Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm

A novel continuation method is presented for solving the inverse medium scattering problem of the Helmholtz equation, which is to reconstruct the shape of the inhomogeneous medium from boundary measurements of the scattered field. The boundary data is assumed to be available at multiple frequencies. Initial guesses are chosen from a direct imaging algorithm, multiple signal classification (MUSIC), along with a level set representation at a certain wavenumber, where the Born approximation may not be valid. Each update via recursive linearization on the wavenumbers is obtained by solving one forward and one adjoint problem of the Helmholtz equation.     

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.     

Shape reconstruction of metallic objects from intensity scattered field data only

We address an inverse scattering problem using only amplitude information of the scattered field. In particular, we are concerned with the reconstruction of the shape of a metallic planar obstacle, located in a known plane (aperture plane), starting from the knowledge of the intensity of the scattered field over two measurement planes in near zone and parallel to the aperture plane, when a single-frequency incident plane wave is exploited. The formulation of the inverse scattering problem is given under the physical optics approximation, and thus the resulting phase retrieval problem is quadratic. This allows us to apply some phase retrieval techniques already developed for antenna diagnostics. Reconstruction results with synthetic data indicate the feasibility of the proposed approach.     

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.     

Numerical solution of inverse scattering for near-field optics

A novel regularized recursive linearization method is developed for a two-dimensional inverse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium located on a substrate from data accessible through photon scanning tunneling microscopy experiments. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to weak scattering at a low frequency, and each update is obtained by continuation on the wavenumber from solutions of one forward problem and one adjoint problem of the Helmholtz equation.     

Reconstructing the adhesion stiffness distribution in a laminated elastic plate: Exact and approximate inverse scattering solutions

This paper formulates and solves a time harmonic inverse scattering problem to reconstruct the effective stiffness distribution of an adhesive bond in a layered elastic plate. The motivation is based on the assumption that localized adhesion flaws that diminish bond stiffness also tend to diminish bond strength. The formulation is based on the invariant imbedding method, applies to isotropic and anisotropic elastic layers, and is essentially that of identifying embedded acoustic sources in elastic layered structures. This paper presents two solutions for the inverse problem: the Born approximation and the exact solution. The example calculations compare the two solutions and show that when imperfections are too large in either magnitude or extent the accuracy of the Born approximation breaks down. The impact of noise and uncertainties in the background properties in the inversion is also investigated. A regularization strategy is introduced in the exact solution that controls solution sensitivity in regions with low signal to noise ratio.     

Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium

We propose an analytical model of the spatio-temporal structure of a short laser pulse transmitted through a layer of an optically inhomogeneous medium with high anisotropy of scattering. The light-field brightness in the medium is represented as a finite series in terms of multiplicities of the small-angle scattering, while the contribution from the higher-order scattering is allowed for as a quasi-diffuse component. The scattered-pulse structure is calculated on the basis of solving the radiative-transfer equation in the small-angle approximation with allowance for the effect of multipath light propagation. Compared with the first approximation of the multiple-scattering theory (attenuated nonscattered light plus the diffuse component), this approach makes it possible to describe more correctly the transformation of the spatio-angular distribution of light in the medium when passing from the single-scattering to multiple-scattering regime, as well as specify the temporal profile of the scattered pulse. The temporal profile of the femtosecond pulse transmitted through a layer of model scattering medium with various concentrations of scatterers is studied experimentally. The blurred-pulse structure is studied with the help of nonlinear optical gating in the case of noncollinear generation of the second harmonic. Good agreement between the theoretical and experimental time profiles of the scattered pulse is shown for the optical-thickness intervals corresponding to both the predominantly low multiplicity scattering and multiple small-angle scattering, which allows us to use the proposed analytical model for solving the inverse problem of the pulse sounding of a homogeneous turbid medium. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 4, pp. 333–348, April 2008.     

A Monte Carlo approach for the calculation of polarized light: application to an incident narrow beam

Monte Carlo approaches to compute multiple scattering of polarized light are examined. A Backward Monte Carlo (BMC) method is developed to solve the Stokes vector of the multiple scattered light for an inhomogeneous scattering medium with boundaries. A generalized form of the BMC method in vector notation is proposed. This method can determine the scattered light with sufficient accuracy in both intensity and polarization compared to the same calculation using the doubling-adding method for a plane parallel medium.For application to a narrow incident beam and an inhomogeneous medium, a modified BMC method is developed, borrowing a concept from the Forward Monte Carlo (FMC) method for the first scattering events. Furthermore, a modification of the total scattering matrix, i.e., the combination of the derived scattering matrix with its time inverse, is discussed. This BMC method can be used successfully for model calculations of lidar and other laser measurements of polarized light.     

Scattering of spatially inhomogeneous sound waves by a spherical particle

The problem of monopole, dipole, and rotational scattering of a spatially inhomogeneous time-harmonic sound field by an arbitrary spherical particle is solved for the cases of the medium being a viscous compressible liquid or an isotropic elastic medium. Equations for the spherical mean fields at the particle are obtained. These equations are used to derive the formulas for the scattered fields. Different limiting cases of particle behavior are considered. In particular, it is shown that the dipole scattering is determined by two components of particle oscillations, one of which corresponds to translational oscillatory motion and the other to oscillations of two antiphase monopoles. For these types of particle oscillations, a scattering matrix, which determines the scattering of an arbitrary field by a particle, is constructed. The matrix allows the formalization of the processes of multiple sound scattering by particles and is valid for any distances between the particles down to their contact.     

Index to volume 72

A numerical analysis of wave propagation in one dimensional inhomogeneous media is presented in this paper, the first (Part I) of two companion papers. The aim is to provide a natural and physical basis for the solution of the inverse scattering problem in one dimension. The paper includes (i) a discussion of some aspects of the numerical solution of the direct problem, (ii) typical results of calculation and (iii) a comparison between exact solutions and forward scattering approximation (FSA) solutions of the direct problem. It is shown that the FSA allows accurate calculations of the reflected wave and thus constitutes a sound basis for the solution of the inverse problem treated in Part II.     

Scattering of surface waves by a sub-surface crack

The scattering of Rayleigh waves by a two-dimensional sub-surface crack, which is perpendicular to the free surface of an elastic half-space, is investigated. The boundary-value problem for the scattered field is stated, and reduced to an uncoupled system of integral equations which are solved numerically. At large distances from the crack the scattered field is shown to consist of outgoing Rayleigh waves and cylindrical body waves. Graphs of the variation of the amplitude and phase of the forward and backward scattered Rayleigh waves with the frequency and the geometrical parameters of the crack are presented. The stress intensity factors at both crack tips have also been obtained.     

Fractal characteristics investigation on electromagnetic scattering from 2-D Weierstrass fractal dielectric rough surface

A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. The variance of scattering intensity is presented to study the fractal characteristics through theoretical analysis and numerical calculations. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated as a slope of linear equation. This conclusion will be applicable for solving the inverse problem of reconstructing rough surface and remote sensing.     

Angular structure of radiation scattered by monolayer of polydisperse droplets of nematic liquid crystal

We considered light scattering by a polydisperse ensemble of droplets of a nematic liquid crystal. To model light scattering by a monolayer of polymer-dispersed spherical droplets of a nematic liquid crystal with cylindrical symmetry of its internal structure, we proposed a semianalytical modeling method. The method is based on interference approximation of the theory of multiple wave scattering, anomalous diffraction approximation, and effective-medium approximation. The method takes into account cooperative optical effects in concentrated, partially ordered layers and can be used to analyze the small-angle structure of the intensity of scattered radiation in relation to the concentration, size, polydispersity of liquid crystal droplets, orientation of their optical axes, and refractive indices of the liquid crystal and polymer. The obtained relations can be applied to solving direct and inverse problems of light scattering in composite liquid crystal materials using data of polarization measurements. We present graphical results of solving the direct problem for components of the polarization vector of scattered wave. These results illustrate the formation of an angular structure for monolayers with a high concentration of polydisperse droplets of the liquid crystal in the range of small scattering angles (0 < θ _s ≤ 8°).     

Similarity effects in multiple scattering of coherent radiation: Phenomenology and experiments

Based on phenomenological concepts of statistics of effective optical paths for multiple scattering of coherent radiation, an analysis is carried out of similarity effects observed for the dependences of statistical moments of the scattered field on the relaxation parameters with a dimension of reciprocal length. Within the framework of the diffusion approximation, expressions are obtained that describe the autocorrelation function of fluctuations of the scattered-field amplitude, the degree of polarization, and the normalized intensity of scattered light for media with a finite absorption length in the case of forward scattering of coherent radiation in a plane layer of an isotropic scattering medium. The results of the analysis show the similarity of the dependences of these quantities on the corresponding spatial scales. Experiments with model scattering media (aqueous suspensions of polystyrene spherical particles) supported the existence of similarity effects in multiple scattering. An experimental study was made of the relation between the characteristic depolarization length and the transport length for multiple scattering of coherent radiation in a plane layer. The effective value of the radiation diffusion coefficient providing the best agreement between the experimental and the calculated values of parameters of the scattered field is shown to be independent of the absorption coefficient of a medium.     

Temperaturabhängigkeit des Streuuntergrundes im Elektroneninterferenzdiagramm polykristalliner Silberschichten

The filtered and unfiltered intensities of electrons (30 to 50 kev) scattered in thin polycrystalline silver foils (300 to 1400 Å) were measured by means of a retarding field apparatus in the temperature range from 170 to 380 °K. Further the absorption coefficients due to the inelastic and the different elastic scattering processes were determined. It is shown, that the diffuse elastic scattering intensity increases with temperature for all values of thickness and electron energy. From the measured absorption coefficients the contributions of the different elastic scattering processes to the total background intensity are calculated. It turns out, that the main contribution is due to multiple scattering processes including both Bragg reflections and thermal diffuse scattering. Furthermore it is shown, that in contrary to the filtered diagram the background intensity of the unfiltered diagram increases with temperature only for larger scattering angles. In the small angle region a reversed temperature behaviour is observed. This effect is caused by that part of inelastically scattered electrons at low angles coming directly from the primary beam and from the low order diffraction rings.     

Elastic-wave identification of penetrable obstacles using shape-material sensitivity framework

This study deals with elastic-wave identification of discrete heterogeneities (inclusions) in an otherwise homogeneous “reference” solid from limited-aperture waveform measurements taken on its surface. On adopting the boundary integral equation (BIE) framework for elastodynamic scattering, the inverse query is cast as a minimization problem involving experimental observations and their simulations for a trial inclusion that is defined through its boundary, elastic moduli, and mass density. For an optimal performance of the gradient-based search methods suited to solve the problem, explicit expressions for the shape (i.e. boundary) and material sensitivities of the misfit functional are obtained via the adjoint field approach and direct differentiation of the governing BIEs. Making use of the message-passing interface, the proposed sensitivity formulas are implemented in a data-parallel code and integrated into a nonlinear optimization framework based on the direct BIE method and an augmented Lagrangian whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible (or overly distorted) trial inclusion configurations. Numerical results for the reconstruction of an ellipsoidal defect in a semi-infinite solid show the effectiveness of the proposed shape-material sensitivity formulation, which constitutes an essential computational component of the defect identification algorithm.     

Scattering of sound by an elastic spherical shell immersed in a waveguide with a fluid bottom

The paper is devoted to modeling an acoustic field scattered by an elastic spherical shell. The shell is immersed in a waveguide with a fluid attenuating bottom. Modal analysis is applied to calculations of both the point source field in a free waveguide and the scattering coefficients of the shell. The occurring integrals along a branch cut are expressed via the probability integral. Cases are analyzed when the source frequency differs from the critical frequencies of normal waves and when it coincides with one of them. The method is applied to estimate the effect of the field scattered by a thin elastic shell filled with air with transmission loss of the total acoustic field with distance and its depth distribution.     

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.     

Characteristics of exact retrieval of the structure of plane-stratified media using reflected and transmitted pulse signals

We consider the exact iterative algorithm for solution of the direct and inverse scattering problems in 1D inhomogeneous half-space using piecewise-constant approximation of the constitutive parameter profiles (e.g., refractive index). A novel modification of the Gelfand–Levitan equation is derived for solution of the inverse scattering problem. Numerical examples are given and the effects related to a finite frequency band of the sounding pulse are discussed.     

Scattering of time-harmonic elastic waves by an elastic inclusion with quadratic nonlinearity

This paper considers the scattering of a plane, time-harmonic wave by an inclusion with heterogeneous nonlinear elastic properties embedded in an otherwise homogeneous linear elastic solid. When the inclusion and the surrounding matrix are both isotropic, the scattered second harmonic fields are obtained in terms of the Green's function of the surrounding medium. It is found that the second harmonic fields depend on two independent acoustic nonlinearity parameters related to the third order elastic constants. Solutions are also obtained when these two acoustic nonlinearity parameters are given as spatially random functions. An inverse procedure is developed to obtain the statistics of these two random functions from the measured forward and backscattered second harmonic fields.