Regions of validity of geometric optics and Kirchhoff approximations for reflection from Gaussian random rough dielectric surfaces

In this paper, the effects of characteristics of incident light and the geometrical parameters to the reflectivity of dielectric Gaussian random rough surfaces are presented. The behaviors of the reflectivity vs. several parameters are quantified using approximate methods: the geometric optics approximation (GOA) and the Kirchhoff approximation (KA) and an exact method called integral method (IM). Finally, we determine the limits of validity of approximate methods by comparisons with IM results. The regions of validity of approximate methods depending of many geometrical and physical parameters: roughness, Brewster and shadowing effects, multiple reflections, surface materials, and nature of polarization. The broader domain of validity (DV) is for KA, at normal TE-polarized incident light, for the higher dielectric permittivity. However, the narrowed DV is for GOA, at normal TM-polarized incident light for lower dielectric permittivity.     

Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation

This paper presents a theory of scattering from very rough metallic and dielectric surfaces using the first- and second-order Kirchhoff approximations (KA) modified with the angular and propagation shadowing. The shadowing functions limit the single and double scattered waves which are illuminated and not shadowed by the surface. The theoretical results are compared with the Monte Carlo simulations showing the range of validity of the theory. The theory is applicable to the range where the RMS height is close to a wavelength and the RMS slope is close to unity, and the second medium is lossy. The second-order scattering includes two waves travelling in opposite directions on the surface, giving a physical explanation of the enhanced backscattering.     

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.     

Radar backscattering from Gerstner's sea surface wave

In the framework of a two-scale scattering model, radar backscattering from the rough sea surface was considered. The sea surface was modelled as a superposition of a nonlinear, large-scale Gerstner's wave and small-scale resonant Bragg scattering ripples. The zero-order diffracted field was found by a geometrical optics approach, with shadowing taken into account, and by an 'exact' solution of the diffraction problem obtained numerically. For vertical and horizontal polarizations, the spatial distribution of specific scattering cross sections along the large-scale wave was obtained. The spatially averaged specific backscattering cross sections, as well as the mean Doppler frequency shifts at both polarizations, obtained by the geometrical optics approach are compared with those obtained by using the 'exact' solution of the large-scale diffraction problem. The roles of shadowing and multiple wave scattering processes are discussed, and qualitative explanations of the difference between these two approaches are given.     

Stability in Debye series calculation for light scattering by absorbing particles and bubbles

The Debye-series decomposition is of importance for understanding of light scattering features and for the validity of the geometrical optics approximation to light scattering. The numerical stability and accuracy for calculating light scattering with Debye series is studied and an improved algorithm is proposed in this work. The ratios of the Riccati-Bessel functions and the logarithmic derivatives of the Riccati-Bessel functions are employed and calculated with proper recurrences. Exemplifying results are provided to show the improvement of the algorithm.     

Shadowing effects on quark and gluon production in ultrarelativistic heavy ion collisions

A framework for including shadowing effects to quark and gluon production in very high energy nucleus-nucleus collisions is given. A formalism for impact parameter dependent nuclear structure functions of partons is introduced and, using two models to describe the average shadowing corrections to the parton number densities, the average numbers and the transverse energy spectrum of hard partons produced inU+U collisions at RHIC and LHC energies are computed. We conclude that shadowing and also the impact parameter dependence of structure functions should be taken into account at very high energies.     

Geometry of the Saxon-Hutner theorem

Making use of the fact that any real localized potential of both the Dirac and Schrödinger equations on the line can be viewed either as a vector in the three-dimensional vector space or as a point on the single-sheeted hyperboloid we are able to derive a fairly general condition which ensures the validity of the Saxon-Hutner theorem that is spelled out in purely geometrical terms.     

Bounds for the complex dielectric constant of a two-phase composite

We study the effective dielectric constant of a two-phase composite for the case where the dielectric constants of the two components are complex. Milton has derived a sequence of narrowing bounds in the complex plane limiting the possible values of the effective dielectric constant, assuming that certain geometrical coefficients characterizing the medium are known. We cast Milton's bounds in a different form, making use of auxiliary functions introduced by Bergman. We derive explicit expressions for the bounds up to third order and compare values for the corresponding geometrical coefficient derived for different geometries.     

Nuclear shadowing and Drell-Yan production on heavy nuclei

The suggestion that nuclear shadowing in deep inelastic scattering is primarily a parton phenomenon is extended to the Drell-Yan reaction on heavy nuclei. The nuclear structure functions required to describe the latter process are found to be compatible with those obtained from an analysis of the shadowing data, and lend weight to the hypothesis that this is indeed a parton phenomenon.     

Shadowing effect in clusters of opaque spherical particles

The shadowing effect is studied for clusters of opaque spherical particles. The present modeling allows geometric optics computations of cluster scattering phase functions and shadowing effects with internal accuracy better than 1%. Three types of cluster structures are treated—uniform, ballistic, and hierarchical (physical fractal)—and three types of elementary surface scattering laws are examined—Lambertian, flux-isotropic, and specular. All structures investigated give rise to an opposition effect, that is, a nonlinear brightening toward zero phase angle. The amplitude and width of the opposition effect depend on the cluster parameters. For uniform clusters, the volume fraction and number of particles are the parameters that characterize the shadowing effect. The opposition effect becomes sharper with increasing number of constituent particles and with decreasing particle volume fraction. For ballistic clusters, the only parameter is the number of particles: when it increases, the opposition effect becomes sharper. For hierarchical clusters, the number of cluster structural levels plays a crucial role. With increasing number of cluster levels, the opposition behavior of brightness becomes markedly more nonlinear, mostly due to the decreasing particle volume fraction. It is notable, however, that the opposition effects of the hierarchical clusters and the uniform clusters with the same particle volume fraction differ from each other underscoring the importance of the detailed cluster structure on shadowing. It is shown that, with reasonable accuracy, the cluster scattering phase functions can be factorized as the products of the corresponding single-particle phase function and the so-called shadowing factor almost independently of the elementary surface scattering law. While the opposition effect due to shadowing is presently confirmed, it is typically wider than the opposition effect due to coherent backscattering, an interference mechanism in multiple scattering. The present work helps us to understand, e.g., the opposition effects of the Moon, asteroids, and other atmosphereless celestial bodies.     

Analytical approach for the approximate solution of the longitudinal structure function with respect to the GLR-MQ equation at small x

We show that the nonlinear corrections to the longitudinal structure function can be tamed the singularity behavior at low x values, with respect to GLR-MQ equations. This approach can determined the shadowing longitudinal structure function based on the shadowing corrections to the gluon and singlet quark structure functions. Comparing our results with HERA data show that at very low x this behavior completely tamed by these corrections.     

Photon-hadron interactions revisited in deep inelastic muon scattering

New results for deep inelastic muon scattering are reviewed. These include shadowing results tox _bj ∼10⁻⁵, nucleon structure functions measurements, the Gottfried sum rule and the use of deep inelastic scattering to obtain information on gluon distributions. The future of muon scattering in the HERA era is discussed.     

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.     

Effect of parton shadowing on the distributions of global observables

The effect of parton shadowing on the distributions of global observables (total transverse energy E _T and charged-particle multiplicity n _ch) is investigated on the basis of the HIJING model of nucleus-nucleus interactions. It is shown that, for the interaction of lead nuclei at $\sqrt s = 5$ TeV per nucleon, the shadowing effect results in a considerable reduction of the cross section for (mini)jet production (approximately by a factor of 4); this in turn reduces the total transverse energy and the charged-particle multiplicity by a factor of 2.7. Upon taking into account the QCD evolution of nuclear structure functions in Q ², the shadowing effect becomes less pronounced (by a factor of 1.9 for PbPb interactions). It is shown that global observables can be used to test models of parton shadowing.     

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.     

Three dimensional analyses of scattering by pressure-release sinusoidal surfaces

Scattering by pressure-release sinusoidal surfaces in three dimensions is analyzed using the Fresnel phase approximation and realistic source and receiver directivity approximations. Geometrical shadowing and second-order scattering are explicitly included to explore the validity of the Kirchhoff approximation. No restrictions on the surface heights and slopes are made. The "goodness" of the resulting expressions is verified by requiring the pressure scattered by a sinusoidal surface to reduce to the image solution as the surface amplitude goes to zero. The first-order scattered pressure achieves a very good approximation to the image solution, and the second-order scattered pressure goes to zero, as expected, under this requirement. The theory is compared with available experimental scattering measurements, and the agreement is good. Because the slopes on the experimental surface are very steep, shadowing corrections are indispensible to achieving accurate first and second order scattering results. Shadowing has a greater impact on the scattering prediction than the second-order scattering contribution. This suggests that the Kirchhoff approximation may be more robust when incorporated into a theory including a detailed shadowing treatment as well as the Fresnel phase approximation and a good directivity approximation.     

Directional hemispherical radiative properties of random dielectric rough surfaces

In this paper the directional hemispherical reflectivity and transmissivity of one-dimensional, randomly rough, dielectric surfaces are determined by the use of the integral method. This method is derived from electromagnetic theory without any restrictive hypotheses. Since this exact approach is computationally very intensive, a geometric optics approximation method is also developed. Curves displaying radiative properties versus the correlation length for a constant mean square deviation of the surface from flatness are presented. In this respect, the influence on the validity of the approximate method of multiple scattering, the shadowing effect and the real index of refraction of the dielectric have been investigated. Transverse electric and transverse magnetic polarized incident plane waves are considered. For the latter, our interest is focused on the influence of roughness on the reflected and transmitted intensities for an angle of incidence close to the Brewster angle.     

Directional hemispherical radiative properties of random dielectric rough surfaces

In this paper the directional hemispherical reflectivity and transmissivity of one-dimensional, randomly rough, dielectric surfaces are determined by the use of the integral method. This method is derived from electromagnetic theory without any restrictive hypotheses. Since this exact approach is computationally very intensive, a geometric optics approximation method is also developed. Curves displaying radiative properties versus the correlation length for a constant mean square deviation of the surface from flatness are presented. In this respect, the influence on the validity of the approximate method of multiple scattering, the shadowing effect and the real index of refraction of the dielectric have been investigated. Transverse electric and transverse magnetic polarized incident plane waves are considered. For the latter, our interest is focused on the influence of roughness on the reflected and transmitted intensities for an angle of incidence close to the Brewster angle.