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.     

A study of the higher-order small-slope approximation for scattering from a Gaussian rough surface

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.     

On the high-frequency limit of the second-order small-slope approximation

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

On the high-frequency limit of the second-order small-slope approximation

Abstract

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

An extension of the small-slope approximation for rough surface scattering

Solutions are derived for scattering from a rough one-dimensional pressure-release surface in the form of a functional series in the surface slope. These solutions are obtained by solving an integral equation of the first kind for the surface potential to obtain a representation for the scattering amplitude. It is shown that the subsequent expansion of terms occurring in the scattering amplitude to obtain a functional series in the slope does not yield a unique result. The result obtained contains a free parameter that may be arbitrarily selected. Thus, this result is an extension or generalization of the small-slope approximation of Voronovich (1985 Sov. Phys.-JETP 62 65-70) that differs at second order in the slope from his result. It is also shown that the free parameter can be selected such that each term of the functional series is reciprocal and exhibits a limiting grazing angle dependence consistent with the requirements of flux conservation and the absence of boundary waves. A new formulation of the leading terms of the small-slope expansions is derived and is used to explore the conditions under which the two expansions reduce to the Kirchhoff approximation. Finally, a numerical example is presented to demonstrate that the extended approximation provides corrections that are important for near grazing scatter.     

The effect of the modulation of Bragg scattering in small-slope approximation

Small-slope approximation (SSA) belongs to a class of 'unifying' scattering theories which reproduce small perturbations and semiclassic (Kirchhoff) results within appropriate limits. However, the most stringent test for such theories involves a two-scale situation when a small-scale roughness is located on a tilted plane. A 'unifying' theory should properly account for the effects of modulation of the scattering cross section associated with a large-scale tilt. This paper shows that SSA does properly take into account these modulation effects.     

The effect of the modulation of Bragg scattering in small-slope approximation

Abstract

Small-slope approximation (SSA) belongs to a class of ‘unifying’ scattering theories which reproduce small perturbations and semiclassic (Kirchhoff) results within appropriate limits. However, the most stringent test for such theories involves a two-scale situation when a small-scale roughness is located on a tilted plane. A ‘unifying’ theory should properly account for the effects of modulation of the scattering cross section associated with a large-scale tilt. This paper shows that SSA does properly take into account these modulation effects.     

Generalized Veneziano model from the point of view of manifestly crossing-invariant parametrization

The construction of the Veneziano-typen-point function with manifestly crossing-invariant parametrization is reviewed in a heuristic way. Here, one meets a divergence problem, the solution of which is discussed in detail. Equivalence to previous works and its asymptotic behaviour are rederived with the help of affine group representation.     

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.     

A hard sphere model of chemical reactions theory from the point of view of many-particle approach

On the basis of the previously obtained non-Markovian kinetic equation for irreversible chemical reaction A+B→C+B, we have examined a model of contact reaction between hard (impenetrable) spheres (singular force potential). The nature of the additional contribution to the rate constant is explained. As is found, for reactants with a hard core it is necessary to allow for the presence of initial correlations between them. A class of initial conditions for arbitrary form of the pair potential energy is established.     

Geometry from the spectral point of view

In this Letter, we develop geometry from a spectral point of view, the geometric data being encoded by a triple (A. H. D.) of an algebraA represented in a Hilbert spaceH with selfadjoint operatorD. This point of view is dictated by the general framework of noncommutative geometry and allows us to use geometric ideas in many situations beyond Riemannian geometry.

This paper is dedicated to the memory of J. Schwinger     

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.     

Stochastic resonance in the FitzHugh-Nagumo model from a dynamic mutual information point of view

Stochastic resonance(SR) in a FitzHugh-Nagumo neuron model is investigated based on a dynamic mutual information (DMI) between the input and the corresponding output signals. The DMI is expressed in terms of the (cross)power spectra of the input and output time series. Both stochastic-periodic and aperiodic SR are treated based on the DMI and our results are in good accord with the SR measured by the signal to noise ratio(SNR) for the case of the stochastic-periodic input and the power norm for the case of the aperiodic input.     

CP invariance: A point of view

That the longlived componentL ofK ₀ has bothcp=+1 andcp=−1 modes of decay is often cited as evidence of violation ofcp invariance. The careful ones find the compelling evidence to be the non-dilution of the regeneration interference pattern when the incidentK ₀ beam is mixed even substantially with . However the two phenomena comprehensively imply thatL has acp=+1 componentL ₊ and acp=−1 componentL ₋ and that the longlived component of bothK ₀ and are one and the sameL. This does not demand abandoningcp invariance. It does imply that is not thecp conjugate ofK ₀.     

Master-slave synchronization from the point of view of global dynamics

We present a mathematical framework for the theory of a synchronization phenomenon for dynamical systems discovered by Pecora and Carroll [Phys. Rev. Lett. 64, 821-824 (1990)]. From this perspective, we can synchronize, using a single coordinate, an open dense set of linear systems. We use our insights to synchronize nonlinear systems which were not previously recognized as being synchronizable. (c) 1995 American Institute of Physics.