Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

Formal tilt invariance of the local curvature approximation

Abstract

Tilt invariance is a stringent but necessary condition that a second-order wave scattering model must satisfy in order to qualify for a broad range of applications. This invariance expresses the fact that the scattering model is unchanged whether the tilting of the scattering surface is implemented before or after its reduction to the limit of the small-perturbation method (SPM). Our scattering model is based on a second-order kernel which is quadratic in its lowest order with respect to successive derivatives of the rough surface. Hence, it is termed the local curvature approximation (LCA). We have previously demonstrated that the LCA is approximately tilt invariant in the quasi-specular and quasi-backscattering geometries. In this contribution, LCA is made formally tilt invariant up to first order in the tilting vector. It will be shown that this formal tilt invariance is achieved mainly through inclusion of polarization mixing due to out-of-plane tilt. Even though the LCA formally reduces to the SPM and Kirchhoff limits in addition to tilt invariance, its curvature kernel stays reasonably concise and practical to implement in both analytical and numerical evaluations. This curvature kernel may also be used in the other two formulations of our model, namely the non-local curvature approximation and the weighted curvature approximation.     

Analytical comparison between the surface current integral equation and the second-order small-slope approximation

Abstract

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 previous formulation where the polarization is independent of surface roughness. The second-order small slope approximation (SSA-2), which satisfies the SPM-1 and second-order small perturbation method (SPM-2) limits by construction, was shown by Voronovich to converge toward the tangent plane approximation of the Kirchhoff model under high frequency conditions. In the present paper, we show that, in addition to the linear orders in our previous model, one must now include cross-terms between slope and surface elevation to ensure convergence toward both high frequency and small perturbation limits. With the inclusion of these terms, our new formulation becomes comparable to the SSA-2 (second-order kernel) without the need to evaluate all the quadratic order slope and elevations terms. SSA-2 is more complete, however, in the sense that it guarantees convergence toward the second-order Bragg limit (SPM-2) in the fully dielectric case in addition to both SPM-1 and Kirchhoff. Our new generalization is shown to explain correctly extra depolarization in specular conditions to be caused by surface curvature and surface autocorrelation for incoherent and coherent scattering, respectively. This result will have large repercussions on the interpretation of bistatically reflected signals such as those from GPS.     

Analytical comparison between the surface current integral equation and the second-order small-slope approximation

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 previous formulation where the polarization is independent of surface roughness. The second-order small slope approximation (SSA-2), which satisfies the SPM-1 and second-order small perturbation method (SPM-2) limits by construction, was shown by Voronovich to converge toward the tangent plane approximation of the Kirchhoff model under high frequency conditions. In the present paper, we show that, in addition to the linear orders in our previous model, one must now include cross-terms between slope and surface elevation to ensure convergence toward both high frequency and small perturbation limits. With the inclusion of these terms, our new formulation becomes comparable to the SSA-2 (second-order kernel) without the need to evaluate all the quadratic order slope and elevations terms. SSA-2 is more complete, however, in the sense that it guarantees convergence toward the second-order Bragg limit (SPM-2) in the fully dielectric case in addition to both SPM-1 and Kirchhoff. Our new generalization is shown to explain correctly extra depolarization in specular conditions to be caused by surface curvature and surface autocorrelation for incoherent and coherent scattering, respectively. This result will have large repercussions on the interpretation of bistatically reflected signals such as those from GPS.     

Local and non-local curvature approximation: a new asymptotic theory for wave scattering

Abstract

We present a new asymptotic theory for scalar and vector wave scattering from rough surfaces which federates an extended Kirchhoff approximation (EKA), such as the integral equation method (IEM), with the first and second order small slope approximations (SSA). The new development stems from the fact that any improvement of the ‘high frequency’ Kirchhoff or tangent plane approximation (KA) must come through surface curvature and higher order derivatives. Hence, this condition requires that the second order kernel be quadratic in its lowest order with respect to its Fourier variable or formally the gradient operator. A second important constraint which must be met is that both the Kirchhoff approximation (KA) and the first order small perturbation method (SPM-1 or Bragg) be dynamically reached, depending on the surface conditions. We derive herein this new kernel from a formal inclusion of the derivative operator in the difference between the polarization coefficients of KA and SPM-1. This new kernel is as simple as the expressions for both Kirchhoff and SPM-1 coefficients. This formal difference has the same curvature order as SSA-1 + SSA-2. It is acknowledged that even though the second order small perturbation method (SPM-2) is not enforced, as opposed to the SSA, our model should reproduce a reasonable approximation of the SPM-2 function at least up to the curvature or quadratic order. We provide three different versions of this new asymptotic theory under the local, non-local, and weighted curvature approximations. Each of these three models is demonstrated to be tilt invariant through first order in the tilting vector.     

Theory of radio wave scattering at a rough sea surface

The problem of the reflection of a plane single-frequency electromagnetic wave from a statistically rough dielectric boundary with arbitrary is solved in the perturbation approximation. The statistical characteristics (scattering cross section, change of polarization, and frequency spectrum) of a radar signal reflected from a rough sea surface are investigated. The model used for the surface—a small ripple superimposed on large waves—enables the perturbation theory approach to be extended to the decimeter and centimeter wave band.Izvestiya VUZ. Radiofizika, Vol. 9, No. 5, pp. 876–887, 1966     

Evaluation of times and arrival angles of surface prereverberation in the ocean

Acoustic prereverberation caused by sound scattering from the rough sea surface is considered. For the case of low-frequency scattering described by the first approximation of the small perturbation method, the arrival times and angles of prereverberation signals in the subsurface sound channel are calculated as functions of the wind speed, sound frequency, and distance.     

The role of the frozen surface approximation in small wave-height perturbation theory for moving surfaces

In the standard development of the small wave-height approximation (SWHA) perturbation theory for scattering from moving rough surfaces [e.g., E. Y. Harper and F. M . Labianca, J. Acoust. Soc. Am. 58, 349-364 (1975) and F. M. Labianca and E. Y. Harper, J. Acoust. Soc. Am. 62, 1144-1157 (1977)] the necessity for any sort of frozen surface approximation is avoided by the replacement of the rough boundary by a flat (and static) boundary. In this paper, this seemingly fortuitous byproduct of the small wave-height approximation is examined and found to fail to fully agree with an analysis based on the kinematics of the problem. Specifically, the first order correction term from the standard perturbation approach predicts a scattered amplitude that depends of the source wave number, whereas the kinematics point to a scattered amplitude that depends on the scattered wave number. It is shown that a perturbation approach in which an explicit frozen surface approximation is made before the SWHA is invoked predicts (first order) scattered amplitudes that are in agreement with the analysis based on the kinematics.     

Studies of scattering theory using numerical methods

Abstract

Numerical simulations, using both exact and approximate methods, are used to study rough surface scattering in both the smd and large roughness regimes. This study is limited lo scattcring lrom rough one-dimensional surfaces that obey the Dirichlet boundary condition and have a Gaussian roughness spectrum. For surfdces with small roughness (kh?1, where k is the radiation wavenumber and h is the root-mean-square (RMS) Surface height), perturbation theory is known to be valid. However, it is shown numerically that when kh?1 and kl?6 (where I is the surface correlation length) the Kirchhoffapprorimation is valid except at low grazing angles, and one must sum the first three orders of perturbation theory obtain the correct result. For kh?1 and kl?1, first-order perturbation theory is accurate. In this region, the accuracy of the first two terms of the iterative series solution of the exact integral equation is examined; the first term a1 this series is the Kirchhoff approximation, It is shown numerically that lor very small kh these first two terms reduce to first-order perturbation theory. However, lor this reduction to occur, kh must be made smaller than necessdry lor first-order perturbation theory to be accurate. In the regime of large roughness (kh?1) backscattering enhancement occurs when the RMS slope is on the order of unity. Several investigators have recently shown that the second term of the iterative series solution (the double-scattering term) replicates the properties of backscattering enhancement reasonably well. However, the double-scattering term has a lundamental flaw: predictions lor the scattering cross section per unit length based on the double-scattering term increase as the surfdce length is increased. This is shown here with numerical simulations and with an approximate analytical result based on the high frequency limit. The physical significance of this finding is also discussed. The final topic is the use of the double-scattering approximation to study the mechanism for backscattering enhancement with the Dirichlet boundary condition. This mechanism is usually assumed to be interference between reciprocal scattering paths. When the interlerence between reciprocal scattering paths is removed, the enhancement is eliminated. This shows that interference between reciprocal paths is almost certainly the dominant mechanism for backscattering enhancement in the scattering regime studied.     

Scattering of electromagnetic waves from a surface with small and shallow inhomogeneities

Using the second-order perturbation theory, we obtain an analytical expression for the electromagnetic field scattered from a dielectric surface with inhomogeneities that are shallow and small in comparison with the wavelength. This expression is used to analyze the critical phenomena in thermal radio emission of a periodically rough water surface, and the second order of the specific scattering cross section of radio waves, in the short and medium ranges of wavelengths, from a rough surface (in particular, from a sea surface with waves).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 27, No. 1, pp. 48–55, January, 1984.     

The region of validity of perturbation theory

Abstract

We present a study of the region of validity of perturbation theory applied to rough surface scattering. We solve numerically the case of a periodic surface or grating varying in one dimension. For a statistical ensemble of gratings with a sufficiently long period one may obtain a good approximation of rough surface scattering. We use this to test the validity of perturbation theory.

Only the perfect conductor case was considered. We find that as the grazing angle becomes small the perturbation result for the TE (E horizontal) polarization remains valid, while for the TM (E vertical) polarization it breaks down. The results show that the perturbation results should be used carefully when being compared with experimental data at grazing angles.     

The region of validity of perturbation theory

We present a study of the region of validity of perturbation theory applied to rough surface scattering. We solve numerically the case of a periodic surface or grating varying in one dimension. For a statistical ensemble of gratings with a sufficiently long period one may obtain a good approximation of rough surface scattering. We use this to test the validity of perturbation theory.

Only the perfect conductor case was considered. We find that as the grazing angle becomes small the perturbation result for the TE (E horizontal) polarization remains valid, while for the TM (E vertical) polarization it breaks down. The results show that the perturbation results should be used carefully when being compared with experimental data at grazing angles.     

二维导体微粗糙面与其上方金属平板的复合电磁散射研究

研究了二维导体微粗糙面与其上方金属平板复合电磁散射特征.应用互易性原理使求解二次散射场简化为求解包含平板上的极化电流和微粗糙面散射场的积分方程，从而降低了求解难度.应用物理光学近似和微扰法分别求解了平板上的极化电流和粗糙面的电磁散射场，得到了复合散射截面计算公式并进行了数值计算.尤其对该复合模型后向耦合电磁散射结果进行了详细分析和讨论. 关键词： 互易性原理 复合电磁散射 微粗糙面 平板     

Region of validity of perturbation theory for dielectrics and finite conductors

We present a study of region of validity of first-order perturbation theory applied to rough surface scattering. The scattering problem is solved numerically for the case of periodic surface or gratings varying in one dimension. Scattering of electromagnetic waves from an ensemble of gratings of sufficiently long period will give a good approximation to the case of an infinite rough surface. We use this to test the validity of the first-order perturbation theory. Use of an infinite periodic surface allows us to give results for a range of angle of incidence covering those representing a low grazing angle, near 90° from the mean surface normal. We consider the case for perfect dielectrics and finite conductors. The real and imaginary parts of the refractive index used were limited to less than three due to the numerical instability of the numerical calculation method involved. We find that for perfect dielectrics the first-order small perturbation theory remains for TE polarization valid for all incidence angles, while for TM polarization it seems to fail if the incidence angle approaches the Brewster angle.     

Variational field theoretic approach to relativistic meson-nucleon scattering

Non-perturbative polaron variational methods are applied, within the so-called particle or world-line representation of relativistic field theory, to study scattering in the context of the scalar Wick-Cutkosky model. Important features of the variational calculation are that it is a controlled approximation scheme valid for arbitrary coupling strengths, the Green functions have all the cuts and poles expected for the exact result at any order in perturbation theory and that the variational parameters are simultaneously sensitive to the infrared as well as the ultraviolet behaviour of the theory. We generalize the previously used quadratic trial action by allowing more freedom for off-shell propagation without a change in the on-shell variational equations and evaluate the scattering amplitude at first order in the variational scheme. Particular attention is paid to the s-channel scattering near threshold because here non-perturbative effects can be large. We check the unitarity of a our numerical calculation and find it greatly improved compared to perturbation theory and to the zeroth order variational results.     

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.     

A practical second-order electromagnetic model in the quasi-specular regime based on the curvature of a 'good-conducting' scattering surface

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 therefore it is suitable for a vast array of analytical and numerical applications in quasi-specular applications.     

A practical second-order electromagnetic model in the quasi-specular regime based on the curvature of a ‘good-conducting’ scattering surface

Abstract

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 therefore it is suitable for a vast array of analytical and numerical applications in quasi-specular applications.     

Tilt-invariant theory of rough-surface scattering: I

Abstract

The small-slope approximation (SSA) in rough-surface scattering theory uses the surface slope as a small parameter of expansion. But, from the physical point of view, the slope may not be a restrictive parameter because we can change the slope of a surface simply by tilting the coordinate system. We present the theory of rough-surface scattering in a coordinate-invariant form. The new method, tilt-invariant approximation (TIA), leads to a different expansion that does not require that the slope of a surface be small. For a small Rayleigh parameter this approximate solution provides the correct perturbation theory, for a large Rayleigh parameter it provides the Kirchhoff approximation with several correcting terms.