Boundary element analysis of an integral equation for three-dmensional crack problems

In another paper, the authors proposed an integral equation for arbitrary shaped three-dimensional cracks. In the present paper, a discretization of this equation using a tensor formalism is formulated. This approach has the advantage of providing the displacement discontinuity vector in the local basis which varies as a function of the point of the crack surface. This also facilitates the computation of the stress intensity factors along the crack edge. Numerical examples reported for a circular crack and a semi-elliptical surface crack in a cylindrical bar show that one can obtain good results, using few Gaussian points and no singular elements.     

柱面透射光栅的衍射特性

从理论上分析了较高空间频率的柱面透射光栅的衍射特性，在不作近似的条件下，用计算机模拟出了不同曲率半径的柱面透镜组成的柱面透镜组成的柱面透镜光栅的夫琅和费衍射特性，得出了共衍射光强的分布规律。光学实验给出了由光学塑料制成的空间频率为１０ｌｐ／ｍｍ的柱面透射光栅的衍射光强分布。计算机模拟结果与光学实验结果吻合。     

Analysis of the fiber-matrix cylindrical model with a circumferential crack

An axisymmetrical fiber-matrix cylindrical model with a circumferential crack in the matrix of finite diameter is formulated within elastostatic scope. The problem is considered by means of integral transforms and a singular integral equation with a dominant generalized Cauchy kernel is obtained. Following the numerical solution technique developed by Erdogan, Gupta and Cook, the singular integral equation is reduced to a system of linear equations. By solving the linear equations, stress intensity factors associated with the crack length and the material properties are calculated and discussed. The solutions presented in this study are found to be general, including the solutions as special cases of the present formulation for a homogeneous solid cylindrical bar and a thick-walled shell with an outer circumferential crack. This revised version was published online in August 2006 with corrections to the Cover Date.     

Sudden twisting of an external circular crack in an infinite medium with a cylindrical inclusion

In this paper the torsional impact response of an external circular crack in an infinite medium bonded to a cylindrical inclusion has been investigated. The infinite medium and cylindrical inclusion are assumed to be of different homogeneous isotropie elastic materials. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the results are expressed in terms of a Fredhol integral equation of the second kind. By solving Fredholm integral equation of the second kind the numerical results for the dynamic stress-intensity factor are obtained which measure the load transmission on the crack.     

Spectral integral formulae for the response of acoustic transducers in cylindrically curved configurations

High-frequency spectral integral formulae are derived for the frequency-domain voltage of reciprocal electro-acoustic ultrasonic transducers interacting with a fluid-immersed cylindrically-layered elastic configuration. The transducers may insonify the cylindrical configuration from either the interior or the exterior and may be operated in either pulse-echo or pitch-catch mode. The formulae are arrived at through the use of a traveling wave spectral representation of all pertinent wave fields along the azimuthal and axial directions and use of Plancherel's theorem to convert the conventional surface integral for the transducer voltage into a spectral one. Within this representation, the transducers are expressed in terms of their radiation and reception spectral amplitudes on a cylindrical surface and the interaction of their fields with the cylindrically layered environment is expressed in terms of a composite spectral reflection coefficient. The integral formulae allow for computation of the transducer voltage and transducer beam interaction with the cylindrical environment in one step. The derivation is carried out for three-dimensional beams and specialized to two-dimensional (sheet) beams that extend to infinity along the axial direction. Furthermore, the formulae are applied to the case where complex transducer points are used to simulate transducers with quasi-Gaussian distribution profiles.     

High-order perturbation theory for light scattering from a rough metal surface.

The angular distribution of the mean diffuse intensity scattered from a metal surface with one-dimensional roughness is studied with perturbation theory. From an approach based on the reduced Rayleigh equations in p polarization, exact perturbation terms up to eighth order in the height parameter are developed for surface roughness consistent with a stationary Gaussian process. The theory is evaluated for a number of cases in which surface plasmon polariton excitation is significant and produces effects such as backscattering enhancement. For surface roughness having a wide Gaussian power spectrum, it is found that the high-order terms lead to roughness-induced broadening of the backscattering peak. For rectangular spectra, two cases are studied in which backscattering effects are due to sixth- and eighth-order terms; both cases provide good comparisons with previously unexplained experimental results. Further, because of an eighth-order term, the diffuse intensity is shown to contain a specular peak that also relies on polariton excitation. This new effect is studied in detail and is found to arise from the constructive interference of contributions produced by multiple-scattering processes, although the time-reversed paths that produce backscattering enhancement are not essential to the specular effect.     

Acousto-optic interaction of a Gaussian laser beam with an ultrasonic wave of cylindrical symmetry

We present experimental studies of the interaction between a narrow Gaussian laser beam and a standing cylindrical ultrasonic wave. As a theoretical approach, a Fourier-optics-based successive diffraction model is used. Depending on the ratio of the Gaussian laser beam diameter to the first nodal diameter of the cylindrical ultrasound, light refraction or diffraction is observed. We experimentally investigate the time-averaged light intensity as well as the modulation of light in the far field of light refraction-diffraction by a cylindrical ultrasound. It is revealed that significant focusing appears if the phase front of the incident light is curved. The focusing effects of the acousto-optic system depend on the width of the laser beam and curvature of the phase front. Finally, possible applications are discussed.     

基于SPR的多层金属膜厚在线纳米测量

通过对于首层金属膜的SPR反射光强曲线的分析利用两峰值点标记角度,同时用柱面光与CCD测量系统建立无机械测角转台的反射光强角度分布测量系统。根据CCD接收的光强与位置对应信息经计算机处理换算为光强角度分布曲线,并通过与库内的曲线数据进行匹配而获得膜厚数值。利用该系统对铬钛银铝等常用金属薄膜进行了在线测控研究,得到了纳米级金属薄膜的膜层蒸镀顺序与相应的膜厚测量范围。     

Local spectrum analysis of field propagation in an anisotropic medium. Part I. Time-harmonic fields

The phase-space beam summation is a general analytical framework for local analysis and modeling of radiation from extended source distributions. In this formulation, the field is expressed as a superposition of beam propagators that emanate from all points in the source domain and in all directions. In this Part I of a two-part investigation, the theory is extended to include propagation in anisotropic medium characterized by a generic wave-number profile for time-harmonic fields; in a companion paper [J. Opt. Soc. Am. A 22, 1208 (2005)], the theory is extended to time-dependent fields. The propagation characteristics of the beam propagators in a homogeneous anisotropic medium are considered. With use of Gaussian windows for the local processing of either ordinary or extraordinary electromagnetic field distributions, the field is represented by a phase-space spectral distribution in which the propagating elements are Gaussian beams that are formulated by using Gaussian plane-wave spectral distributions over the extended source plane. By applying saddle-point asymptotics, we extract the Gaussian beam phenomenology in the anisotropic environment. The resulting field is parameterized in terms of the spatial evolution of the beam curvature, beam width, etc., which are mapped to local geometrical properties of the generic wave-number profile. The general results are applied to the special case of uniaxial crystal, and it is found that the asymptotics for the Gaussian beam propagators, as well as the physical phenomenology attached, perform remarkably well.     

Scattering from Gamma-distributed Surfaces

Abstract

We present a theoretical and experimental study of the scattering of light from diffusers with gamma-distributed surface height profiles. The theory is developed using a thin-phase screen model: it is shown that, following a steepest-descent method, the mean scattered intensity as a function of the scattering angle follows a modified Bessel K-function. The theory is compared with experimental data in the two extreme cases of the gamma distribution, namely the negative exponential and Gaussian cases. The surfaces used were made by exposing photoresist-coated plates with laser speckle patterns. For the case of a negative-exponential surface it is shown that it is not possible, in practice at least, to extinguish the specular component.     

Reconstruction of the net emission distribution from the total radiance distribution on a reflecting surface

The problem is considered of reconstructing, from a measurement of the total radiance distribution on an emitting surface, the radiance distribution that would be observed in the absence of reflected radiation. An explicit solution of the implied inverse problem is derived for the case in which the reflective properties of the surface are given in terms of a bidirectional reflection distribution function. Also considered are the limiting cases of diffuse and specular reflection. Practical considerations are discussed for application of the theory to the nonintrusive and remote measurement of temperatures and pressures on concave surfaces, either by traditional radiometry or by the use of thermographic phosphors and temperature- and pressure-sensitive paints.     

Superresolution and increased depth of focus: An inverse problem of vector diffraction

Abstract

Using vector diffraction theory, we have addressed the issue of finding the pupil function when the rate of decay of the intensity in the vicinity of the Gaussian focus along the optical axis is prescribed. The problem posed here reduces to a Fredholm integral equation, which is then solved to obtain the pupil function. We show that the diffraction integrals, as formulated by Wolf, are invariant if they are expressed in terms of the zth (the direction of the optical axis) component of the unit normal to the aberrated wave front. This makes it possible to obtain the pupil function from the solution of the Fredholm integral equation. We present results for lenses with high numerical aperture and show that the depth of focus is significantly increased without any loss of transmitted energy. Results further indicate that the FWHM of the primary lobe is significantly narrower than the clear aperture.     

Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on a planar circular surface coaxial to the disc

Abstract

Average solid angles subtended by an external contour of a given body from several points are required for the calculation of radiation fluxes or particle beams that are incident on the body from multiple-point isotropic emitters of analogous kinds. To consider the refraction effects for fluxes of light propagating through various optical media, knowledge of the angular distributions of such average solid angles is necessary. In this paper the formula describing angular dependence of the average solid angle subtended by a circular disc from uniformly distributed points within the circular surface of a parallel and coaxial disc was derived analytically and used for the calculation of some representative results. The solution has been made in the cylindrical coordinate system. The final and some intermediate formulas were expressed as functions of the polar angle, of the radii of both discs and of the distance between their planes. These formulae were represented by superpositions of simple elementary functions, single integrals of, these superpositions and by incomplete and complete Legendre—Jacobi elliptic integrals of all three kinds. Mathematica 2.2.3 software was used to illustrate graphically the relationships between some computed data. These data indicate that the derived formula is directly applicable in any computer programs calculating the Legrendre—Jacobi elliptic integrals to estimate the fluxes of optical radiation and particle beams propagated within a non-absorbing homogeneous medium. The expressions obtained may also be used to calculate the fluxes of optical radiation propagated through various homogeneous media.     

Diffraction of transient horizontal shear waves by a finite crack at the interface of two bonded dissimilar elastic solids

Scattering of transient horizontal shear waves by a finite crack located at the interface of two bonded dissimilar elastic solids is investigated in this study. Laplace and Fourier transform technique is used to reduce the problem to a pair of dual integral equations. The solution of the dual integral equation is expressed in terms of the Fredholm integral equation of the second kind having the kernel of a finite integration. Dynamic stress intensity factor is obtained as a function of the material and geometric properties and time.     

Luminance-based specular gloss characterization

Gloss is a feature of visual appearance that arises from the directionally selective reflection of light incident on a surface. Especially when a distinct reflected image is perceptible, the luminance distribution of the illumination scene above the sample can strongly influence the gloss perception. For this reason, industrial glossmeters do not provide a satisfactory gloss estimation of high-gloss surfaces. In this study, the influence of the conditions of illumination on specular gloss perception was examined through a magnitude estimation experiment in which 10 observers took part. A light booth with two light sources was utilized: the mirror image of only one source being visible in reflection by the observer. The luminance of both the reflected image and the adjacent sample surface could be independently varied by separate adjustment of the intensity of the two light sources. A psychophysical scaling function was derived, relating the visual gloss estimations to the measured luminance of both the reflected image and the off-specular sample background. The generalization error of the model was estimated through a validation experiment performed by 10 other observers. In result, a metric including both surface and illumination properties is provided. Based on this metric, improved gloss evaluation methods and instruments could be developed.     

Depolarization of diffusely reflecting man-made objects

The polarization properties of light scattered or diffusely reflected from seven different man-made samples are studied. For each diffusely reflecting sample an in-plane Mueller matrix bidirectional reflectance distribution function is measured at a fixed bistatic angle using a Mueller matrix imaging polarimeter. The measured profile of depolarization index with changing scattering geometry for most samples is well approximated by an inverted Gaussian function. Depolarization is minimum for specular reflection and increases asymptotically in a Gaussian fashion as the angles of incidence and scatter increase. Parameters of the Gaussian profiles fitted to the depolarization data are used to compare samples. The dependence of depolarization on the incident polarization state is compared for each Stokes basis vector: horizontal, vertical, 45 degrees, 135 degrees, and right- and left-circular polarized light. Linear states exhibit similar depolarization profiles that typically differ in value by less than 0.06 (where 1.0 indicates complete depolarization). Circular polarization states are depolarized more than linear states for all samples tested, with the output degree of polarization reduced from that of linear states by as much as 0.15. The depolarization difference between linear and circular states varies significantly between samples.     

Computation of the External Magnetic Field, Near-Field or Far-Field, From a Circular Cylindrical Magnetic Source Using Toroidal Functions

A method is developed for computing the magnetic field from a circular or noncircular cylindrical magnetic source. A Fourier series expansion is introduced which yields an alternative to the more familiar spherical harmonic solution, Elliptic integral solution, or Bessel function solution. This alternate formulation coupled with a method called charge simulation allows one to compute the external magnetic field from an arbitrary magnetic source in terms of a toroidal expansion which is valid on any finite hypothetical external observation cylinder. In other words, the magnetic scalar potential or the magnetic field intensity is computed on a exterior cylinder which encloses the magnetic source. Also, one can compute an equivalent multipole distribution of the real magnetic source valid for points close to the circular cylindrical boundary where the more familiar spherical multipole distribution is not valid. This method can also be used to accurately compute the far field where a finite-element formulation is known to be inaccurate     

3D acoustic wave simulation using BEM formulations: Closed form integration of singular and hypersingular integrals

Among the obstacles to applying boundary element techniques to three-dimensional wave propagation problems is the difficulty of accurately representing the singular and hypersingular terms at the points of application of the virtual loads. This paper presents the analytical evaluation of the singular and hypersingular integrals for constant boundary elements. First, the singular integral results are compared with those evaluated by means of a Gaussian quadrature scheme, which uses an enormous amount of sampling points. In the case of hypersingular integrals the comparison makes use of the results provided by the method presented by Terai [T. Terai, On calculation of sound fields around three dimensional objects by integral equation methods, J Sound Vib 69 (1980) 71–100.]. An additional verification is performed by comparing the boundary element method (BEM) results with known analytical solutions for cylindrical inclusions.     

Diffraction of Gaussian beams by periodic and aperiodic rulings

The diffraction of Gaussian beams by periodic and aperiodic rulings is considered. The theory of diffraction is based on the Rayleigh-Sommerfeld integral equation with Dirichlet conditions. The transmitted power and the normally diffracted energy are analyzed as a function of the beam radius. Two methods to determine the Gaussian beam radius by means of periodic and aperiodic lamellar gratings are proposed. One is based on the maximum and the minimum transmitted power, and the other one considers the normally diffracted energy. Small and large Gaussian beam radii can be treated with these two methods.     

Resolution enhancement by extrapolation of the optically measured spectrum of surface profiles

Surface features are measured by the analysis of the diffracted field intensity distribution generated by a scanning Gaussian beam. The measured data are shown to represent the amplitude of the Gabor expansion coefficients of the complex function embossed by the surface onto the reflected wave front. The surface is reconstructed by use of a custom-designed algorithm based on generalized projections, with a resolution exceeding the classical diffraction limit.