Superresolution of three-dimensional optical imaging by use of evanescent waves

We simulate a three-dimensional optical diffraction tomography experiment in which superresolution is achieved by illuminating the object with evanescent waves generated by a prism. We show that accounting for multiple scattering between the object and the prism interface is mandatory to obtain superresolved images. Because the Born approximation leads to poor results, we propose a nonlinear inversion method for retrieving the map of permittivity of the object from the scattered far field. We analyze the sensitivity to noise of our algorithm and point out the importance of using incident propagative waves together with evanescent waves to improve the robustness of the reconstruction without losing the superresolution.     

Numerical solution of an inverse medium scattering problem for Maxwell’s Equations at fixed frequency

Consider a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in R³${\mathbb{R}}^3$. In this paper, well-posedness of the variational problem for the direct scattering is examined. An energy estimate for the scattered field is obtained on which the Born approximation is based. A regularized recursive linearization method for the inverse medium scattering, which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field, is developed. The algorithm requires only single-frequency data. Using an initial guess from the Born approximation, each update is obtained via continuation on the spatial frequency of a two-parameter family of plane waves by solving one direct problem and one adjoint problem of the Maxwell equation.     

On multidimensional ultrasonic scattering in an inhomogeneous elastic background.

This work is concerned with the modeling of elastic wave scattering by solid or fluid-filled objects embedded in an inhomogeneous elastic background. The medium is probed by a monochromatic force and the scattered field is computed (forward problem) or observed (inverse problem) at some known receiver locations. Based on vector integral equations for elastic scattering, a general framework is developed, independent of both the problem geometry and the transmitter-receiver characteristics. This framework encompasses both forward and inverse modeling. In the forward model, a Born approximation for an inhomogeneous background is applied to obtain a closed form expression for the scattered field. In the inverse model, this approximation is also invoked to linearize for the multiparameter characteristic of the object. Finally, an iterative inversion scheme alternating forward and inverse modeling is proposed to improve the resolution and accuracy of the reconstruction algorithm.     

Fast prediction of scattered sound field based on Fourier diffraction theory under second-order born approximation

The directional pattern of sound waves scattered from an object insonified by a plane wave can be efficiently predicted using the Fourier diffraction theorem (FDT). This is achieved by sampling a circle in the discrete Fourier transform of the object/medium distribution. However, the FDT-based approach under the first-order Born approximation is only applicable to weak scattering. To improve the prediction accuracy and expand the method’s scope of applications, we introduce a second-order correction term to the solution, which is obtained by taking the first-order scattered waves as secondary incident sources, and calculate the “scattering” in the same way as in the first-order FDT-based approach. Adding the resulting correction term to the directional pattern based on the first-order Born approximation, the second-order prediction is obtained. Numerical results show that the proposed method can provide improved directional patterns of the scattered waves, and the range of applicability is significantly expanded.     

Inverse problem of the wave equation and the Schwinger approximation

A new method of profile inversion for acoustic waves propagating in a medium with spherical inhomogeneity based on the Schwinger variational method is presented. The wave equation of interest is transformed into a Schrodinger equation, so that the Born approximation and the new method could also be applied at high frequencies. It is shown that the new method is stable and is more accurate than the Born approximation. To illustrate the method, an exactly solvable analytical example is presented. Also numerical examples using synthetic data, with and without additive noise, are given and the corresponding inversion results and the stability of the method are studied.     

Scattering of bulk acoustic waves with different polarizations on a statistically rough free surface of a solid at oblique incidence

In the first Born approximation of the perturbation theory by a Green's function method developed by Maradudin, Mills [7] and Kosachev, Lokhov, Chukov [8,9] the problem of scattering bulk acoustic waves with different polarizations at oblique incidence on a statistically rough free boundary of an isotropic solid was solved. When the correlation function of the surface roughness is of a Gaussian form, the expressions for the transformation energy factor of the incident wave in the scattered volume and surface Rayleigh waves with respect to polarization, frequency and grazing angle of the incident wave as well as the roughness parameters and the Poisson coefficient of the medium were obtained. These results are helpful in accounting for the experiments on residual losses [15–17].     

Scattering of a Rayleigh surface acoustic wave by a small-size inhomogeneity in a solid half-space

We use the Born approximation of the perturbation method to solve the problem of scattering of a harmonic Rayleigh surface acoustic wave by a weak-contrast inhomogeneity that is small compared with the wavelength and is located in a solid half-space near its boundary. The material of the inhomogeneity differs from the material of the half-space only in its density. The Rayleigh wave incident on the inhomogeneity is excited by a monochromatic surface force source acting normally to the half-space boundary. We derive expressions for the displacement fields in the scattered spherical compressional and shear (SV- and SH-polarized) waves. Scattering of the Rayleigh wave into a Rayleigh wave is studied in detail. We find expressions for the vertical and horizontal components of the displacement vector in the scattered Rayleigh wave as well as its radiated power. It is shown that the field of the scattered surface wave is mainly formed by vertical oscillations of the inhomogeneity in the field of the incident wave. In this case, the radiated power for the scattered Rayleigh wave formed by vertical motion of the inhomogeneity in the incident-wave field depends on the depth of the inhomogeneity as the fourth power of the function describing the well-known depth dependence of the vertical displacements in the Rayleigh surface wave. Correspondingly, the dependence of the radiated power for the scattered Rayleigh wave formed by horizontal motion of the inhomogeneity depends on its location depth as the fourth power of the depth dependence of the horizontal displacements in the Rayleigh surface wave. We perform calculations of the ratio between the powers of the scattered and incident Rayleigh waves for different ratios between the velocities of the compressional and shear waves in a solid. It is shown that the radiated power for the scattered surface wave decreases sharply with increasing depth of the subsurface-inhomogeneity location. Thus, the scattering of a Rayleigh wave into a Rayleigh wave is fairly efficient only when the location depth of the inhomogeneity does not exceed about one-third of the wavelength of the shear wave in an elastic medium.     

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。      

An efficient method for computing backscattering from Born objects of arbitrary shape

A method is presented for efficiently computing the propagating pressure field backscattered by an arbitrarily shaped, weakly scattering, three-dimensional object. This is accomplished by drawing upon a previously reported relationship between the boundary condition on a two-dimensional radiating aperture and the pressure propagating along an axis normal to the aperture, and the fundamental theorem of diffraction tomography, which relates the Fourier transform of an object function to its scattered pressure field. Together, these two results are used to derive an integral formula that expresses the pressure field backscattered from an object as a one-dimensional Fourier transform of its scattering amplitude. This formula is then utilized to compute the backscattered pressure field from a uniform fluid sphere in the first Born approximation; the results of which are compared to the rigorous partial wave expansion.     

Characterization of a cylindrical rod by inversion of acoustic scattering data

In this paper, a new approach is proposed for nondestructive characterization of immersed and embedded isotropic rod-shaped samples by inversion of acoustic scattering data. The normal mode expansion technique is used for modelling the scattered field and the compression incident and compression scattered waves are considered. Genetic algorithm is the inversion technique used for estimating the elastic wave velocities and density of the rods from their measured backscattered pressure spectrum. The inversion technique is capable of computing the parameter values that best fit a particular set of data. A perturbation study is conducted on the sensitivity of the resonance frequencies to changes in elastic properties and density of the rods. The numerical results indicate that proper selection of resonance frequencies leads to accurate measurement of elastic constants and density. The proposed approach showed very good convergence and the results obtained were found to agree very well with available data.     

Parametric scattering of high-frequency elastic waves by a small spherical cavity oscillating under the action of a Rayleigh wave

Scattering of high-frequency transverse and longitudinal plane waves incident on a spherical cavity located at a small depth under the surface of a half-space is considered. The cavity oscillates as a whole in the field of a low-frequency Rayleigh surface wave, the oscillation vectors of the longitudinal, transverse, and surface waves being coplanar. The cavity radius is assumed to be small compared to the wavelengths of the sounding wave and the pumping surface wave. The scattered compression and shear waves at the combination frequencies ω±Ω are calculated in the dipole approximation. Expressions obtained describe the qualitative behavior of the combination-frequency signal levels produced at the outputs of horizontally and vertically oriented geophones moving over the free surface of the elastic half-space.     

水中密排圆柱体群的超声多重散射参数

以水中紧密排列的平行圆柱体群为对象,研究平面超声脉冲经多重散射后的透射波性质,通过分析其中头波和散射波的特征获得对应的多重散射参数.对直径随机分布、位置无序排列、数量密度约100个/cm2、面积占空比约0.53的非接触圆柱体群,采用中心频率2.5 MHz的宽带脉冲波入射。为解决透射信号在时域表现出随机性的问题,将散射体尺寸、分布都相同但位置分布不同的多个模型仿真的透射波叠加平均后用于分析.在频域对头波的宽带衰减系数进行分析,并在时域研究散射波声强的时间演化曲线,获得了系统的弹性平均自由程、传输平均自由程等多重散射参数。经多重散射后,透射波中的头波表现出相干性,由不相干近似理论可对其对应的散射参数进行定性描述;散射波是不相干的,其对应的多重散射参数可近似利用扩散近似理论获得。      

Stochastically excited waves in a 1D random medium - the mean field and power spectrum

Scalar wave propagation is examined when both the wave source and the propagation speed are random. Results are derived for the mean field and the power spectrum using the second-order Born approximation. The results depend on whether the source S(x, t) and the propagation speed c(x, t) are correlated or not. When they are uncorrelated, the mean field is zero. When they are correlated, the mean field is non-zero only when the source is non-stationary. The power spectrum is incoherent to leading order. There is a transfer of energy from lower to higher frequencies owing to wave scattering. The corresponding frequency upshift of the power profile in the (k, ω) domain is mainly caused by the cross power between the direct and the twice scattered field, which represents a second-order incoherent power contribution. The results are confirmed using a numerical solution of the wave equation where the scattered field is expanded to fifth order.     

Reciprocity relations for scattering from quasi-homogeneous anisotropic media

Valid within the accuracy of the first-order Born approximation, we obtain the analytical expressions of the three-dimensional spatial Fourier transforms, which show the reciprocity relations for the scattering of a polychromatic plane wave incident upon a Gaussian-correlated, quasi-homogeneous, anisotropic scatterer. We find that the spectral degree of coherence of the scattered field is only related with the effective radius of the scatterer and the normalized spectral density of the scattered field is only related with the correlation length of the scatterer.     

三维可穿透目标远场声波反演的一种指示器样本方法

提出了用时谐声散射场的远场信息来可视化三维可穿透目标的一种指示器样本方法,它是通过析取一个指示器函数在包含可穿透目标的某个样本区域中的支集来实现这种可视化的,其中,这个指示器函数在可穿透目标的内部和外部有显著不同的取值.这个算法的一个特别吸引人的性质是不需要关于障碍物的任何几何和物理的先验信息,并且只需要散射场在某个有限孔径中若干个入射和测量方向上的远场信息,即可得到可穿透目标的一个很理想的可视化.数值算例保证了这个可视化算法是有效和实用的 关键词： 声散射 反问题 物形反演     

The effect of a local field on Raman scattering in a uniaxial crystal

This paper discusses the Raman scattering of light in an anisotropic crystal in the crystal optics approximation, taking into account local fields acting on the molecules. It shows that the effect of the local field reduces to the introduction of the effective Raman polarizability tensor of the molecules, which depends both on the properties of the molecules themselves and on the characteristics of the crystal at the frequencies of the incident and scattered waves. Raman scattering cross sections are obtained in a uniaxial crystal for various types of incident waves. It is shown that, in the case of an extraordinary incident wave, the local field substantially affects how the cross section depends on the direction of incidence. Zh. éksp. Teor. Fiz. 112, 180–191 (July 1997)     

Nonresonant spontaneous bremsstrahlung by a relativistic electron scattered by a nucleus in the field of pulsed light wave

This paper presents theoretical study of nonresonant spontaneous bremsstrahlung by an electron scattered by a nucleus in the field of the pulsed light wave. The electron interaction with a Coulomb potential of a nucleus is considered in the first order of perturbation theory (the Born approximation), and the interaction with an external pulsed field is taken into account accurately. The approximation is examined when pulsewidth is considerably greater than the characteristic time of wave oscillations. The obtained differential cross-section of the process has form of a sum over partial differential cross-sections. Each of them corresponds to processes with emission (absorption) of certain number of wave photons. It is shown, that for spontaneous bremsstrahlung by an electron scattered by a nucleus in the field of a pulsed light wave the distribution of a stimulated emission-absorption probability is determined by the average value of corresponding probabilities in the case monochromatic wave.     

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.     

Surface wave scattering by a nano-object situated on a surface

The scattering of the surface electromagnetic waves by a nano-defect (object) on a surface was calculated. The scattered field has been considered as a field caused by the current generated by the self-consistent local field inside the defect. In turn, the self-consistent local field has been determined as a result of solution of the integral Lippmann-Schwinger equation. The effective susceptibility of the object has been calculated using a self-consistent procedure. The corrections of self-energy part due to direct and indirect electromagnetic interactions, as well as due to interaction with surface wave field are taken into account. The self-energy part is calculated analytically within the framework of the near-field approximation. The scattering indicatrisses in reciprocal space have been computed for different shapes of the scatterer. Strong dependence of the scattered field on geometry of the scatterer has been found and explained.     

Spectral changes of light produced by scattering from disordered anisotropic media

We investigate theoretically changes in the spectrum of polychromatic light scattered by a disordered, birefringent medium. We derive an expression for the spectrum of scattered light for ordinary and extraordinary incident waves within the accuracy of the first Born approximation. Using this result, we analyze the changes in the spectrum of light due to the combined action of disorder and anisotropy in the scattering process.