Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium

This paper investigates the reconstruction of elastic Green's function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation-the resolvent formula-Green's function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell T-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell T-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The T-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green's function reconstruction in a scattering medium is discussed.     

Generalization of the optical theorem for an arbitrary multipole in the presence of a transparent half-space

The optical theorem is generalized to the case of excitation of a local inhomogeneity introduced in a transparent substrate by a multipole of arbitrary order. It is shown that, to calculate the generalized extinction cross section, it is sufficient to calculate the derivatives of the scattered field at a single point by adding a constant and a definite integral. Apart from general scientific interest, the proposed generalization makes it possible to calculate the absorption cross section by subtracting the scattering cross section from the extinction cross section. The latter fact is important, because the scattered field in the far zone contains no Sommerfeld integrals. In addition, the proposed generalization allows one to test computer modules for the case where a lossless inhomogeneity is considered.     

Cancellation of spurious arrivals in Green's function retrieval of multiple scattered waves

The Green's function for wave propagation can be extracted by cross-correlating field fluctuations excited on a closed surface that surrounds the employed receivers. This study treats an acoustic multiple scattering medium with discrete scatterers and shows that for a given source the cross-correlation of waves propagating along most combinations of scattering paths gives unphysical arrivals. Because theory predicts that the true Green's function is retrieved, such unphysical arrivals must cancel after integration over all sources. This cancellation occurs because the scattering amplitude of each scatterer satisfies the generalized optical theorem. The cross-correlation of scattered waves with themselves does not lead to the correct retrieval of scattered waves, because the cross-terms between the direct and scattered waves is essential.     

Optical theorem for multipole sources in wave diffraction theory

The optical theorem is generalized to the case of local body excitation by multipole sources. It is found that, to calculate the extinction cross section, it is sufficient to calculate the scattered field derivatives at a single point. It is shown that the Purcell factor, which is a rather important parameter, can be represented in analytic form. The result is generalized to the case of a local scatterer incorporated in a homogeneous halfspace.     

Brillouin's theorem and the Hellmann-Feynman theorem for Hartree-Fock wave functions

It is shown that there is an intimate relation between the Hellmann-Feynman theorem and Brillouin's theorem. A more general form of Brillouin's theorem is provided, which applies to excited states of arbitrary symmetry and multiplicity. This new form leads to a simple proof of the Hellmann-Feynman theorem. This theorem is valid when all the orbitals that occur in the wave function are determined by a complete, and not a partial, variational procedure. Arguing in the opposite direction it is shown that the complete satisfaction of the generalized Brillouin's theorem provides an alternative scheme for obtaining the Hartree-Fock orbitals.     

等效原理和互易性定理在两个相邻球形目标电磁散射中的应用

基于等效原理和互易性定理研究了两个相邻目标的电磁散射问题，给出了这一复合电磁散射模型的二阶散射结果. 通常平面入射波的一阶散射结果容易求解，但由于耦合效应的复杂性，很难给出二阶散射结果的解析形式. 应用互易性定理给出了求解任意相邻导体/介质目标二阶电磁散射场的公式，并利用等效原理将求解散射场公式中的体积分简化为面积分的形式，从而降低了求解难度. 同时还推导了两个目标的二次散射场之间的关系. 最后应用给出的公式，求解了两相邻球形目标的复合散射场，对双站散射结果进行了讨论，同时与应用时域积分方程法求得的结果进行了比较. 关键词： 等效原理 互易性定理 电磁散射 相邻圆球     

Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering

The far-field acoustic scattering amplitudes for the scattering of plane waves by targets having inversion symmetry obey a generalized optical theorem in the absence of dissipation. The theorem allows a component of the complex scattering amplitude in an arbitrary direction to be expressed in terms of an angular integration involving scattering amplitudes evaluated at different angles. The result reduces to the usual optical theorem in the case of forward scattering. The theorem is applied to the backscattering by a perfectly soft sphere as a numerical example. The relevant integrand is shown to be oscillatory. Some potential applications to inverse problems, multiple scattering, and the verification of numerical algorithms are noted.     

格林互易定理在求解均匀带电圆环的静电势分布中的应用

对于静电场,由普遍的格林互易定理得出特殊情形下的格林互易定理.应用该定理推导出均匀带电圆环的数种在形式上互不相同的静电势分布解式.     

Reciprocity theorem in high-temperature superconductors

This article is devoted to the problem of the validity of the reciprocity theorem in high-temperature superconductors (HTSC). The violation of the reciprocity theorem in zero external magnetic fields has been studied. Experimental data obtained for two different superconducting materials BiSrCaCuO and YBaCuO are presented. Results show that the basic form of the reciprocity theorem (without consideration of any additional anisotropy) is not valid near the critical temperature. We assume that the breakdown of the reciprocity theorem is connected with the existence of an extraordinary transverse electric field originating from additional anisotropy and that a more general form of the reciprocity relations should be valid. However, the origin of this anisotropy is not clear yet. We suggest that the vortex–antivortex dynamics model taking into account vortex guiding can be responsible for the observed effect. Also the explanation based on weak P and T symmetry breaking in HTSC, which is supported by the observation of the spontaneous magnetisation, cannot be excluded.     

The NMR reciprocity theorem for arbitrary probe geometry

It is shown that the NMR reciprocity theorem is a variant of a problem considered by Lorentz in 1895. This formulation is quite general and applies to electric-dipole-based as well as coil-based or resonator-based magnetic resonance probes. The reasoning is related to, but different from, the proof of the reciprocity theorem for radiofrequency networks and for transmit/receive antenna systems in telecommunications. The signal-to-noise ratio of the NMR experiment is also discussed in very general terms.     

The electromagnetic optical theorem revisited

We revisit the optical theorem relevant to the far-field electromagnetic scattering by an arbitrary particle. We compute both the Poynting vector and the energy density of the total field and demonstrate once again that, despite a recent claim to the contrary, the extinction is caused by the interference of the incident and the forward-scattered field. However, caution must be exercised when one describes electromagnetic scattering using a formalism based on the coherency dyad since this approach may lead to unphysical artifacts.     

Retrieval of Green's function and generalized optical theorem for the scattering of complete dyadic fields

Green's function retrieval has been widely used in different research fields due to the fact that the Green's function can be extracted by cross-correlating the records at two receivers. In this paper, the retrieval of the dyadic Green's function is studied by investigating the representation theorem. The generalized optical theorem for the dyadic fields is derived based on the elastic dynamic interferometric equation. By addressing the cross-correlation recorded at two receivers, the important role of the generalized optical theorem and energy equipartition in retrieving the exact Green's function is shown. The presented derivation also shows the Newton-Marchenko equation holdsif the condition of equipartition is not satisfied.     

Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation

A correlation-type reciprocity theorem is used to show that the elastodynamic Green's function of any inhomogeneous medium (random or deterministic) can be retrieved from the cross correlation of two recordings of a wave field at different receiver locations at the free surface. Unlike in other derivations, which apply to diffuse wave fields in random media or irregular finite bodies, no assumptions are made about the diffusivity of the wave field. In a second version, it is assumed that the wave field is diffuse due to many uncorrelated sources inside the medium.     

Alternate proof of the optical theorem for impenetrable targets

The optical theorem is proven for impenetrable targets without using the method of stationary phase. The proof depends upon the divergence theorem, the Helmholtz representation of the scattered field, and the law of cosines.     

A METHOD TO PREDICT ACOUSTIC RADIATION FROM AN ENCLOSED MULTICAVITY STRUCTURE

The acoustical reciprocity theorem can be used to solve the problem of vibroacoustic coupling. However, the theorem can be used only on the presupposition that the scattered sound field of the elastic surface concerned is known. This is the key point and the most difficult point for many complicated surfaces, such as a multicavity structure. A new method, covering-domain method, which transforms the calculation of scattered sound field of an arbitrary-shaped closed shell into that of a series of simply closed spherical shells, is applied in this paper to calculate the scattered sound field of a multicavity structure with elastic surfaces. So the radiated sound pressure of an elastic multicavity structure excited by an external force can be predicted by using the acoustical reciprocity theorem. It is verified to be correct by a corresponding test in this paper.     

Wave scattering functions and their application to multiple scattering problems

Scattering functions arise naturally in standard treatments of the effects of a material object or surface embedded in a uniform field. The most commonly used scattering function describes the far-field modulation imparted at large distances to a spherical wavefront eminating from the scatterer. The purpose of this is to develop the properties of the spectrum of scattered plane waves as an exact generalized scattering function. The linearity of the wave equations guarantees that such a representation exists; moreover, it is possible to derive the generalized scattering function from the far-field scattering function by analytic continuation. Although these properties are known, recent theoretical developments have motivated us to reexplore the interrelations among the far-field scattering function, the Green's function and various forms of the generalized scattering function as well as the symmetry properties of the generalized scattering function imposed by reciprocity. For multiple-scattering objects that can be separated by parallel planes, a system of difference equations is developed that fully accommodates the mutual interaction among the scatterers. The mutual interaction equations were developed earlier, but we show here that they can be transformed into the form that would be obtained by using the Foldy-Lax-Twersky formalism. This reinforces the equivalence between wave-space and configuration space formulations of the scattering problems.     

Rayleigh reciprocity relations:Applications

Classical reciprocity relations have wide applications in acoustics, from field representation to generalized optical theorem. In this paper we introduce our recent results on the applications and generalization of classical Rayleigh reciprocity relation: higher derivative reciprocity relations as a generalization of the classical one and a theoretical proof on the Green's function retrieval from volume noises.     

A unified optical theorem for scalar and vectorial wave fields

The generalized optical theorem is an integral relation for the angle-dependent scattering amplitude of an inhomogeneous scattering object embedded in a homogeneous background. It has been derived separately for several scalar and vectorial wave phenomena. Here a unified optical theorem is derived that encompasses the separate versions for scalar and vectorial waves. Moreover, this unified theorem also holds for scattering by anisotropic elastic and piezoelectric scatterers as well as bianisotropic (non-reciprocal) EM scatterers.     

Electromagnetic scattering from two parallel 2D targets arbitrarily located in a Gaussian beam

In this paper based on the equivalence principle and the reciprocity theorem, the scattered field up to second-order by two parallel 2D targets arbitrarily located in a Gaussian beam is considered. The first-order solution can easily be obtained by calculating the scattered field from isolated targets when illuminated by a Gaussian beam. However, because of the difficulty in formulating the couple scattering field, it is almost impossible to find an analytical solution for the second-order scattered field if the shapes of 2D targets are not canonical geometries. In order to overcome this problem, in this paper, the second-order solution is derived by using the technique based on the reciprocity theorem and the equivalence principle. Meanwhile, the relation between the secondary scattered field from target #1 and target {\#}2 is obtained. Specifically, the bi- and mono-static scattering of Gaussian beam by two parallel adjacent inhomogeneous plasma-coated conducting circular cylinders is calculated and the dependence of attenuation of the scattering width on the thickness of the coated layer, electron number density, collision frequency and radar frequency is discussed in detail.     

An extinction theorem for optical scattering

From a many-body theory of optical scattering we derive a relation for the collective behaviour of incoherently scattered light emerging from a molecular fluid. The relation involves a surface integral and has precisely the mathematical form of the Ewald-Oseen extinction theorem; it is exact in the sense of that theorem. We prove from the relation that scattered radiation is refracted and reflected at the surface of the scattering medium according to rules inferred from macroscopic optics. The relation provides a construction of contributions to the scattering with any number of internal reflections; it applies also to scattering restricted by stops. Explicit expressions for the surface dependent scattering in terms of a Fresnel transmissivity correct a previously reported theory of bulk scattering.