Double passage analysis in random media using two-scale random propagators

In location and remote sensing experiments there arise a number of effects related to the double passage of the backscattered field through the same random inhomogeneities as the incident one. To account for the correlation of the forward-backward propagating events, there is a need for a measure in which the random information along the propagation path is preserved. For the generation of even statistical moments, the relevant measure defined in the recently formulated stochastic geometrical theory of diflraction is the two-point random function (TPRF)—a paired field measure which is propagated along the geometrical rays of the deterministic background medium. From this function all even statistical moments can be generated. Here we present an approximate analytical solution for the high-frequency propagator obtained by applying the multiscale expansion asymptotic procedure to the partial differential equation governing the propagation a1 the TPRF. The test of the solution is performed on canonical backscattering problems based on point source-point scatterer and paint source-plane mirror configurations, which justifies its further application for construction of the coherence measures of the rctrareflected field. Coherence properties of the plane and spherical wavefields reflected backward by a plane mirror were investigated. Further, we investigated the intensity enhancement effects observed in the double passage of a Gaussian beam retroreflected from a plane mirror. Asymptotic expressions lor the retroreflected intensity are obtained, and their computations show good agreement with the direct numerical evaluations.     

Backscattering response from periodic spatial patterns in random media

Abstract

In wave-based remote sensing or radio-location of distant objects in a random medium, a high-frequency electromagnetic wave is scattered by object discontinuities, and portions of the scattered radiation can traverse the same random inhomogeneities as the initial incident field. The statistical dependence of the forward–backward travelling events results in an anomaly in the backscattered intensity pattern that carries information about the scattering object. The quality of this information depends on the ability to resolve the fine-structure elements. In this work we investigate the resolving properties of periodic spatial objects by using the random propagators of the stochastic geometrical theory of diffraction.     

Off-axis-Gaussian beam intensity in random medium

The receiver intensity profile of an off-axis-Gaussian beam travelling in random medium is formulated. By examining the related exponential terms of this intensity expression, the rules governing the receiver plane displacements are deduced. Off-axis-Gaussian beam is characterized by introducing into a Gaussian beam, complex displacement parameters that exhibit transverse source coordinate dependent attenuation and phase shifts. Our results are applied to turbulent horizontal links. Intensity plots describing the dependence on the source and propagation parameters both on the source and the receiver planes are provided. Even though the normalized intensities of the off-axis-Gaussian beam having the same source sizes but differing displacements in x- and y-directions may look the same on the source plane, they will differentiate after propagation. The views from the progress of an off-axis-Gaussian beam along the turbulent link length show that a source displaced beam will act according to rules set by related exponential terms. An asymmetrical (ellipsoidal) off-axis-Gaussian beam will initially be converted into symmetric (circular) shape at the intermediate link lengths, then it will start to expand in the other direction, thus reverting to an ellipsoid shape whose major axis is now along the transverse coordinate opposite to that of the source plane.     

Backscattering from different objects in turbulent media

Abstract

The connection between diffraction characteristics of the scatterer and distribution of average backscattered intensity of a spherical wave is considered. In experiments with an ‘infinite’ plane mirror it is shown that the distribution of average backscattered intensity coincides with the correlation function of the intensity fluctuation of a virtual point source located at the mirror and observed from the real source plane. Non-monotonic dependence (with a minimum at the Fresnel number of scattered mirror?1) between the enhancement factor and the size of reflected mirror is observed in experiments.     

Intensity fluctuations in a moving random medium

We consider the intensity fluctuations arising when a point source of radiation moves in a randomly inhomogeneous scattering medium. The medium itself can also move with a velocity whose component normal to the direction of propagation can have an arbitrary distribution. We derive an expression for the space–time autocorrelation function of the intensity fluctuations transverse to the direction of propagation. The result is analysed for some particular cases and it is shown how the resulting information can be useful in examining the behaviour of random media in situations of practical interest.     

Reference-wave solutions for the high-frequency fields in inhomogeneous-background random media

Ray theory plays an important role in determining the propagation properties of high-frequency fields and their statistical measures in complicated random environments. For computations of the statistical measures it is therefore desirable to have a solution for the high-frequency field propagating along an isolated ray trajectory. A new reference wave is applied to obtain an analytic solution of the parabolic wave equation that describes propagation along the ray trajectory of the deterministic-background medium. The methodology is based on defining a paired-field measure as a product of an unknown field propagating in a disturbed medium and the complex-conjugate component propagating in a medium without random fluctuations. When a solution of the equation for the paired-field measure is obtained, the solution of the deterministic component can be extracted from the paired solution to determine the solution of the unknown field in an explicit form.     

Reflection of waves from a strong scatterer buried in a random medium

Reflection of waves from a mirror covered by a random layer of isotropic, absorbing scatterers is studied and the angular distribution of the scattered intensity is calculated both analytically and numerically. It is shown that backscattering enhancement as well as an enhancement of the incoherent signal in the specular direction take place even in the singly scattered random field. The dependence of the retroreflected intensity is shown to be a non-monotonic function of the depth of the mirror, with a maximum at a depth of the order of the scattering mean free path. Possibilities for employing the results obtained to detect buried strong scatterers and to retrieve parameters of the random media are discussed. In particular, it is shown that in the case of strong absorption the reflecting plane manifests itself by the presence of a peak in the retroreflected intensity which is missing from the scattering diagram of a free-standing or an infinitely thick random layer.     

Ray theory for a locally layered random medium

We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is modelled by a smooth three-dimensional background that is modulated by stratified random fluctuations. We refer to such media as locally layered .

We show that, when the pulse is observed relative to its random arrival time, it stabilizes to a shape determined by the slowly varying background convolved with a Gaussian. The width of the Gaussian and the random travel time are determined by the medium parameters along the ray connecting the source and the point of observation. The ray is determined by high-frequency asymptotics (geometrical optics). If we observe the pulse in a deterministic frame moving with the effective slowness , it does not stabilize and its mean is broader because of the random component of the travel time. The analysis of this phenomenon involves the asymptotic solution of partial differential equations with randomly varying coefficients and is based on a new representation of the field in terms of generalized plane waves that travel in opposite directions relative to the layering.     

The mutual coherence function and the backscatter amplification effect for a reflected Gaussian-beam wave in atmospheric turbulence

The influence of a modified spectrum of refractive-index fluctuations (that includes a high wavenumber rise as well as inner- and outer-scale parameters) on the backscatter amplification effect, arising from double passage of an optical wave through statistically dependent inhomogeneities of a random medium, is studied for the case of a Gaussian-beam wave reflected by a mirror of finite size. A formal expression is first developed for the mutual coherence function, which subsequently leads to tractable analytic models for the mean irradiance in the strictly backward direction. When the inner scale and Fresnel zone are of comparable size, the modified spectrum predicts significantly larger values of the enhancement factor than predicted by the Kolmogorov power-law spectrum. It is also shown in this analysis that by varying the focal length of the mirror the enhancement effects can be greater or less than those of a plane mirror, depending on focus adjustment. All calculations are based on weak irradiance fluctuations using complex ABCD ray-matrix representations for the propagation channel and a generalized spectral representation theory for the complex phase perturbations.     

Singular perturbation method for short waves in random media

Abstract

The problem of wave propagation in a randomly inhomogeneous medium is considered on the basis of the parabolic equation approximation. The method of asymptotic expansions construction in powers of the radius of correlation of the random media for the moments of the wave field are proposed.     

Stochastic differential equation analysis for sound scattering by random internal waves in the ocean

The scattering of a weakly divergent narrow sound beam by random inhomogeneities of a fluctuating ocean is considered in the coupled-mode approximation. The random index of sound refraction is described using the Garrett-Munk internal wave spectrum. The problem is solved using the stochastic differential equations for the first-and second-order statistical moments of the acoustic field. The equations are formulated according to the cumulant expansion method. The existence of weakly divergent narrow sound beams in long-range sound propagation was one of the last discoveries of L.M. Brekhovskikh, to which he attached much importance. The concentration of sound into narrow beams away from the axis of the underwater sound channel was first observed experimentally and then explained by Brekhovskikh and his former students Goncharov, Kurtepov, and Petukhov. In the present paper, the scattered field intensity of a sound beam is calculated for different frequencies and source depths. Analytical expressions are obtained for the coefficients of the differential equation. The intermode energy transfer that accompanies the long-range propagation of a weakly divergent sound beam is analyzed. A comparison with the conventionally used Monte Carlo simulation in the parabolic equation approximation is performed.     

Joint moments of the wave beam amplitude and inverse power in an absorbing random medium with lens properties

Abstract

This paper presents a derivation of a system of closed equations for joint moments of the amplitude and inverse power of a wave beam propagating in a regularly inhomogeneous dissipative random medium. The radiation transfer in the medium is characterized by non-conservation of the total radiation energy flux and by the existence of power fluctuations. The statistics of the wave beam power fluctuations have been studied. Information on the power statistical characteristics is applied to close the system of equations for joint moments. For task parameters which are not very strict (an effective radius of the wave beam should be considerably less than the outer scale of the turbulence) a system of independent equations for arbitrary joint moments has been obtained. The equations for the first two lower joint moments of the beam intensity and inverse power have been solved analytically. With the solutions obtained the effective wave beam parameters were calculated, i.e. the beam mean displacement, effective broadening and tremble variance (the beam wandering variance) for the propagation of radiation in the refractive channel of an absorbing turbulent medium. Radically new characteristics of the behaviour of the effective parameters in random absorbing and transparent media have been revealed.     

Statistical properties of nonlinear diffracting <Emphasis Type="Italic">N</Emphasis>-wave behind a random phase screen

Propagation of high amplitude N-wave behind a random phase screen is modeled based on the Khokhlov-Zabolotskaya-Kuznetsov equation. One-dimensional random phase screens with Gaussian power spectrum density are considered. The effects of nonlinear propagation, random focusing, and diffraction on the statistical properties of the acoustic field behind the screen, including propagation through caustics and beyond caustics, are analyzed. Statistical distributions and mean values of the acoustic field parameters obtained within the developed diffraction model and using nonlinear geometrical acoustics approach are compared.     

Scintillation index of a stochastic electromagnetic beam propagating in random media

We study the behavior of the scintillation index (the normalized variance of fluctuating intensity) of a wide-sense statistically stationary, quasi-monochromatic, electromagnetic beam propagating in a homogeneous isotropic medium. In particular, we show that in the case when the beam is treated electromagnetically apart from the correlation properties of the medium in which the beam travels not only its degree of coherence but also its degree of polarization in the source plane can affect the values of the scintillation index along the propagation path. We find that, generally, beams generated by unpolarized sources have reduced level of scintillation, compared with beams generated by fully polarized sources, provided they have the same intensity distribution and the same state of coherence in the source plane. An example illustrating the theory is considered which examines how the scintillation index of an electromagnetic Gaussian Schell-model beam propagates in the turbulent atmosphere. These results may find applications in optical communications through random media and in remote sensing.     

Statistics evolution of Gaussian wavefields in random media

Abstract

A phenomenon of the competition between the normalization and denormalization of the initially normal (Gaussian) wavefield statistics when waves propagate in random media is theoretically investigated. A model for the quasi-Gaussian field, which can be used to describe the multipoint field statistics under conditions of stochastic multipathing, is proposed. The behaviour of the field statistics is demonstrated by examples of weak and strong fluctuating screens. Using this model, uniform (over the propagation distance) representation for the 2nth-order statistical moments of the quasi-Gaussian field in extended random media is obtained, and the relation between the normalization and denormalization processes is considered.     

Backscattering enhancement analysis for targets in continuous random media based on wave polarization

Owing to the double passage effect, the phenomenon of backscattering enhancement arises in which the radar cross-section (RCS) in a random medium is twice that in free space. In a previous study, it was proved that the enhancement in radar cross-section (ERCS) deviates from two and has large and anomalous fluctuations, sometimes as a result of the wave polarization and other parameters, especially for targets in strong random media. Linear, including horizontal and vertical, polarizations were considered. In this paper, a numerical analysis is presented to show that the fluctuations can be reduced and make ERCS dependent almost only on the double passage effect under certain conditions. Therefore, we will have a better detection technique of targets of large sizes in continuous random media. In doing that, the linear and circular polarizations of incident waves are considered. We assume the case where a directly incident wave is produced by a line source in the far field distributed uniformly along the axis parallel to the conducting cylinder (target) axis.     

Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects

Abstract

This paper presents a theory of the radar cross section (RCS) of objects in multiple scattering random media. The general formulation includes the fourthorder moments including the correlation between the forward and the backward waves. The fourth moments are reduced to the second–order moments by using the circular complex Gaussian assumption. The stochastic Green's functions are expressed in parabolic approximation, and the objects are assumed to be large in terms of wavelength; therefore, Kirchhoff approximations are applicable. This theory includes the backscattering enhancement and the shower curtain effects, which are not normally considered in conventional theory. Numerical examples of a conducting object in a random medium characterized by the Gaussian and Henyey–Greenstein phase functions are shown to highlight the difference between the multiple scattering RCS and the conventional RCS in terms of optical depth, medium location and angular dependence. It shows the enhanced backscattering due to multiple scattering and the increased RCS if a random medium is closer to the transmitter.     

The efficiency of the reversed wavefront field with a phase shift focusing in random media

The effectiveness of reversed wave focusing in a random atmosphere was studied by using the phase approximation of geometrical optics as well as in an oceanic waveguide using the method of normal modes and horizontal rays. Calculations were made for the turbulence fluctuation obeying a structural function with a 2/3 power law. To provide focusing of the field radiated by a point-like source into a given point, a correction phase factor, determined for a homogeneous medium, was introduced. We estimated that the size of a region near the location of the source wherein the intensity of a field, focused in accordance with the free-space algorithms, decreased due to random inhomogeneities to no less than half of its original size. The minimal number of reference sources necessary for effective focusing at any point in the region under consideration was estimated.     

Optimal beams for propagation through random media

The problem of maximizing the intensity that is transferred from a transmitter aperture to a receiver aperture is considered in which the propagation medium is random. Two optimization criteria are considered: maximal expected intensity transfer and minimal scintillation index. The beam that maximizes the expected intensity is shown to be fully coherent. Its coherent mode is determined as the principal eigenfunction for a kernel that is determined through the second-order moments of the propagation Green's function. The beam that minimizes the scintillation index is shown to be partially coherent in general, with its coherent modes determined by minimizing a quadratic form that has nonlinear dependence on the coherent-mode fields, and on the second- and fourth-order moments of the propagation Green's function.