Improving the Born approximation for the scattering of ultrasound in elastic solids

It has recently been demonstrated that the Born approximation for predicting the scattering response of flaws can be improved through the use of simple modifications called the "doubly distorted Born approximation". In this paper the doubly distorted Born approximation itself is modified with phase and amplitude corrections that further improve the Born scattering results for isotropic elastic media. The reliability of this new modification of the Born approximation has been evaluated by comparison with the exact solution for spherical inclusions obtained with the method of separation of variables. Unlike the ordinary Born approximation which works well only for very weak scattering inclusions, our modification of the doubly distorted Born approximation gives improved scattering results for both weak and strong scattering inclusions.     

Convergence of solutions to the rough-surface scattering problem

Abstract

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.     

The Fokker-Planck equation in the second-order pitch angle approximation and its exact solution

The diffusive particle propagation and its pitch angle scattering is studied using kinetic equation of the Fokker-Planck form. The case is considered when charged particles preferable propagate along the strong mean magnetic field direction and undergo the pitch angle scattering with respect to it. The paper deals with solution of the equation for particle distribution function in the second-order approximation in the pitch angle. The exact analytical solution is obtained in an integral form. The well-known solution in the first-order pitch angle approximation can be restored performing the small time limit in the result. Unlike the first-order solution the obtained solution in the second approximation rightly shows that the pitch angle diffusion is closely connected with the particle transport along the mean magnetic field. The expression for particle density for the point instantaneous unidirectional source also has been obtained.     

Three dimensional analyses of scattering by pressure-release sinusoidal surfaces

Scattering by pressure-release sinusoidal surfaces in three dimensions is analyzed using the Fresnel phase approximation and realistic source and receiver directivity approximations. Geometrical shadowing and second-order scattering are explicitly included to explore the validity of the Kirchhoff approximation. No restrictions on the surface heights and slopes are made. The "goodness" of the resulting expressions is verified by requiring the pressure scattered by a sinusoidal surface to reduce to the image solution as the surface amplitude goes to zero. The first-order scattered pressure achieves a very good approximation to the image solution, and the second-order scattered pressure goes to zero, as expected, under this requirement. The theory is compared with available experimental scattering measurements, and the agreement is good. Because the slopes on the experimental surface are very steep, shadowing corrections are indispensible to achieving accurate first and second order scattering results. Shadowing has a greater impact on the scattering prediction than the second-order scattering contribution. This suggests that the Kirchhoff approximation may be more robust when incorporated into a theory including a detailed shadowing treatment as well as the Fresnel phase approximation and a good directivity approximation.     

Quasiclassical green function in an external field and small-angle scattering processes

A representation is obtained for the quasiclassical Green functions of the Dirac and Klein-Gordon equations allowing for the first nonvanishing correction in an arbitrary localized potential which generally possesses no spherical symmetry. This is used to obtain a solution of these equations in an approximation similar to the Furry-Sommerfeld-Maue approximation. It is shown that the quasiclassical Green function does not reduce to the Green function obtained in the eikonal approximation and has a wider range of validity. This is illustrated by calculating the amplitude of small-angle scattering of a charged particle and the amplitude of Delbrück forward scattering. A correction proportional to the scattering angle was obtained for the amplitude of charged particle scattering in a potential possessing no spherical symmetry. The real part of the Delbrück forward scattering amplitude was calculated in a screened Coulomb potential.     

Three-dimensional analysis of scattering by pressure-release plane surfaces and the validity of the image solution

Because of the complexity of the scattering integrals in three dimensions, numerous approximations are used to obtain closed-form solutions. By considering the scattering by an infinite, pressure-release plane surface, the effects of various phase approximations and source directivity approximations can be examined independently of the surface roughness. Calculations are carried out using the Fraunhofer and Fresnel phase approximations, and two directivity approximations. It has been shown experimentally that the image solution is valid for the reflection of an acoustic beam by an infinite, pressure-release plane surface if the plane is in the farfield of the source. Consequently, the image solution is used to compare analytical solutions obtained using various phase and directivity approximations, and it is found that both the Fresnel phase approximation and a realistic directivity approximation are required to achieve a good fit. The solution produced by the Fraunhofer phase approximation is obtained as an asymptotic limit of the modified Fresnel solution. Criteria for the validity of the Fraunhofer and Fresnel phase approximations are developed. The Fresnel phase approximation is valid under fairly broad conditions, but the Fraunhofer phase approximation is never valid for an infinite plane surface that must be in the farfield of the source.     

A new bistatic model for electromagnetic scattering from perfectly conducting random surfaces: numerical evaluation and comparison with SPM

This paper is a companion to our previous contribution deriving a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. We evaluate this model numerically and compare it with an 'exact' numerical solution of the scattering problem. This comparison shows good agreement between our approximation and numerical solution for a wide range of incident and scattering angles. However, for horizontal-incident horizontal-scattered polarization (HH-pol), the model exhibits strong deviation from the 'exact' solution for near-grazing scattering angles. The model shows a similar divergence at HH-pol when compared with the small-perturbation method (SPM). The cause of this divergence is explained. During the SPM comparison, we noticed that the integral equation method model also does not reproduce the SPM limit except for forward and backscatter geometries. We propose in this paper a simple modification of our model to ensure agreement with the bistatic SPM approximation when applicable, and show that the modified model also yields close agreement with numerical computations even when the surface roughness does not satisfy the SPM condition.     

二维导体粗糙面电磁散射的分形特征研究

采用二维归一化带限Brown分形函数来模拟二维分形粗糙面,主要利用基尔霍夫近似给出了该导体粗糙面的电磁散射场.导出了平均散射场、平均散射系数和散射强度方差的计算公式.讨论了散射场分布与分维的关系,获得了散射场波峰拟合线的斜率与分维D满足线性关系这一重要结论. 关键词：     

Modification to first-order Born approximation for improved prediction of scattered sound from weakly scattering objects

When deriving the Fourier diffraction theorem based on the first-order Born approximation,the difference between wave number of the scattering object and that of the surrounding medium is ignored,causing substantial errors in sound scattering prediction.This paper modifies the Born approximation by taking into account the amplitude and phase changes between the scattering object and the water due to the wave number difference.By changing the radius and center position of the sampling circle in the Fourier domain,accuracy of the predicted sound scattering is improved.With the modified Born approximation,the computed far-field directional pattern of the scattered sound from a circular cylinder is in good agreement with the rigorous solution.Numerical calculations for several objects with different shapes are used to show applicability and effectiveness of the proposed method.     

Finite element approximation of the Fokker-Planck equation for diffuse optical tomography

In diffuse optical tomography, light transport theory is used to describe photon propagation inside turbid medium. A commonly used simplification for the radiative transport equation is the diffusion approximation due to computational feasibility. However, it is known that the diffusion approximation is not valid close to the sources and boundary and in low-scattering regions. Fokker-Planck equation describes light propagation when scattering is forward-peaked. In this article a numerical solution of the Fokker-Planck equation using finite element method is developed. Approach is validated against Monte Carlo simulation and compared with the diffusion approximation. The results show that the Fokker-Planck equation gives equal or better results than the diffusion approximation on the boundary of a homogeneous medium and in turbid medium containing a low-scattering region when scattering is forward-peaked.     

Modified geometric optics: Accounting for exponentially small scattered fields

We propose a modified geometrical-optics approximation (MGO). It is based on the solution of the Helmholtz equation in the form of a series in which the main term describes the zero-order geometrical-optics approximation. The proposed method allows one to take into account an exponentially small backscattering in smoothly inhomogeneous media and thus improves the geometrical-optics approximation. In the case of forward scattering, the solution obtained is reduced to the conventional geometrical-optics series over negative powers of the wavenumber. We estimate the backscattering using the perturbation technique for rays and determine the conditions under which the backscattered field agrees with the Born approximation of scattering theory. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 2, pp. 106–114, January 2000.     

激光在不均匀随机媒质中的传播——多重散射的m-n阶矩方程的解

本文给出了受到多重散射的激光,在波传播方向上的物理参数的涨落是不均匀的、而垂直于传播方向上的物理参数的涨落又是均匀的随机媒质中传播时,当波受到前向小角度散射时,具有不同波数不同位置的场的矩方程的解析解.同时讨论了方程的解在激光传播研究中的一些应用.     

修正一阶Born近似改进水中弱散射目标的散射声场预报

在推导Fourier衍射定理中运用一阶Born近似时忽略了弱散射体内外的波数差异,使散射声场方向特性的预报产生较大误差,针对这一问题,对一阶Born近似进行修正。考虑散射体内外波数差异引起的幅度和相位误差,通过调整频域采样圆弧半径并移动圆心位置得到了修正的一阶Born近似解。由于更准确地反映了目标与周围水介质的声学性质,有效提高了散射声场的预报精度。根据修正的Born近似计算了弱散射条件下无限长圆柱目标的散射远场指向性,结果与严格解相吻合,对其它形状截面的柱状目标也得到了合理的计算结果。     

Delta-P1近似漫反射光学模型的二阶参量灵敏度

基于Delta-P₁近似漫反射光学模型, 推导了双点源近似下空间分辨漫反射解对于Henyey-Greenstein散射相函数的二阶参量γ的灵敏度表达式, 研究了参量γ对漫反射分布的影响. 研究表明, 与漫射近似模型相比, Delta-P₁近似漫反射光学模型不仅含有散射相函数的二阶参量γ, 而且在弱吸收或强吸收的情况下, 参量γ对于光源附近(ρ<1.0 mm)的空间分辨漫反射都存在较大的影响(极值大于30%). 因此利用γ研究漫反射光分布对于获取组织光学信息具有重要的意义.     

Scattering in a Pekeris waveguide from a rough bottom using a two-way coupled mode approach

Scattering from a rough ocean bottom is described numerically with a two-way coupled-mode formalism that contains scattering effects to all orders and provides an exact solution to the wave equation. Both scattered field and direct blast components are computed within the formalism framework. A comparison of the scattered component solution from the coupled mode with the Born approximation (BA) solution for scattering from a rough bottom Pekeris waveguide shows that the BA predicts correctly the scattered field levels but not detailed structure. The transition from direct blast to scattered field dominance is identified in the total field time series.     

An exact multiple scattering calculation in comparison with the Fresnel approximation

An exact solution to the problem of nucleon scattering from two fixed potentials is compared with its Fresnel approximation. The differential cross sections are found to be in good agreement over the entire range of scattering angles.     

Diffractional Nuclear Processes Involving Composite Particles

The diffractional picture of nuclear processes at high energies involving composite particles is given. The solution of the Lippmann-Schwinger equation for the problem on scattering in the scattering center systems is considered in the high-energy approximation. The general formulas for the cross section of different diffractional processes with particle transmission are obtained. The effects of multiple scattering and their mutual compensation are discussed. The applicability conditions of the diffractional approach are discussed. The diffractional processes during the interaction of high-energy deuterons with nuclei are considered in detail. Without the assumption of deuteron size smallness compared to nuclear size, the cross sections of different diffractional processes are calculated with nuclear boundary diffusion taken into account. The nonmonotonic dependence of the diffractional disintegration cross section for deuterons on the mass number is shown due to the influence of nuclear boundary diffusion. The role of the diffractional and the Coulomb mechanisms of interaction depending on the nuclear mass number is discussed. The calculated dependences are compared with experimental data. The interference during the multiple scattering and its relationship with the diffractional structure of the angular scattering during the scattering on the composite systems are considered. The comparison the between diffractional approximation and different variations of the impulse approximation is made.     

Coherent transmission and angular structure of light scattering by monolayer films of polymer dispersed liquid crystals with inhomogeneous boundary conditions

We consider monolayer polymer films with oriented droplets of a nematic liquid crystal (LC). Relations for the coherent transmission coefficients of a layer of oriented ellipsoidal droplets and for the intensity of light scattered by monolayers of spherical and spheroidal droplets have been obtained. The amplitude-phase screen model and the interference approximation of the theory of multiple wave scattering have been used. To describe light scattering by an individual ellipsoidal droplet with inhomogeneous surface binding, we have developed an anomalous diffraction approximation. For monolayers of spherical LC droplets, the coherent scattering coefficients and the angular scattering structure have been analyzed. The internal structure of nematic droplets have been calculated by the relaxation method based on the solution of the minimization problem of the free energy volume density. We have studied basic regular features of light scattering by a monolayer with homogeneous and inhomogeneous boundary conditions at the LC-polymer interface. We show that, for films that contain droplets with inhomogeneous boundary conditions of the tangentially normal type, the angular structure of the scattered light is asymmetric with respect to the polar scattering angle.     

Microwave propagation and scattering in a dense distribution of non-tenuous spheres: experiment and theory

Controlled laboratory experimental results of coherent microwave propagation through a random medium are reported. The medium consisted of layers of styrofoam with spherical glass beads embedded at predetermined random positions generated by computer. The magnitude and phase of the transmitted field was measured over the frequency range 18-20.4 GHz for media with volume fractional densities ranging from 0.5% to 11%. The results are compared with independent scattering, Foldy's approximation, and the quasicrystalline approximation (QCA) using the solution of the Percus-Yevick (PY) equation for the pair distribution function. The effects of a size distribution are included. Experimental results indicate that at low densities, the measured extinction rate increases linearly with concentration in agreement with independent scattering. As concentration further increases, the extinction curve turns convex and is lower than independent scattering. However, it is higher than that predicted by QCA-PY. Using the known particle positions the authors have also computed the pair correlation function and good agreement is obtained with the Percus-Yevick approximation.     

Tilt-invariant theory of rough-surface scattering: I

The small-slope approximation (SSA) in rough-surface scattering theory uses the surface slope as a small parameter of expansion. But, from the physical point of view, the slope may not be a restrictive parameter because we can change the slope of a surface simply by tilting the coordinate system. We present the theory of rough-surface scattering in a coordinate-invariant form. The new method, tilt-invariant approximation (TIA), leads to a different expansion that does not require that the slope of a surface be small. For a small Rayleigh parameter this approximate solution provides the correct perturbation theory, for a large Rayleigh parameter it provides the Kirchhoff approximation with several correcting terms.