Multiple small-angle scattering—A review

Small-angle scattering (SAS) is a powerful experimental technique in condensed matter physics for studying structural features of inhomogeneities of colloidal dimensions. So far the technique has been largely exploited to study thin samples for which the single scattering approximation, for the radiation-matrix interaction, holds good. The single scattering approximation is invalid when the thickness of the sample exceeds the scattering mean free path. This situation calls for a guideline to analyse the scattering data having significant contribution from multiple scattering. Since multiple scattering broadens the scattering profile, the beam broadening nature of multiple scattering can also be exploited, by making the sample suitably thick, to study large size inhomogeneities which are otherwise inaccessible to a small-angle scattering set up because of its resolution constraints. The present article presents a review and extension of the theoretical basis for analysing multiple scattering data from the point of view of a recent formalism on multiple small-angle scattering. The formalism is valid for both monodisperse and polydisperse scattering media characterized by the presence of large size inhomogeneities in the matrix. It is shown that multiple scattering from a polydisperse sample can be described by a system of coupled integrodifferential equation. However, multiple scattering from a monodisperse sample can be described by a Fokker-Planck type of equation. These equations have been analysed with an emphasis laid on the nature of the structural information pertaining to the inhomogeneities which is extractable from the multiple scattering profile. When the linear dimension of inhomogeneities becomes comparable to the scattering mean free path of the radiation in the sample, the statistical nature of the medium becomes pronounced. The statistical nature of the medium modulates the scattering profile. The modulation effect could be broadening or narrowing of the profile depending upon the nature of the inhomogeneities and their population distribution. The limiting regimes of validity and the implications of various approximations, frequently used to analyse the scattering data, have been indicated.     

Sound scattering by random fractal inhomogeneities in the ocean

Sound scattering by random volume inhomogeneities (fluctuations of the refraction index in a medium) with an arbitrary anisotropy is considered using the small perturbation method (Born’s approximation). Surfaces (boundaries) of the inhomogeneities are deemed to be fractal ones: the energy spectra of the refraction index fluctuations follow the power law with a nonintegral exponent. Formulas are obtained for the volume scattering coefficient. Frequency and angular dependences of the scattering coefficient and their relations to the fractal dimension of inhomogeneities with different kinds of anisotropy and different sizes (on the sound wavelength scale) are presented. The fractal dimension of the inhomogeneities is estimated.     

Investigation of multiple ultrasmall-angle neutron scattering with the aid of a double-crystal diffractometer

A method of multiple small-angle neutron scattering (MSANS), used to obtain information about inhomogeneities of substances, is considered. Experimental schemes and theory of a double-crystal diffractometer intended for measuring multiple small-angle neutron scattering are described. Methods used to approximate MSANS angular distributions at a low concentration of inhomogeneities and based on Moliére’s theory are reviewed. Their applicability is demonstrated for the examples ofMSANS spectra of samples from ferromagnetic iron-nickel alloys, aluminum powders, and high-T _c superconductor ceramic materials. Theoretical and experimental investigations of interference effects that manifest themselves in multiple small-angle neutron scattering on systems containing high concentrations of scatterers are reported.     

Interference of speckle fields in the diffraction zone of a focused spatially modulated laser beam by a random phase screen

The processes of formation of average-intensity interference fringes upon diffraction by a random-phase object of a laser beam having interference fringes and focused on the surface of the object are considered. The dependences of the fringe contrast on the parameters of scattering inhomogeneities of the object and the parameters of the focused laser beam are established in analytical form for various diffraction regimes. Practical possibilities of a method of probing of scattering objects in problems of measuring the parameters of inhomogeneities and problems of interference-pattern formation in optical systems with scattering media are discussed.     

Single scattering of waves by random strong anisotropic inhomogeneities

Research is carried out into scattering of waves by random strong anisotropic inhomogeneities when the inhomogeneities are in the distant zone according to one (transverse) scale and at the same time they are in the near zone according to another (longitudinal) scale. To analyse the formulas of the single scattering the stationary phase method in the longitudinal coordinate integral is used. It is shown that the angle sensitivity of strong anisotropic scattering, unlike weak anisotropic scattering, strongly depends on the longitudinal statistical homogeneity of the medium.     

The scattering of flexural waves by random fractal inhomogeneities in a thin plate

The scattering of flexural waves by small statistical fractal inhomogeneities in a thin plate is considered. An expression for the average intensity of the fluctuations of the scattered wave field is obtained. A relation of the intensity to the plate parameters and to the fractal dimension of the inhomogeneities is determined. An expected frequency dependence of the attenuation of flexural waves in a plate due to the scattering by fractal inhomogeneities is discussed.     

Localization of inhomogeneities in diffuse optical tomography based on late arriving photons

An algorithm for localizing inhomogeneities in pulsed diffuse optical tomography is proposed and implemented. A distinctive feature of this technique is the formation of an initial approximation to the spatial distributions of the absorption and scattering coefficients in a biomedical object under study based on the angle-dependent homogeneity index, HI(a). The method allows one to determine the approximate optical structure of the object using late arriving photons and thus solve more rapidly the inverse problem. The suggestion that all absorbing and scattering inhomogeneities in an object under study are spherical also simplifies and enhances image reconstruction.     

Multiple scattering of electromagnetic waves by the internal surfaces of turbulent bodies of revolution

Using the method of statistical modeling, we have solved the problem of multiple scattering of electromagnetic waves by THE internal surfaceS of bodies of revolution: a cone and paraboloids of the second and fourth orderS along whose generatrices turbulent inhomogeneities move. The angular and frequency spectra of the scattered signal are obtained and their dependence on the shape of the scattering surface, on the reradiation pattern of turbulent inhomogeneities, and on the variance of their travel velocity is analyzed.Siberian Physical and Technical Institute, State University, Tomsk. Translated from izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 9, pp. 906–912, September, 1995.     

The angular dependence of the coefficient of sound reflection from anisotropic volume inhomogeneities of the ocean

The small perturbation method is used to consider the scattering of sound by random volume inhomogeneities of the ocean. The inhomogeneities are assumed to be horizontally anisotropic with their horizontal dimensions far exceeding the vertical ones. Expressions are obtained for the angular dependence of the scattering coefficient. Vertical and horizontal sections of the scattering pattern are presented, and the angular widths of its maxima are analytically estimated.     

Laser excitation of flexural waves and their scattering by fractal inhomogeneities in a thin plate

The excitation of flexural waves in a thin plate (film) by harmonically modulated laser radiation and their scattering by small fractal inhomogeneities are considered. An expression for the mean fluctuation intensity for the scattered wave field is obtained. A relationship between the intensity, parameters of the laser radiation and the plate, and the fractal dimension of inhomogeneities is found. The expected frequency dependence of the flexural wave attenuation in the plate due to their scattering by fractal inhomogeneities is discussed.     

Formation Mechanisms for the Spectral Characteristis of Low-Frequency Reverberations and Predictive Estimates

The paper considers the problem of monostatic scattering of low-frequency sound waves by nearsurface volumetric inhomogeneities under conditions of intense wind waves. We calculate the expected shape of the scattered signal spectrum taking into account the distribution of the volumetric inhomogeneities over the surface and their quasiperiodic motion in three-dimensional space under the action of wind waves. For deep-ocean conditions, a carrier frequency of 228 Hz, and a pulse duration longer than 100 s, we compare the experimental data on the shape of the reverberation spectrum with theoretical estimates. We compare the spectral levels of subsurface scattering with similar data on sound scattering directly on the wind-roughed surface.     

An experimental study of forward and backward scattering of acoustic waves from a thin bubble layer

Forward and backward scattering of acoustic signals from a thin chaotic bubble screen is studied experimentally. The concentration of inhomogeneities in the bubble layer is estimated by an acoustic method, and the presence of collective scattering effects is revealed. The transverse correlation radius of the scattered field amplitude is determined, and its value is used to estimate the correlation scales of inhomogeneities in the bubble layer.     

On the theory of refractive radio-wave scattering

We consider in detail the frequency correlation of radio-wave fluctuations in one or several thick layers with strong large-scale inhomogeneities of turbulent origin. General expressions are obtained for the space-frequency fluctuation correlation of the radio-waves received. We analyze particular cases of radio wave scattering in turbulent media with inhomogeneities described by power-law spectra with indices p2 and p3. It is shown, in particular, that the coherence band of signals propagated in media with strong large-scale inhomogeneities is critically dependent on the spectral type of thOse inhomogeneities. The occurrence of an additional strongly scattering layer, which has radically different properties compared to the first layer, on the radio-wave path can increase or decrease considerably the frequency correlation of the radio waves received.Radiophysical Research Institute, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 10, pp. 1012–1022, October, 1995.     

On the low-frequency sound scattering in a moving microinhomogeneous medium

The Rayleigh law that governs low-frequency sound attenuation due to the scattering by inhomogeneities in a microinhomogeneous medium is generalized to the case of particles moving in a flow or falling under gravity. Corrections to the scattering’s cross section that adjust the Rayleigh law to the case of a potential flow around inhomogeneities are calculated. It is shown that, when microinhomogeneities are moving in a viscous medium, the characteristics of discrete scatterers may considerably deviate from the Rayleigh law. Based on the data on the velocity and size distribution of falling drops of water in air, refinements are proposed for the laws of low-frequency sound scattering by rain.     

Role of light-induced scattering in the optical limitation of laser radiation on the basis of fullerene-containing media

A theoretical and experimental study of the contribution of light-induced scattering to nonlinear optical limitation is made. It is shown that light-induced scattering makes a considerable contribution to the optical limitation of radiation by fullerene-containing solutions. Radiation is scattered by small-scale (1–10 μm) inhomogeneities of density, and the corresponding increments are rather large. Because of this, scattering can be initiated by small-scale inhomogeneities of the input beam. Numerical simulation of nonlinear scattering is made, and the angular distribution and limitation of radiation in optical limiters are calculated. The calculation results are compared with the experiment.     

Evaluation of light scattering in attenuated total reflection configuration

A theoretical investigation of light scattering in first-order Born approximation in order to analyse the different scattering sources is presented for the case of the so-called Kretschmann attenuated total reflection (ATR) configuration. This three-layer system consists of a high permittivity superstrate acting as a prism, a thin metallic layer and air as substrate. The interesting aspect of this system is the occurrence of a resonance for both the exciting and the scattered light caused by surface plasmon polaritons (SPPs) at the metal-air interface. The scattered light will be calculated under the assumption of different illumination and detection conditions. The scattering contributions are assumed to be caused by the roughness of both interfaces and the volume inhomogeneities in the metallic layer. The calculated curves show the principal possibility of analysing the different scattering sources by an appropriate experimental strategy.     

Studies of the structure of defects in In4Se3 crystals by small-angle neutron scattering

Small-angle neutron scattering is used to study the layered semiconductor In₄Se₃. It is observed that the samples contain inhomogeneities of colloidal dimensions, which are attributed to the precipitation of indium. Annealing reduces the concentration of inhomogeneities. Modeling the system using an ensemble of primary spherical inhomogeneities showed that a single cluster of a rectangular superlattice (2-2-6) contains 24 primary inhomogeneities of radius 13.5±1,5 Å and the distance between them is 70.9±1.5 Å.     

Computer modeling of sound propagation in the ocean with fractal inhomogeneities

The wave-field computer code based on the wide-angle parabolic equation is modified and adapted to the problems of sound scattering in a medium with anisotropic inhomogeneities of fractal type. To verify the computer code, a model numerical experiment on determining the angular dependence of the scattered sound field is performed for different anisotropy coefficients of the sound speed inhomogeneities. The comparison of the computed data with the theoretical dependences shows their rather good agreement and indicates that the computer code can be applied to calculations of sound propagation in the ocean with fine-structure inhomogeneities possessing fractal properties.     

Scattering of radiation by a quasiperiodic two-dimensional medium

We study the scattering of radiation by a medium presenting inhomogeneities distributed in a quasiperiodic way. We show the existence of quasiperiodic solutions of the two-dimensional stationary wave equation, under certain conditions on the index of refraction, using a technique based on Dinaburg-Sinai method for one-dimensional Schrödinger equation with a quasiperiodic potential. Moreover we show that the energy spctrum contains a nonempty absolutely continuous component, with a subset having high degeneracy, provided the inhomogeneities are small enough.     

Selective laser heating and nonlinear light scattering in a homogeneous medium

Observation of light scattering on the laser-produced temperature inhomogeneities, the centres of which are connected with some absorbing molecules, is reported. The experimentally observed scattering intensity agrees well theory.