性分析.     

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.     

Rigorous and approximate methods for modeling wave scattering from a locally perturbed perfectly conducting surface

Rigorous and approximate methods are considered for solving the problem of harmonic plane wave scattering from a plane surface arbitrarily perturbed along one dimension on a finite interval. This problem is treated using the Fredholm integral equations of the second kind and the Kirchhoff and Rayleigh approximations. The estimates of the computational efficiency of the integral equation method and the Rayleigh approximation are compared by calculating fields scattered from random rough surfaces in the resonance region (i.e., when the roughness height is comparable to or smaller than the incident wavelength) for an arbitrary incidence of a plane wave. Scattering patterns calculated using the integral equations and the Kirchhoff approximation are discussed in the case of large-scale random rough surface scattering. Particular attention is paid to scattering at near-grazing incidence.     

An analytical technique for calculation of distributed resistance in a rectangular domain

A Green's function approach is used to obtain an integral representation for the electric potential in a multi-terminal distributed resistive structure. This integral representation for the potential is then used to formulate a coupled system of Fredholm integral equations of the first kind, in which the normal components of current density over the different terminal contacts are the unknowns. A procedure for solving this system of equations is presented and the obtained results for the unknown normal components of the current density are used to express the electric potential and current density vector at any point in the domain of the distributed resistive structure. The indefinite admittance matrix relating the terminal currents to the applied terminal voltages, is a by-product of the solution process. Explicit expressions for the Green's function and complete orthonormal sets of functions which are required to apply the solution to a rectangular domain are given. Applications to three-terminal structures are considered in order to illustrate the method.     

Angular quadrature perturbations in radiative transfer theory

A new numerical method is presented for solving the general equation of radiative transfer. The approximation, which replaces the integral term over angle in the transfer equation by a quadrature sum, is studied; an estimate of the error involved is obtained and this error, which may be thought of as a further source or sink of photons (depending upon the sign), can then be used to evaluate a corection to the radiation field originally determined. This process may then be continued as a perturbation series. The method is found to give a final solution, when starting from the Eddington approximation, at least as accurate as that obtained using variable Eddington factors. Furthermore, the technique involves very little extra computing over that required using the Eddington approximation, and may be trivially generalized to any radiative transfer problem. It can also be used in conjunction with any of the existing methods for solving the equation of transfer. Examples are given in the context of spectral line formation in slab geometry.