Theoretical study of the Kirchhoff integral from a two-dimensional randomly rough surface with shadowing effect: application to the backscattering coefficient for a perfectly-conducting surface

In this paper, the backscattering coefficient of a two-dimensional randomly rough perfectly-conducting surface is investigated using the Kirchhoff approach with a shadowing function. The rough surface height/slope correlations assumed to be Gaussian are accounted for in this analysis. The scattering coefficient is then formulated in terms of a characteristic function for the integrations over the surface heights, in terms of expected values for the integrations over the surface slopes. Numerical comparisons of Kirchhoff's approach (KA) with the stationary-phase (SP) approximation are made with respect to the choice of the one-dimensional surface height autocorrelation function and the shadowing effect. For an isotropic surface the results show that SP underestimated the incoherent backscattering coefficient compared with KA. Moreover, when the correlation between the slopes and the heights is neglected, the shadowing effect may be ignored.     

Study on the shadowing effect for optical wave scattering from randomly rough surface

Based on the Kirchhoff approximation for rough surface scattering and by calculating the shadowing function of the rough surface, the formula of the scattering cross section of the dielectric rough surface is presented with consideration of the shadowing effect for the optical wave incidence. It is obtained that in comparison with the conventional Kirchhoff solution, the shadowing effect should not be neglected for the optical wave scattering from the rough surface. The influence of the shadowing effect for different incidence angle, surface root mean square slope, and surface roughness on the scattering cross section is discussed in detail.     

Scattering of a scalar wave from a two-dimensional randomly rough Neumann surface

We present a reciprocity and unitarity preserving formulation of the scattering of a scalar plane wave from a two-dimensional, randomly rough surface on which the Neumann boundary condition is satisfied. The theory is formulated on the basis of the Rayleigh hypothesis in terms of a single-particle Green's function G(q_‖|k_‖) for the surface electromagnetic waves that exist at the surface due to its roughness, where k_‖ and q_‖ are the projections on the mean scattering plane of the wave vectors of the incident and scattered waves, respectively. The specular scattering is expressed in terms of the average of this Green's function over the ensemble of realizations of the surface profile function (G(q_‖|k_‖)). The Dyson equation satisfied by (G(q_‖|k_‖)) is presented, and the properties of the solution are discussed, with particular attention to the proper self-energy in terms of which the averaged Green's function is expressed. The diffuse scattering is expressed in terms of the ensemble average of a two-particle Green's function, which is the product of two single-particle Green's functions. The Bethe-Salpeter equation satisfied by the averaged two-particle Green's function is presented, and properties of its solution are discussed. In the small roughness limit, and with the irreducible vertex function approximated by the sum of the contribution from the maximally-crossed diagrams, which represent the coherent interference between all time-reversed scattering sequences, the solution of the Bethe-Salpeter equation predicts the presence of enhanced backscattering in the angular dependence of the intensity of the waves scattered diffusely.     

Scattering of a scalar wave from a two-dimensional randomly rough Neumann surface

Abstract

We present a reciprocity and unitarity preserving formulation of the scattering of a scalar plane wave from a two-dimensional, randomly rough surface on which the Neumann boundary condition is satisfied. The theory is formulated on the basis of the Rayleigh hypothesis in terms of a single-particle Green's function G(q_‖|k_‖) for the surface electromagnetic waves that exist at the surface due to its roughness, where k_‖ and q_‖ are the projections on the mean scattering plane of the wave vectors of the incident and scattered waves, respectively. The specular scattering is expressed in terms of the average of this Green's function over the ensemble of realizations of the surface profile function (G(q_‖|k_‖)). The Dyson equation satisfied by (G(q_‖|k_‖)) is presented, and the properties of the solution are discussed, with particular attention to the proper self-energy in terms of which the averaged Green's function is expressed. The diffuse scattering is expressed in terms of the ensemble average of a two-particle Green's function, which is the product of two single-particle Green's functions. The Bethe-Salpeter equation satisfied by the averaged two-particle Green's function is presented, and properties of its solution are discussed. In the small roughness limit, and with the irreducible vertex function approximated by the sum of the contribution from the maximally-crossed diagrams, which represent the coherent interference between all time-reversed scattering sequences, the solution of the Bethe-Salpeter equation predicts the presence of enhanced backscattering in the angular dependence of the intensity of the waves scattered diffusely.     

Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface: II. Multiple scattering

Abstract

The integral equation model (IEM) has been developed over the last decade and it has become one of the most widely used theoretical models for rough-surface scattering in microwave remote sensing. In the IEM model the shadowing function is typically either omitted or a form based on geometric optics with single reflection is used. In this paper, a shadowing function for one-dimensional rough surfaces which incorporates multiple scattering, finite surface length and both monostatic and bistatic configurations is developed. For any uncorrelated process, the resulting equation can be expressed in terms of the monostatic statistical shadowing function with single reflection, derived in the preceding companion paper. The effect of correlation between the surface slopes and heights for a Gaussian surface is studied to illuminate the range over which such correlations can be ignored. It is found that while the correlation between surface slopes and heights in the monostatic statistical shadowing function with single reflection can be ignored, when calculating the average shadowing function with double reflection the correlation between slopes and heights between points must be incorporated.     

Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface: II. Multiple scattering

The integral equation model (IEM) has been developed over the last decade and it has become one of the most widely used theoretical models for rough-surface scattering in microwave remote sensing. In the IEM model the shadowing function is typically either omitted or a form based on geometric optics with single reflection is used. In this paper, a shadowing function for one-dimensional rough surfaces which incorporates multiple scattering, finite surface length and both monostatic and bistatic configurations is developed. For any uncorrelated process, the resulting equation can be expressed in terms of the monostatic statistical shadowing function with single reflection, derived in the preceding companion paper. The effect of correlation between the surface slopes and heights for a Gaussian surface is studied to illuminate the range over which such correlations can be ignored. It is found that while the correlation between surface slopes and heights in the monostatic statistical shadowing function with single reflection can be ignored, when calculating the average shadowing function with double reflection the correlation between slopes and heights between points must be incorporated.     

A time domain rough surface scattering model based on wedge diffraction: application to low-frequency backscattering from two-dimensional sea surfaces

A time domain method for calculating the acoustic impulse response of impenetrable, rough, two-dimensional (2D) surfaces is presented. The method is based on an extension of the wedge assemblage (WA) method to 2D surfaces and objects. Like the WA method for one-dimensional (1D) surfaces, the approach for 2D surfaces uses Biot's and Tolstoy's exact solution for the impulse response of an infinite impenetrable wedge [J. Acoust. Soc. Am. 29, 381-391 (1957)] as its fundamental building block. The validity of the WA method for backscattering from 2D sea surfaces is assessed through comparisons with calculations based on Milder's operator expansion (OE) method [J. Acoust. Soc. Am. 89, 529-541 (1991)]. Average intensities for backscattering from 2D fully developed seas (20 m/s wind speed) were computed by the WA and OE methods using 50 surface realizations and compared at 11 frequencies between 100 and 200 Hz. A single, moderately low grazing angle of incidence (20 degrees) and several scattered grazing angles (90 degrees, 45 degrees, 20 degrees , and 10 ) were considered. Excellent overall agreement between the two models was obtained. The utility of the WA method as a tool to describe the physics of the scattering process is also discussed.     

二维随机粗糙面上导体目标复合瞬态散射的混合算法

提出用时域积分方程法(TDIE)与时域基尔霍夫近似法(TDKA)的混合算法来求解二维导体随机粗糙面及其上方二维导体目标的复合瞬态散射,推导出了在TM波入射情形下显式及隐式格式的时间步进方程.将粗糙面与目标分别进行TDKA和TDIE计算,并考虑目标与粗糙面之间的耦合,对TDKA和TDIE进行混合迭代,既大大降低了粗糙面求解的复杂度,又保证了计算精度.数值算例中,考虑了角反射器(开放体)和圆柱(封闭体)两种目标,分别计算了目标表面电流响应和电场远场响应.计算结果表明,和单纯TDIE法相比,本文混合方法计算效率 关键词： 随机粗糙面 复合瞬态散射 时域积分方程法和时域基尔霍夫近似法 混合算法     

Bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase and scalar approximation with shadowing effect: comparisons with experiments and application to the sea surface

In this paper, the bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase method and scalar approximation with shadowing effect is investigated. Both of these approaches use the Kirchhoff integral. With the stationary phase, the bistatic cross section is formulated in terms of the surface height joint characteristic function where the shadowing effect is investigated. In the case of the scalar approximation, the scattering function is computed from the previous characteristic function and in terms of expected values for the integrations over the slopes, where the shadowing effect is analysed analytically. Both of these formulations are compared with experimental data obtained from a Gaussian one-dimensional randomly rough perfectly-conducting surface. With the stationary-phase method, the results are applied to a two-dimensional sea surface.     

Bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase and scalar approximation with shadowing effect: comparisons with experiments and application to the sea surface

Abstract

In this paper, the bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase method and scalar approximation with shadowing effect is investigated. Both of these approaches use the Kirchhoff integral. With the stationary phase, the bistatic cross section is formulated in terms of the surface height joint characteristic function where the shadowing effect is investigated. In the case of the scalar approximation, the scattering function is computed from the previous characteristic function and in terms of expected values for the integrations over the slopes, where the shadowing effect is analysed analytically. Both of these formulations are compared with experimental data obtained from a Gaussian one-dimensional randomly rough perfectly-conducting surface. With the stationary-phase method, the results are applied to a two-dimensional sea surface.     

Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. Single scattering

When solving electromagnetic rough-surface scattering problems, the effect of shadowing by the surface roughness often needs to be considered, especially as the illumination angle approaches grazing incidence. This paper presents the Ricciardi-Sato, as well as the Wagner and the Smith formulations for calculating the monostatic and bistatic statistical shadowing functions from a one-dimensional rough stationary surface, which are valid for an uncorrelated Gaussian process with an infinite surface length. In this paper, these formulations are extended to include a finite surface length and any uncorrelated process. The inclusion of a finite surface length is needed to extend the single-reflection shadowing function to the more general multiple-reflection case, presented in the following companion paper. Comparisons of these shadowing functions with the exact numerical solution for the shadowing (using surfaces with Gaussian and Lorentzian autocorrelation functions for a Gaussian process) shows that the Smith formulation without correlation is a good approximation, and that including correlation only weakly improves the model. This paper also presents a method to include the shadowing effect in the electromagnetic scattering problem.     

High-frequency diffraction corrections to backscattering cross sections from a rough three-dimensional surface

Diffraction corrections to scalar wave fields at perfectly free and rigid rough surfaces were derived by two iterations of the corresponding integral equations. These diffraction corrections to the pressure or normal velocity (which, in the geometrical optics limit, are doubled at perfectly rigid and free surfaces, respectively) were obtained with an accuracy of approximately 1k(2), where k is the wave number of incidence radiation. Based on these corrections to the surface fields, the backscattering cross sections at normal incidence from the statistically rough Gaussian surfaces were derived. It was found that for the gentle roughness, diffraction results in effective "smoothing" of roughness for rigid and free surfaces and increasing of the backscattering cross sections, but for a rigid surface with steep roughness, the "fictitious" surface can be more rough than the real one, and the diffraction corrections become negative.     

Calculation of the ellipsometric parameters characterizing a randomly rough surface by means of the Stratton-Chu-Silver integral

Considering some simplifying assumptions (especially a moderately rough surface) we present a method for calculating the ellipsometric parameters δ and Ψ characterizing a randomly rough surface. The Stratton-Chu-Silver integral has been used in our approximation. We present also a comparison of the present results with those obtained earlier by means of the Kirchhoff scalar theory.     

Calculation of three-dimensional acoustic scattering from Gaussian rough under-ice surface using the Kirchhoff approximation method with modified reflection coefficient

The Kirchhoff approximation with a modified reflection coefficient is used to calculate the three-dimensional acoustic scattering of a Gaussian rough under-ice surface.The concept of a local statistical average reflection coefficient of an under-ice surface is proposed in the calculation model.The scattered sound field of a two-dimensional Gaussian rough under-ice surface is divided into coherent scattering and incoherent scattering.A formula is derived for the scattering coefficient of each sca...     

修正反射系数的Kirchhoff近似方法计算高斯起伏冰面三维声散射

针对随机起伏冰面的声散射计算问题,利用修正反射系数的Kirchhoff近似方法计算了高斯起伏冰面的三维声散射。在计算模型中引入了冰面局部统计平均反射系数的概念,将二维高斯起伏冰面的散射分为相干散射和非相干散射,分别得到两类散射成分的散射系数公式,计算了高斯起伏冰面三维声散射的散射强度。分析了散射强度与随机起伏冰面的均方根高度、声波入射角度及频率的关系。通过实验室水池中高斯起伏冰面的散射强度测量实验,对理论模型的计算结果进行了验证。将实验结果分别与采用冰面局部统计平均反射系数的模型计算结果和文献中采用平整冰面镜反射系数的模型计算结果进行了对比,采用冰面局部统计平均反射系数的模型计算结果与实验测量值吻合较好。     

随机粗糙表面上相关点的散射

随机粗糙面上邻近两点的高度与坡度是相关的.由这种相关性对散射的贡献是明显的.在计算粗糙表面散射强度时,应包括这一相关性引起的贡献.本文提供了一种解法和修正后的一维散射强度的解析公式,以及数值计算的实例.     

Wave scattering from a random rough surface: reciprocal theorem and backscattering enhancement

The present paper gives a more complete treatment of the scattering from a two-dimensional random surface than previous works. Reciprocal theorems for the stochastic wave field and the incoherent scattering distribution (bistatic cross section) are derived and the presence of backscattering enhancement in the case of a slightly random Neumann surface is demonstrated. A physical interpretation of the backscattering enhancement associated with the presence of anomalous scattering on a slightly random Neumann surface is given. Some numerical calculations are performed to show the incoherent scattering distribution and the backscattering enhancement.     

Amplification of enhanced backscattering from a dye-doped polymer bounded by a rough surface

We report the experimental study of the enhanced backscattering from a random rough surface through a laser dye-doped polymer. The sample is a slice of pyrromethene-doped polymer coupled with a two-dimensional rough gold layer with a large slope. When the sample is illuminated with an s-polarized He-Ne laser and pumped by a cw argon-ion laser, amplified backscattering is observed. The enhanced backscattering peak increases sharply and its width narrows for a sample with low dielectric constant |?(2)|.     

Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces

Using the unitarity and reciprocity preserving formulation of Brown et al a perturbation treatment, correct to fourth order in the surface profile function, is given for the scattering of electromagnetic waves from a weakly rough, two-dimensional, random metal surface. In this formulation the boundary conditions on the electromagnetic fields are satisfied using the extinction theorem in conjunction with the Rayleigh hypothesis and the vector equivalent of the Kirchhoff integral. The theory is applied to, and results are presented for, several different types of rough surfaces which are characterized by power spectra that are extensions to two-dimensional random surfaces of the power spectrum of some one-dimensional random surfaces recently fabricated by West and O'Donnell. These surfaces, which can be realized experimentally, favor coherent, interferent, multiple scattering of electromagnetic waves via surface plasmon polaritons in intermediate states, and clearly exhibit enhanced backscattering caused by the surface plasmon polariton mechanism. Theoretical results are presented for silver surfaces at optical wavelengths.