T-matrix formulation and generalized Lorenz-Mie theories in spherical coordinates

There has been recently a growing interest in the development of what is usually known as the T-matrix method (better to be named: T-matrix formulation), in connection with studies concerning light scattering by nonspherical particles. Another line of research has been devoted to the development of generalized Lorenz-Mie theories dealing with the interaction between arbitrary electromagnetic shaped beams and some regular particles, allowing one to solve Maxwell’s equations by using a method of separation of variables. Both lines of research are conjointly considered in this paper. Results of generalized Lorenz-Mie theories in spherical coordinates (for homogeneous spheres, multilayered spheres, spheres with an eccentrically located spherical inclusion, assemblies of spheres and aggregates) are modified from scalar results in the framework of the Bromwich method to vectorial expressions using vector spherical wave functions (VSWFs) in order to match the T-matrix formulation, and to express the T-matrix. The results obtained are used as a basis to clarify statements, some of them erroneous, concerning the T-matrix formulation and to provide recommendations for better terminologies.     

近场条件下的扫描成象方程

本文考察了传统(远场)扫描成象方程,发现它们在近场条件下不成立.通过引入一个探测效率因子,我们建立了对近场条件与远场条件都成立的统一的扫描成象方程.同时我们还分别采用标量Kirchhof模型和矢量Bethe-Bouwkamp模型对探测效率因子进行了计算,计算结果表明它们都与探针孔径成正比,但后者的分布要比前者宽约6倍.     

Diffraction of spherical waves at an annular aperture in the use of the boundary diffraction wave theory: A comparison of different diffraction integral approaches

The integral expression for divergent spherical waves diffracted at an annular aperture is derived based on the theory of the boundary diffraction wave. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the axial field are treated as the special cases of our general one. Numerical calculation results for axial and transversal intensity distributions are given to compare our results with the Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals. As expected, our results are in agreement with those in the use of the Kirchhoff diffraction integral, but the computer time is reduced greatly by using the boundary diffraction wave theory. The four diffraction formulae are shown to be consistent for axial and transversal intensity distributions, if the source and observation points are positioned equally from the aperture, or the observation point is located enough far from the aperture. Otherwise, the mean value of the first and second Rayleigh diffraction integrals is equal to the result of the boundary diffraction wave theory.     

Direct computation of the noise radiated by a subsonic cavity flow and application of integral methods

The goal of this paper is to investigate the acoustic field generated by the flow over a cavity using two different and complementary numerical methods. First, a Direct Numerical Simulation of the 2-D compressible Navier-Stokes equations is performed to obtain directly the radiated noise. The results of the acoustic and aerodynamic fields are compared to the experimental data in the literature. Second, this reference solution is compared to solutions provided by hybrid methods using the flowfield computed inside the cavity combined with an integral formulation to evaluate the far-field noise. Numerical issues of three integral methods are studied: the Ffowcs Williams and Hawkings analogy that extends Lighthill's theory to account for solid boundaries and two Wave Extrapolation Methods from a control surface, the Kirchhoff and porous Ffowcs Williams and Hawkings methods. All methods show a good agreement with the Direct Numerical Simulation, but the first one is more expensive owing to an additional volume integral. However, the analogy can help in the analysis of wave patterns, by separating the direct waves from the reflected ones. The wave extrapolation methods from a surface are more efficient and provide a complementary tool to extend Computational Aeroacoustics near field to the very far field.     

The IEM2M rough-surface scattering model for complex-permittivity scattering media

The integral equation model (IEM) was developed in the late 1980s and arguably became the most cited and implemented rough-surface scattering model in the field of radar remote sensing for Earth observation. It was derived by applying a second-order iteration in the incident electromagnetic field to the integral equations of the surface fields as given by Poggio and Miller. It is thus an extension of the first-order, Born approximation of these equations that produce the classical Kirchhoff approximation. The IEM has been subject to numerous amendments and variations over the last 20 years due to the imperfect introduction and handling of the Weyl representation of the spherical wave in its first version. The work presented here is a further development of the contribution made by the same author in 2001 (IEM2M), which was the first version of IEM able to blend analytically both the Kirchhoff and the small-perturbation approximations for the bistatic case. The improvement reported in this article is concerned with the inclusion of evanescent waves in the formulation of the model and the extension of the range of applicability of the second-order scattering terms to interfaces with complex-permittivity scattering media.     

Local and non-local curvature approximation: a new asymptotic theory for wave scattering

Abstract

We present a new asymptotic theory for scalar and vector wave scattering from rough surfaces which federates an extended Kirchhoff approximation (EKA), such as the integral equation method (IEM), with the first and second order small slope approximations (SSA). The new development stems from the fact that any improvement of the ‘high frequency’ Kirchhoff or tangent plane approximation (KA) must come through surface curvature and higher order derivatives. Hence, this condition requires that the second order kernel be quadratic in its lowest order with respect to its Fourier variable or formally the gradient operator. A second important constraint which must be met is that both the Kirchhoff approximation (KA) and the first order small perturbation method (SPM-1 or Bragg) be dynamically reached, depending on the surface conditions. We derive herein this new kernel from a formal inclusion of the derivative operator in the difference between the polarization coefficients of KA and SPM-1. This new kernel is as simple as the expressions for both Kirchhoff and SPM-1 coefficients. This formal difference has the same curvature order as SSA-1 + SSA-2. It is acknowledged that even though the second order small perturbation method (SPM-2) is not enforced, as opposed to the SSA, our model should reproduce a reasonable approximation of the SPM-2 function at least up to the curvature or quadratic order. We provide three different versions of this new asymptotic theory under the local, non-local, and weighted curvature approximations. Each of these three models is demonstrated to be tilt invariant through first order in the tilting vector.     

圆形球面虚共焦非稳腔的三维数值研究

 通过对1/2共焦腔和稳定腔的计算,考查了同三维菲涅耳积分方程相应的C 8程序的可靠性。对激光在三维圆形球面虚共焦非稳腔中的传输作了计算和分析。取相同的远场传输距离且放大率由大变小时，强度分布由近场的投影分布逐渐趋于远场的高斯、超高斯分布。     

Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an arbitrary location

A complete theory of Gaussian beam scattering by a sphere is exposed. It is a generalization of the Lorenz-Mie Theory to the case of Gaussian beam illumination. The spherical, isotropic and homogeneous scatterer may be located anywhere with respect to the beam. The Bromwich Scalar Potentials are used to solve the scattering problem and expressions are obtained for the scattered field (both in the near field and far field regions), the scattered intensities and the phase angle. In the limit of special cases the expressions agree with previous works restricted to more particular problems.     

Derivatives of light scattering properties of a nonspherical particle computed with the T-matrix method

Based on the T-matrix formalism, we analytically calculate derivatives of light scattering quantities by a nonspherical particle with respect to its microphysical parameters. Illustrative computations are performed for a spheroid, and the results agree with those obtained by finite differencing. The proposed formalism also predicts correctly derivatives for a sphere obtained by linearized Lorenz-Mie theory.     

离心球对高斯波束的光散射及应用

基于广义米氏理论研究了在轴离心球对高斯波束光散射特性.入射高斯波束的波束因子用积分区域近似法计算,散射场的展开系数由矢量球面波函数的加法定理并求解边界条件得到.以离心球为模型研究了单核生物细胞对高斯波束的散射特性并给出了相关数值模拟,讨论了离心距、波束的束腰半径和核的大小对散射强度角分布和散射、消光系数的影响.     

A reformulation of the one-dimensional surface field integral equations

By starting from the matrix forms of the two coupled, inhomogeneous integral equations for the values of the magnetic field and its normal derivative on a one-dimensional, rough metal surface, or for the values of the electric field and its normal derivative on such a surface, we have obtained an equivalent pair of equations for these quantities in which the inhomogeneous terms are just the Kirchhoff approximations to them. The new pair of equations for the surface values of the magnetic field and its normal derivative is solved iteratively to generate a multiple-scattering expansion for the scattering amplitude when p-polarized light is scattered from a large RMS height, large RMS slope, one-dimensional, random silver surface, with the plane of incidence perpendicular to the generators of the surface. It is shown that the Kirchhoff approximation to the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light displays no evidence of enhanced backscattering. However, the pure double-scattering contribution already displays this effect, stamping it as a multiple-scattering phenomenon.     

On the sound field of a shallow spherical shell in an infinite baffle

A method is presented for calculating the far field sound radiation from a shallow spherical shell in an acoustic medium. The shell has a concentrated ring mass boundary condition at its perimeter representing a loudspeaker voice coil and is excited by a concentrated ring force exerted by the end of the voice coil. A Green's function is developed for a shallow spherical shell, which is based upon Reissner's solution to the shell wave equation [Q. Appl. Math. 13, 279-290 (1955)]. The shell is then coupled to the surrounding acoustic medium using an eigenfunction expansion, with unknown coefficients, for its deflection. The resulting surface pressure distribution is solved using the King integral together with the free space Green's function in cylindrical coordinates. In order to eliminate the need for numerical integration, the radiation (coupling) integrals are solved analytically to yield fast converging expansions. Hence, a set of simultaneous equations is obtained which is solved for the coefficients of the eigenfunction expansion. These coefficients are finally used in formulas for the far field sound radiation.     

Acoustic scattering by arbitrary distributions of disjoint, homogeneous cylinders or spheres

A T-matrix formulation is presented to compute acoustic scattering from arbitrary, disjoint distributions of cylinders or spheres, each with arbitrary, uniform acoustic properties. The generalized approach exploits the similarities in these scattering problems to present a single system of equations that is easily specialized to cylindrical or spherical scatterers. By employing field expansions based on orthogonal harmonic functions, continuity of pressure and normal particle velocity are directly enforced at each scatterer using diagonal, analytic expressions to eliminate the need for integral equations. The effect of a cylinder or sphere that encloses all other scatterers is simulated with an outer iterative procedure that decouples the inner-object solution from the effect of the enclosing object to improve computational efficiency when interactions among the interior objects are significant. Numerical results establish the validity and efficiency of the outer iteration procedure for nested objects. Two- and three-dimensional methods that employ this outer iteration are used to measure and characterize the accuracy of two-dimensional approximations to three-dimensional scattering of elevation-focused beams.     

A reformulation of the one-dimensional surface field integral equations

Abstract

By starting from the matrix forms of the two coupled, inhomogeneous integral equations for the values of the magnetic field and its normal derivative on a one-dimensional, rough metal surface, or for the values of the electric field and its normal derivative on such a surface, we have obtained an equivalent pair of equations for these quantities in which the inhomogeneous terms are just the Kirchhoff approximations to them. The new pair of equations for the surface values of the magnetic field and its normal derivative is solved iteratively to generate a multiple-scattering expansion for the scattering amplitude when p-polarized light is scattered from a large RMS height, large RMS slope, one-dimensional, random silver surface, with the plane of incidence perpendicular to the generators of the surface. It is shown that the Kirchhoff approximation to the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light displays no evidence of enhanced backscattering. However, the pure double-scattering contribution already displays this effect, stamping it as a multiple-scattering phenomenon.     

基于递归T矩阵的离散随机散射体散射特性研究

本文根据电磁场矢量球波函数多极点展开原理及矢量叠加定理提出了递归T矩阵算法的矢量形式,并且基于矢量递归T矩阵算法建立了多散射球模拟离散随机散射体散射的三维电磁散射模型.通过计算不同尺寸、随机分布散射球的散射以及分析散射球间的高阶散射效应,结果表明:矢量递归T矩阵算法具有很高的计算精度,算法中包含多散射体间的高阶散射效应,因此能够精确计算多散射体总的散射效应.本文所建模型可应用于土壤湿度探测工程中评估地表下掩埋离散随机散射体散射对雷达回波信号产生的影响.     

The far field asymptotics in the problem of diffraction of an acoustic plane wave by an impedance cone

The work deals with the far field asymptotics of the classical solution for the problem of diffraction by an impedance cone. The incident acoustic plane wave completely illuminates the semi-infinite conical surface. The scattered field contains different components in the asymptotics, namely, the spherical wave from the vertex of the cone, the reflected waves, and, under some conditions, also the surface waves of Rayleigh type. We give integral representations for the scattering diagram of the spherical wave. The uniform (with respect to the observation direction) asymptotic expression for the wave field is also addressed and described by the parabolic cylinder ansatz. Dedicated to the memory of Vladimir Borovikov     

Scattering of a spherical wave from the coupling of two half-planes

Summary  The diffraction of a spherical acoustic wave from the juncture of pressure release (soft) and locally reacting (absorbing) half-planes in a fluid moving at subsonic velocity is examined. This consideration is important because the point sources are regarded as better substitutes for real sources than line sources/plane waves. The integral representation of the field is obtained using integral transforms and the Wiener-Hopf technique. The factorization of the kernel function in the Wiener-Hopf functional equation is accomplished. The analytic solution of the integrals is obtained by employing asymptotic methods and the far field is presented. The effect of the Mach number is shown explicitly on the diffracted field.     

Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces

Using the unitarity and reciprocity preserving formulation of Brown et al a perturbation treatment, correct to fourth order in the surface profile function, is given for the scattering of electromagnetic waves from a weakly rough, two-dimensional, random metal surface. In this formulation the boundary conditions on the electromagnetic fields are satisfied using the extinction theorem in conjunction with the Rayleigh hypothesis and the vector equivalent of the Kirchhoff integral. The theory is applied to, and results are presented for, several different types of rough surfaces which are characterized by power spectra that are extensions to two-dimensional random surfaces of the power spectrum of some one-dimensional random surfaces recently fabricated by West and O'Donnell. These surfaces, which can be realized experimentally, favor coherent, interferent, multiple scattering of electromagnetic waves via surface plasmon polaritons in intermediate states, and clearly exhibit enhanced backscattering caused by the surface plasmon polariton mechanism. Theoretical results are presented for silver surfaces at optical wavelengths.     

Modified null field method for elastic wave scattering from a partially debonded fiber-SH-waves

Debonded fibers influence the macro-mechanical behavior of fiber-reinforced composite materials. Debonded fibers contribute to the initiation and growth of cracks at the fiber/matrix interface. To examine such problems, the scattering of elastic SH-waves (problem of anti-plane strain) from debonded fibers is studied with a numerical method which can handle the mixed boundary conditions on the partially bonded fiber. A modification of the null field of T-matrix method is developed for this purpose. The modification is achieved by the introduction of a mathematical surface. The simultaneous solution of the integral representations of the field scattered by the mathematical surface and the actual fiber surface give rise to sufficient equations that permit solution of the debonded fiber problem. The scattering cross-section and the far field amplitude are calculated as a function of frequency, fiber properties and debonding area. This will be of interest in structural applications where such cross-sections can be used to compute the dynamical effective properties of damaged composites.     

Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systems: I. General formulation

The generalized Lorenz-Mie theory in the strict sense describes the interaction between an illuminating arbitrary shaped beam and a homogeneous sphere characterized by its diameter d and its complex refractive index m. It relies on the method of separation of variables expressed in spherical coordinates. Other generalized Lorenz-Mie theories (for other kinds of scatterers) expressed in spherical coordinates are available too. In these theories, the illuminating beam is expressed by using expansions with expansion coefficients depending on some fundamental coefficients named beam shape coefficients, more specifically spherical beam shape coefficients. In this paper we present a general formulation for the transformation of spherical beam shape coefficients through rotations of coordinate systems.