Saddek Afifi

On the Co-Polarized Phase Difference of Rough Layered Surfaces: Formulae Derived From the Small Perturbation Method

We determine the statistical distribution of the co-polarized phase difference of fields scattered from a stack of two two-dimensional rough interfaces in the incidence plane. The electromagnetic fields are represented by Rayleigh expansions and a perturbation method is used to solve the boundary value problem and to determine the first-order scattering amplitudes. For slightly rough interfaces with infinite length and Gaussian height distributions, we show that the probability density function is only a function of two parameters. For a sand layer on a granite surface in backscattering configurations, we study the influence of the incidence angle, the layer thickness, the cross-spectral density and the wave frequency upon both parameters of the probability law.

Scattering by Anisotropic Rough Layered 2D Interfaces

We propose a statistical study of the scattering of an incident plane wave by a stack of two two-dimensional rough interfaces. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. The electromagnetic fields are represented by Rayleigh expansions, and a perturbation method is used for solving the boundary value problem and determining the first-order scattering amplitudes. For slightly rough interfaces with a finite extension, we show that the modulus of the co- and cross-polarized scattering amplitudes follows a Hoyt law and that the phase is not uniformly distributed. For interfaces with an infinite extension, the modulus follows a Rayleigh law and the phase is uniformly distributed. We show that these results are true for correlated or uncorrelated interfaces and for isotropic or anisotropic interfaces.

Electromagnetic Scattering From 3D Layered Structures With Randomly Rough Interfaces: Analysis With the Small Perturbation Method and the Small Slope Approximation

We propose a theoretical study on the electromagnetic wave scattering from three-dimensional layered structures with an arbitrary number of rough interfaces by using the small perturbation method and the small slope approximation. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. They can be correlated or not, isotropic or not. The electromagnetic field in each medium is represented by a continuum of plane waves and a perturbation theory is used for solving the boundary value problem and determining the first-order scattering amplitudes by recurrence relations. The scattering amplitudes under the first-order small slope approximation are deduced from results derived from the first-order small perturbation method. We analyze with the small slope approximation model the combined influence of the anisotropy and cross-correlation upon the electromagnetic signature of a natural stratified structure.

Electromagnetic Wave Scattering from Rough Layered Interfaces: Analysis with the Small Perturbation Method and the Small Slope Approximation

We propose a theoretical study on the electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces by using the small perturbation method and the small slope approximation. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. They can be correlated or not. The electromagnetic fleld in each medium is represented by a Rayleigh expansion and a perturbation method is used for solving the boundary value problem and determining the flrst-order scattering amplitudes by recurrence relations. The scattering amplitude under the flrst-order small slope approximation are deduced from results derived from the flrst-order small perturbation method. Comparison between these two analytical models and a numerical method based on the combination of scattering matrices is presented. The study of electromagnetic wave scattering from rough layered interfaces has many applications in remote sensing, communication techniques, civil engineering, geophysics and optics. Several models give the average scattered fleld and the average intensity. Analytical methods are based on physical approximations and give closed-form formulae for the flrst- and second-order moments of the scattered fleld. Exact methods estimate the average scattered fleld and the average intensity from the results over many realizations of rough layered interfaces. In this paper, we propose a theoretical study on the electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces by using two analytical models: the flrst-order small perturbation method (SPM) and the flrst-order small slope approximation (SSA). Elson was one of the flrst authors to develop a vector theory of scattering from a stratifled medium. This vector theory allows the angular distribution of scattered light to be determined and can be used with correlated or uncorrelated surface roughness (1,2). The SPM has been used for the study of light scattering from multilayer optical coatings (1{5) and many authors have also implemented a perturbative theory for analyzing remote sensing problems (6{12). The small slope approximation (SSA1) has an extended domain of applicability (13{15) which includes the domain of the small-perturbation method that is only valid for surfaces with small roughness (16) and the domain of the Kirchhofi approximation that is applicable to surfaces with long correlation length (17,18). In the present paper, the structure under consideration is a stack of several rough one-dimensional interfaces. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. The electromagnetic fleld in each region is represented by a continuous spectrum of plane waves, the amplitudes of which are found by matching the boundary conditions

On the Co-Polarized Scattered Intensity Ratio of Rough Layered Surfaces: The Probability Law Derived From the Small Perturbation Method

We determine the probability law of the ratio between the co-polarized intensities scattered from a stack of two two-dimensional rough interfaces in the incidence plane. Calculations are carried out within the framework of the first-order small perturbation method. For slightly rough interfaces with infinite length and Gaussian height distributions, we show that the probability density function is only a function of two parameters and has an infinite average and an infinite variance. For a sand layer on a granite surface in backscattering configurations, we study the influence of the incidence angle and the cross-spectral density upon this probability law.

Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method

We present a statistical study of electromagnetic wave scattering by a stack of two random rough interfaces that are characterised by Gaussian distributed heights and by exponential correlation functions. These interfaces can be correlated or not. The coherent and incoherent intensities and the statistical distribution of the scattered field in modulus and phase are obtained using the Rayleigh expansion and the small perturbation method. For a structure of finite extension, we show that the modulus follows a Hoyt law and the phase is not uniform. For a structure of infinite extension, whether the interfaces are correlated or not, the modulus of the field follows a Rayleigh law while the phase is uniform.

Statistical study of radiation loss from planar optical waveguides: the curvilinear coordinate method and the small perturbation method.

This article presents an original method for the theoretical analysis of the intensity radiated by a dielectric waveguide with rough walls. The method is based on Maxwells equations under their covariant form written in nonorthogonal coordinate systems adapted to the geometry of the waveguide. The solutions are found by using a perturbation method starting from a guide with smooth walls. The statistical characteristics of the radiant intensity, the mean value, and the probability density function are analytically determined.

Analytical derivation of the stationarity and the ergodicity of a field scattered by a slightly rough random surface

In the framework of the small perturbation method, we present a new theoretical derivation of the statistical and spatial properties of a field scattered by a one-dimensional slightly rough random surface. The work concerns the intermediate field zone where the scattered field is reduced to the contribution of the progressive plane waves. The surface is assumed to be stationary, ergodic and Gaussian. First, from a statistical point of view, we demonstrate that under oblique incidence the scattered field is not stationary while it is strictly stationary under normal incidence. For an infinite surface, the scattered field modulus obeys to Hoyt law and the phase is not uniform. Second, from a spatial point of view, for a given altitude and under all incidences, we show that the scattered field is ergodic. Under oblique incidence, the phase is spatially uniform and the modulus is given by a Rayleigh law. Under normal incidence, the phase is not uniform and the modulus is given by a Hoyt law. Third, from a practical point of view, we show that the field measured by a directional antenna is ergodic and stationary if the angular transfer function of the antenna does not contain the specular direction.

On the Stationarity of Signal Scattering by Two-Dimensional Slightly Rough Random Surfaces

We present statistical properties of signal scattering by two-dimensional slightly rough random surface. The work concerns the intermediate-field zone. Calculations are carried out within the framework of the first-order small perturbation method. The surface is assumed to be ergodic and stationary and the height distribution to be Gaussian. Under an oblique incidence, we demonstrate that the first-order scattered field is not wide-sense stationary. For a given altitude, under the normal incidence, the scattered field is a strictly stationary random process. By using the stationary phase method, we show that the scattered field becomes asymptotically a strictly stationary random process when increasing the altitude of the observation point. We also define the condition that must be satisfied by an antenna transfer function so that the measured scattered field becomes stationary and ergodic.

Statistical Distributions of the Copolarized and Cross-Polarized Phase Differences of Stratified Media

We determine the statistical distributions of the copolarized and cross-polarized phase differences of fields scattered from a stack of two 2-D rough interfaces in any observation plane. This communication is carried within the framework of the first-order small perturbation method. The probability laws are established for Gaussian distributions of heights and for infinite extension interfaces. We show that, outside the incidence plane, the cross-polarized phase difference is not uniformly distributed and contains information about the stratified medium.