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Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects
Ishimaru A., Jaruwatanadilok S., Kuga Y.
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.
A critical survey of approximate scattering wave theories from random rough surfaces
Elfouhaily T.M., Guérin C.
This review is intended to provide a critical and up-to-date survey of the analytical approximate methods that are encountered in scattering from random rough surfaces. The underlying principles of the different methods are evidenced and the functional form of the corresponding scattering amplitude or cross-section is given. The reader is referred to the original papers in order to obtain the explicit expressions of the coefficients and kernels. We have tried to identify the main strengths and weaknesses of the various theories. We provide synthetic tables of their respective performances, according to a dozen important requirements a valuable method should meet. Both scalar acoustic and vector electromagnetic theories are equally addressed.
Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere
Salem M., Korotkova O., Dogariu A., Wolf E.
In this paper, we study the effects of turbulent atmosphere on the degree of polarization of a partially coherent electromagnetic beam, which propagates through it. The beam is described by a 2 × 2 cross-spectral density matrix and is assumed to be generated by a planar, secondary, electromagnetic Gaussian Schell-model source. The analysis is based on a recently formulated unified theory of coherence and polarization and on the extended Huygens–Fresnel principle. We study the behaviour of the degree of polarization in the intermediate zone, i.e. in the region of space where coherence properties of the beam and the atmospheric turbulence are competing. We illustrate the analysis by numerical examples.
High-frequency asymptotic expansions of a backscattering cross-section and an HH/VV polarization ratio for smooth two-dimensional surfaces
Fuks I.M.
High-frequency asymptotic expansions of electrical and magnetic fields are obtained at a perfectly conducting smooth 2D surface illuminated by a plane incident wave in two cases of TE and TM polarization. Corrections to the geometrical optics backscattering HH and VV cross-sections are derived and the polarization ratio HH/VV is estimated for the specular points of a general form. It is shown that the coefficient of the first term in this series ∼ 1/k2 depends not only on the local surface curvature radius at the specular reflecting points,but also on higher derivatives up to the sixth order. For backscattering from a statistically rough surface, averaging over surface random derivatives at the specular points was performed. It was shown that for statistically uniform (stationary on space variables) random surfaces, the polarization ratio HH/VV (dB) is positive on average and proportional to ∼ 1/k2
The calculated performance of forest structure and biomass estimates from interferometric radar
Treuhaft R.N., Siqueira P.R.
Vertical structure and biomass are key characteristics of the forest random medium. This paper calculates the power and interferometric synthetic aperture radar (InSAR) sensitivity to tree height and vegetation density as it manifests in extinction, using a homogeneous, random-volume model of the forest medium, and accounting for speckle and thermal noise. Tree height and extinction are both related to biomass within the context of this simple model. Signal and noise calculations show that InSAR coherence and phase are more sensitive than radar power to structure and biomass to 10% variations in structure parameters over a wide range of medium to high density forests. For example, for extinctions of 0.2 db m−1 and other parameters as noted in the text, the sensitivity of InSAR coherence to 10% changes in tree height exceeds observation errors for trees shorter than about 37 m, as opposed to 14 m for radar power. InSAR phase sensitivity to 10% structural and associated biomass changes exceeds obse...
Correlation length facilitates Voigt wave propagation
Mackay T., Lakhtakia A.
Under certain circumstances, Voigt waves can propagate in a biaxial composite medium even though the component material phases individually do not support Voigt wave propagation. This phenomenon is considered within the context of the strong--permittivity--fluctuation theory. A generalized implementation of the theory is developed in order to explore the propagation of Voigt waves in any direction. It is shown that the correlation length--a parameter characterizing the distributional statistics of the component material phases--plays a crucial role in facilitating the propagation of Voigt waves in the homogenized composite medium.
The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory
Young1 C., Masino A., Thomas F., Subich C.
The Rytov perturbation method can be used to derive analytic expressions governing statistical quantities of an optical wave propagating through the Earth's atmosphere. It is generally accepted that the validity of these expressions is restricted to the weak fluctuation regime, and that the wave structure function for plane and spherical waves obtained via the Rytov method is valid in all fluctuation regimes, for sufficiently small separation distances. Data from experimental results for the wave structure function as a function of the fluctuation strength for a fixed value of the separation distance indicate that the Rytov method does not accurately model the behaviour of the wave structure function in moderate to strong fluctuation regimes. This is similar to what is observed for the scintillation index. Recently, however, it was shown that the integral definition of the scintillation index obtained via the Rytov perturbation yields analytic expressions that are valid in all fluctuation regimes when a f...
Quantum graphs: I. Some basic structures
Kuchment P.
A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.
Modified spherical harmonics method for solving the radiative transport equation
Markel V.
A new effective approach to solving the three-dimensional radiative transport equation with an arbitrary phase function is proposed. The solution depends on eigenvectors and eigenvalues of several symmetrical tridiagonal matrices of infinite size. The matrices must be truncated and diagonalized numerically. Then, given eigenvectors and eigenvalues of these matrices, the dependence of the solution on position and direction is found analytically. The approach is based on expanding the angular part of the specific intensity in q-dependent spherical functions for each spatial Fourier component characterized by the vector q. Apart from the truncation of the matrices, no other approximations are made.
Nonlinear photonic crystals: III. Cubic nonlinearity
Babin A., Figotin A.
Weakly nonlinear interactions between wavepackets in a lossless periodic dielectric medium are studied based on the classical Maxwell equations with a cubic nonlinearity. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the cubic interaction phase function based on the dispersion relations of the underlying linear periodic medium. It turns out that only a few quadruplets of modes have significant nonlinear interactions. They are singled out by a system of selection rules including the group velocity, frequency and phase matching conditions. It turns out that the intrinsic symmetries of the cubic interaction phase stemming from assumed inversion symm...
Microwave sea return at moderate to high incidence angles
Plant W.J.
Bragg scattering is widely recognized as the dominant mechanism by which the ocean surface backscatters microwave radiation, but efforts to identify other, non-Bragg sources of this scattering have been pursued for many years. Non-Bragg backscattering from the sea surface is known to occur at incidence angles close to 0° and 90°. In this paper Bragg scattering is shown to explain most features of sea surface backscatter for incidence angles between about 20° and 80°, except when it predicts very small mean cross sections. The often-quoted evidence for non-Bragg scattering in this incidence angle range is that σ o (HH) is occasionally found to be larger than or equal to σ o (VV) for short integration times. We show that because of fading this is not evidence of non-Bragg scattering. For incidence angles up to about 50°, standard Bragg/composite surface scattering theory yields probabilities of finding σ o (HH)>σ o (VV) that are only slightly smaller than those found experimentally. As the incidenc...
Travel-time statistics for signals scattered at a rough surface
Godin O.A., Fuks I.M.
The travel time of signals reflected or refracted by a rough surface is investigated in the geometrical optics approximation. It is shown that surface roughness typically decreases the mean travel time in the case of large-scale roughness, when only one specularly reflecting point moves randomly around its unperturbed position, resulting in a negative travel-time bias (toward early echoes). In the opposite limiting case of multipath propagation, when many specular points exist on a random surface, the travel-time bias is always positive. General results are illustrated by two examples related to ocean remote sensing which involve sound scattering from the ocean surface and bottom.
Lasing in random media
Cao H.
Abstract A random laser is a non-conventional laser whose feedback mechanism is based on disorder-induced light scattering. Depending on whether the feedback supplied by scattering is intensity feedback or amplitude feedback, random lasers are classified into two categories: random lasers with incoherent feedback and random lasers with coherent feedback. A brief survey of random lasers with incoherent feedback is presented. It is followed by a review of our recent experimental work on random lasers with coherent feedback, including measurement of the lasing threshold, lasing spectra, emission pattern, dynamical response, photon statistics, speckle pattern and the investigation of relevant length scales. Large disorder leads to spatial confinement of the lasing modes, that is the foundation for the micro random laser. Some theoretical models of random lasers with coherent feedback are briefly introduced. The study of random lasers improves our understanding of the interplay between light localization and coherent amplification.
A practical second-order electromagnetic model in the quasi-specular regime based on the curvature of a ‘good-conducting’ scattering surface
Elfouhaily T., Guignard S., Thompson D.R.
This letter presents an approximate second-order electromagnetic model where polarization coefficients are surface dependent up to the curvature order in the quasi-specular regime. The scattering surface is considered ‘good-conducting’ as opposed to the case for our previous derivation where perfect conductivity was assumed. The model reproduces dynamically, depending on the properties of the scattering surface, the tangent-plane (Kirchhoff) or the first-order small-perturbation (Bragg) limits. The convergence is assumed to be ensured by the surface curvature alone. This second-order model is shown to be consistent with the small-slope approximation of Voronovich (SSA-1+SSA-2) for perfectly conducting surfaces. Our model differs from SSA-1 + SSA-2 in its dielectric expression, to correct for a full convergence toward the tangent-plane limit under the ‘good-conducting’ approximation. This new second-order formulation is simple because it involves a single integral over the scattering surface and t...
Analytical comparison between the surface current integral equation and the second-order small-slope approximation
Elfouhaily T., Joelson M., Guignard S., Thompson D.R.
This paper is the third in a series discussing a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. Our previous approach supplemented the Kirchhoff model through the addition of new terms involving linear orders in slope and surface elevation differences that arise naturally from a second iteration of the surface current integral equation. This completion of the Kirchhoff was shown to provide the correct first-order small perturbation method (SPM-1) in the general bistatic context. The agreement with SPM-1 was achieved because differences of surface heights are no longer expanded in powers of surface slope. While consistent with SPM, our previous formulation fails to reconverge toward the Kirchhoff model, at some incidence and scattered angles, when the illuminated surface satisfies the high frequency roughness condition. This weakness is also shared with the first-order small slope approximation (SSA-1) which is structurally equivalent to our...
Seismic scattering from a spherical inclusion eccentrically located within a homogeneous, spherical host: theoretical derivation
Videen G.
The scattered fields from a spherical body eccentrically located within an otherwise homogeneous host sphere are derived by satisfying the boundary conditions at both interfaces simultaneously. The source, which may be composed of any linear combination of S and P waves, is also located arbitrarily within the host sphere. The scattering system may have applications in seismic scattering, by cavities or dense bodies located near the Earth surface or to the Slichter mode.
A study of the higher-order small-slope approximation for scattering from a Gaussian rough surface
Gilbert M.S., Johnson J.T.
Results from the first three terms of the small-slope approximation (SSA) for incoherent electromagnetic scattering from a penetrable randomly rough interface are discussed. Surface roughness is characterized as a Gaussian random process with an isotropic Gaussian correlation function. Sample results illustrate parameter spaces for which each correction term is appreciable. Reduction of the SSA to the physical optics theory is also discussed for both perfectly conducting and dielectric surfaces.
Nonlinear photonic crystals: II. Interaction classification for quadratic nonlinearities
Babin A., Figotin A.
Weakly nonlinear interactions between wavepackets in lossless periodic dielectric media are studied based on the classical nonlinear Maxwell equations. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the Bloch modes and the dispersion relations of the underlying linear periodic medium. It turns out that only a few triads of modes have significant nonlinear interactions. They are singled out by the frequency and phase matching conditions and, as we show, by an additional selection rule: the group velocity matching condition. The latter condition is the most important selection rule for the nonlinear regimes. We give a complete quantitative classific...
Graph models for waves in thin structures
Kuchment P.
A brief survey on graph models for wave propagation in thin structures is presented. Such models arise in many areas of mathematics, physics, chemistry and engineering (dynamical systems, nanotechnology, mesoscopic systems, photonic crystals etc). Considerations are limited to spectral problems, although references to works with other studies are provided.
Focusing of time-reversed reflections
Sølna K.
Recently time-reversal techniques have emerged as a new, important and fascinating discipline within wave propagation. Many of the problems involved can best be understood, analysed and optimized based on a random field model for the medium. Here we discuss stable refocusing of second-order time-reversed reflections. This phenomenon may appear as surprising at first. However, we show how it can be understood in very simple terms viewing the wavefield as a stochastic process. We give sufficient conditions on the Green function of the propagation problem for the phenomenon to happen. In particular we discuss acoustic wave propagation in the regime of weak random medium fluctuations and give explicitly the derivation of stable refocusing in this case, illustrating it with numerical examples. (Some figures in this article are in colour only in the electronic version)
Photon density wave for imaging through random media
Jaruwatanadilok S., Ishimaru A., Kuga Y.
The passage of a photon density wave through random media has been investigated extensively for medical imaging based on the diffusion approximation. In this paper, the photon density wave is studied based on the exact time-dependent vector radiative transfer theory. Both continuous and pulse photon density waves are analysed in a plane parallel medium using Mie scattering and the discrete ordinates method. The photon density wave shows superior properties over regular waves in several aspects. It has a narrower angular spectrum and maintains the original pulse shape. It also preserves the degree of polarization and increases the cross-polarization discrimination. These properties of a photon density wave suggest its potential for improving imaging. Thus, we apply the photon density wave to an imaging problem and show that it improves the quality of the images compared to other conventional imaging techniques.
New results in applied scattering theory: the physical-statistics approach, including strong multiple scatter versus classical statistical-physical methods* and the Born and Rytov approximations versus exact strong scatter probability distributions
Middleton D.
In order to carry out effective signal processing for signal detection and estimation in scatter-dominated environments, it is necessary to obtain the needed probability distributions and probability densities (PDFs) of the received scatter. In general, the received scatter is non-Gaussian and often strongly so. It is also dominated by multiple-scatter contributions. This is particularly the case for (radar) scatter off random interfaces at small angles, e.g.ocean wave surfaces and terrain, as well as acoustically (sonar) off the ocean surface and bottom. For ‘classical’ theory, based on statistical-physical (S-P) models and methods, the analytical construction of the required PDFs is beyond the general reach of S-P theory. The present paper reviews, amplifies and extends with new material the author's recently developed physical-statistical (P-S) alternative equivalent to classical (S-P) formulations. Here the fundamental innovation, starting with the basic Langevin equation of propagation in op...
Design of one-dimensional Lambertian diffusers of light
Maradudin A.A., Simonsen I., Leskova T.A., Méndez E.R.
Theoretical and computational aspects of scattering from periodic surfaces: two-dimensional transmission surfaces using the spectral-coordinate method
Desanto J., Erdmann G., Hereman W., Krause B., Misra M., Swim E.
We consider the scattering from and transmission through a two-dimensional periodic surface. We use the spectral-coordinate (SC) method for all the computations. It was the fastest method for one-dimensional problems and proved optimal for scattering from two-dimensional surfaces where computation time can prove to be excessive. In particular, we can avoid approximation methods and solve the exact equations. The SC equations are derived for an infinite surface and reduced to coupled equations for a periodic surface which are solved numerically for the two boundary unknowns. Solutions of the SC equations for various periodic sinusoidal surface examples are studied. The surfaces vary in roughness and period. Extensive computations are included in terms of the maximum roughness slope which can be computed using the method with a fixed maximum error as a function of azimuthal angle of incidence, polar angle of incidence, wavelength-to-period ratio, density ratio and wavenumber ratio. Examples of refl...
Theoretical and computational aspects of scattering from periodic surfaces: one-dimensional transmission interface
Desanto J., Erdmann G., Hereman W., Misra M.
