Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces

Abstract

Analytical expressions for the two-frequency mutual coherence function and angular correlation function of the scattered wave from rough surfaces based on the Kirchhoff approximation are presented. The coherence bandwidth depends on the illumination area as well as on the incident and scattered angles and the surface characteristics. Scattered pulse shapes are calculated as the Fourier transform of the two-frequency mutual coherence function. Calculations based on analytical solutions are compared with millimetre wave experimental data and Monte Carlo simulations showing good agreement.     

Pulse broadening of enhanced backscattering from rough surfaces

Recently, we presented a study of pulse scattering by rough surfaces based on the first-order Kirchhoff approximation which is applicable to rough surfaces with RMS slope less than 0.5 and correlation distance l≳λ. However, there has been an increased interest in enhanced backscattering from rough surfaces, study of which requires inclusion of the second-order Kirchhoff approximation with shadowing corrections. This paper presents a theory for the two-frequency mutual coherence function in this region and shows that the multiple scattering on the surface gives rise to an additional pulse tail in the direction of enhanced backscattering. The theory predicts pulse broadening approximately 20% greater than that caused by single scattering alone for a delta-function incident pulse and typical surface parameters. Analytical results are compared with Monte Carlo simulations and millimetre-wave experiments for the one-dimensional rough surface with RMS height 1λ and correlation distance 1λ, showing good agreement.     

Pulse broadening of enhanced backscattering from rough surfaces

Abstract

Recently, we presented a study of pulse scattering by rough surfaces based on the first-order Kirchhoff approximation which is applicable to rough surfaces with RMS slope less than 0.5 and correlation distance l?λ. However, there has been an increased interest in enhanced backscattering from rough surfaces, study of which requires inclusion of the second-order Kirchhoff approximation with shadowing corrections. This paper presents a theory for the two-frequency mutual coherence function in this region and shows that the multiple scattering on the surface gives rise to an additional pulse tail in the direction of enhanced backscattering. The theory predicts pulse broadening approximately 20% greater than that caused by single scattering alone for a delta-function incident pulse and typical surface parameters. Analytical results are compared with Monte Carlo simulations and millimetre-wave experiments for the one-dimensional rough surface with RMS height 1λ and correlation distance 1λ, showing good agreement.     

Two-frequency mutual coherence function for electromagnetic pulse propagation over rough surfaces

The propagation of a transient electromagnetic pulse over irregular terrain is considered. We model the wave propagation using the parabolic wave equation, which is valid for near-horizontal propagation. We model the effect of scattering from the rough terrain by introducing a surface-flattening coordinate transform. This coordinate transform simplifies the boundary condition of our problem, and introduces an effective refractive index into our wave equation. As a result, the problem of propagation over an irregular surface becomes equivalent to the problem of propagation through random media. The parabolic equation is solved analytically using the path integral method. Both vertically polarized and horizontally polarized signals are treated. Cumulant expansion is introduced to obtain an approximate expression for the two-frequency mutual coherence function. From the mutual coherence function, spatial and temporal dependence of the propagating signal can be determined. It can be shown that scattering from the irregular surface can cause broadening of the transient signal. This can have a significant impact on the performance of radio communication systems.     

Two-frequency mutual coherence function for electromagnetic pulse propagation over rough surfaces

The propagation of a transient electromagnetic pulse over irregular terrain is considered. We model the wave propagation using the parabolic wave equation, which is valid for near-horizontal propagation. We model the effect of scattering from the rough terrain by introducing a surface-flattening coordinate transform. This coordinate transform simplifies the boundary condition of our problem, and introduces an effective refractive index into our wave equation. As a result, the problem of propagation over an irregular surface becomes equivalent to the problem of propagation through random media. The parabolic equation is solved analytically using the path integral method. Both vertically polarized and horizontally polarized signals are treated. Cumulant expansion is introduced to obtain an approximate expression for the two-frequency mutual coherence function. From the mutual coherence function, spatial and temporal dependence of the propagating signal can be determined. It can be shown that scattering from the irregular surface can cause broadening of the transient signal. This can have a significant impact on the performance of radio communication systems.     

Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: I

In this paper, the modal expansion theory is presented as a new analytical approach together with the resulting new physical parameters. In particular, the features of an arbitrary power-law structure function are investigated. The exact expression for the Gaussian spectrum is rederived. An approximate analytical expression for the two-frequency coherence function evaluated at equal positions for the Kolmogorov spectrum is presented and comparison with the numerical solution in the literature exhibits a remarkable agreement. As a result of the modal decomposition, general properties for a transversally homogeneous and isotropic medium are demonstrated, such as the exponential decay of the amplitude of the solution and the linear phase behaviour at large propagation distances.     

Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II

Abstract

In a previous publication (part I) it has been shown that for an arbitrary statistically isotropic and homogeneous medium the parabolic equation for the two-frequency mutual coherence function can be separated and thereby expressed as a superposition of modes. A parameterization based on the longitudinal part of this representation has also been treated. This paper explores the transverse structure and parameterization of the field solution by employing dimensional, variational and the modified WKB procedures for solving the eigenfunction/eigenvalue problem. General expressions are derived first for a general structure function and then specialized for a power-law structure function with emphasis on quadratic and Kolmogorov media.     

Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II

In a previous publication (part I) it has been shown that for an arbitrary statistically isotropic and homogeneous medium the parabolic equation for the two-frequency mutual coherence function can be separated and thereby expressed as a superposition of modes. A parameterization based on the longitudinal part of this representation has also been treated. This paper explores the transverse structure and parameterization of the field solution by employing dimensional, variational and the modified WKB procedures for solving the eigenfunction/eigenvalue problem. General expressions are derived first for a general structure function and then specialized for a power-law structure function with emphasis on quadratic and Kolmogorov media.     

Modal theory for the two-frequency mutual coherence function in random media: beam waves

Pulse propagation in a random medium is mainly determined by the two-frequency mutual coherence function which satisfies the parabolic equation. It has recently been shown that this equation can be solved by separation of variables, thereby reducing the solution for any structure function to the solution of ordinary differential equations. In this paper, the method is applied for a beam-wave excitation in a random medium. The exact solution for a quadratic medium is derived. For non-quadratic power-law media an analytical expression at equal positions is presented.     

Modal theory for the two-frequency mutual coherence function in random media: point source

The recently introduced modal expansion representation for the two-frequency mutual coherence function is applied here to the solution of a point-source field in a random medium. This approach reduces the solution for any structure function to an eigenvalue problem for an ordinary differential equation. For the initial point source it is shown here that the modal expansion yields a result similar to that for the initial plane wave, modified by a spherical free-space phase which contains a weighted coordinate that does not interact with the medium. Having established these general characteristics, special attention is paid to power-law media and, in particular, to a quadratic medium, for which a new exact solution is derived. Via a collective summation of this new modal solution, we rederive the alternative exact solution which exists in the literature. We also discuss the new parameterization implied by the new modal solution.     

The depolarization of light scattered from rough metal surfaces

To describe the depolarization of the electromagnetic field scattered from rough metal surfaces, a finite, non-vanishing value of the conductivity has to be considered. To reveal the modifications imposed by a complex dielectric constant, the Beckmann formula is expanded in power series of slope terms. We then arrive at an analytic expression which is convenient for evaluation as a function of the parameters involved. With this expression, plots of the cross-polarization ratio vs. angle of incidence and vs. slope are presented.     

弹性波在固体自由粗糙平界面上的散射

利用散射幅度矩阵的概念来处理弹性波在粗糙界面弹性介质中的散射问题。利用微扰近似解边界方程,对散射幅度进行求解,得到了散射幅度的0阶、1阶和2阶解。同时分析了粗糙起伏高度符合高斯分布时,散射幅度的数学期望值和方差,它们分别代表平均场和声场在偏离镜面方向的起伏。最后进行了实验。     

Speckle correlations in the light scattered from weakly rough random metal surfaces

Abstract

Diagrammatic perturbation theory and computer simulation methods are used to compute the angular intensity correlation function C(q, k|q′,k′)=([I(q|k) - (I(q|k))] × [I(q′|k′) - (I(q′|k′))]) for p-polarized light scattered from a weakly rough, one-dimensional random metal surface. I(q|k) is the squared modulus of the scattering matrix for the system, and q, q′ and k, k′ are the projections on the mean scattering surface of the wavevectors of the scattered and incident light, respectively. Contributions to C include: (a) short-range memory effect and time-reversed memory effect terms, C ⁽¹⁾; (b) an additional short-range term of comparable magnitude C ⁽¹⁰⁾; (c) a long-range term C ⁽²⁾; (d) an infinite-range term C ⁽³⁾; and (e) a term C ^(1.5) that along with C ⁽²⁾ displays peaks associated with the excitation of surface plasmon polaritons. The diagrammatic methods are also extended to treat the angular intensity correlation function for the scattering of p to p, p to s, s to p, and s to s polarizations of light from a two-dimensional randomly rough surface. These correlations are again described in terms of C ⁽¹⁾, C ⁽¹⁰⁾, C ^(1.5), C ⁽²⁾, and C ⁽³⁾ contributions to C for the two-dimensional surfaces. Short-range memory and time-reversed memory effects are observed in the two-dimensional C ⁽¹⁾ correlations, and peaks associated with the excitation of surface polaritons are observed in the two-dimensional C ^(1.5) and C ⁽²⁾ correlations. Most of the results for the one- and two-dimensional systems are presented for incident electromagnetic plane waves. In addition, results for one-dimensional systems are presented for incident electromagnetic beams of finite width. Some of the results for one-dimensional surfaces are corroborated by means of computer simulation techniques.     

Two-frequency mutual coherence function and its applications to pulse scattering by random rough surface

An analytic expression of the two-frequency mutual coherence function (MCF) was derived for a two-dimensional random rough surface. The scattered field was calculated by the Kirchhoff approximation, which is valid in the case that the radius of curvature of the surface is much larger than the incident wave length. The scattering problem of narrowband pulse was investigated to simplify the analytic expression of the two-frequency MCF. Numerical simulations show that the two-frequency MCF is greatly dependent on the root-mean-square (RMS) height, while less dependent on the correlation length. The analytic solutions were compared with the results of Monte Carlo simulation to assess the accuracy and computational efficiency. Supported by the National Natural Science Foundation of China (Grant No. 60571058) and the National Defense Foundation of China (Grant No. 51403020505DZ0111)     

Modulation effects of a sound wave on the mutual coherence function of light

A theoretical investigation is described on coherence effects in a light-sound interaction. The mutual coherence function can be modulated in time and space by electronic means using a simple travelling sound wave. The modulation factor is expressed in terms of the zeroth-order Bessel function whose argument is a function of the two-point separation of interest and the time delay between the two light waves to be correlated. The spectral characteristics obtained from the space-time Fourier transform of the mutual coherence function coincides with an earlier statement in the Raman-Nath theory.     

Brewster's scattering angle in scattered waves from slightly rough metal surfaces

Brewster's scattering angle in electromagnetic wave scattering from slightly random metal surfaces is investigated by means of the stochastic functional approach. While there are dips due to Brewster's scattering angle in scattering profiles from dielectric surfaces, Brewster's scattering angle does not exist in scattering from metal surfaces. However, the dips can exist in scattering from rough metal surfaces with the optically denser medium to convert evanescent wave into radiative wave.     

Non-local small-slope approximation for wave scattering from rough surfaces

A new general analytical approach to solving the problems of wave scattering from rough surfaces, referred to as the non-local small-slope approximation (NLSSA), is suggested. It is formulated in the general form both for vector and scalar waves. This approach is valid for an arbitrary wavelength of radiation provided that the slopes of the undulations are small enough. The NLSSA represents a generalization of the small-slope approximation to situations where double scattering (in the optical sense) appears. It is demonstrated that with appropriate approximations the NLSSA of the lowest order reduces to the small-slope approximation of the second order.     

Propagation of the two-frequency coherence function in an inhomogeneous background random medium

The spatial and temporal structures of time-dependent signals can be appreciably affected by random changes of the parameters of the medium characteristic of almost all geophysical environments. The dispersive properties of random media cause distortions in the propagating signal, particularly in pulse broadening and time delay. When there is also spatial variation of the background refractive index, the observer can be accessed by a number of background rays. In order to compute the pulse characteristics along each separate ray, there is a need to know the behaviour of the two-frequency mutual coherence function. In this work, we formulate the equation of the two-frequency mutual coherence function along a curved background ray trajectory. To solve this equation, a recently developed reference-wave method is applied. This method is based on embedding the problem into a higher dimensional space and is accompanied by the introduction of additional coordinates. Choosing a proper transform of the extended coordinate system allows us to emphasize 'fast' and 'slow' varying coordinates which are consequently normalized to the scales specific to a given type of problem. Such scaling usually reveals the important expansion parameters defined as ratios of the characteristic scales and allows us to present the proper ordering of terms in the desired equation. The performance of the main order solution is demonstrated for the homogeneous background case when the transverse structure function of the medium can be approximated by a quadratic term.