Effective probability density function of rough surface slopes when strong shadowing is present

It is shown that at low grazing angles, the slope probability density function (PDF) of the nonshadowed part of a rough surface can differ significantly from the slope PDF of the overall surface, if surface heights and slopes are functionally dependent. If the surface steepness has a tendency to increase with height, the effective slopes of the illuminated part of the surface can be significantly steeper than the average slope of the surface as a whole. This fact can play a crucial role in any theoretical interpretation of experimental results concerning radar scattering by the sea surface at low grazing angles.     

Electromagnetic propagation in a curved two-dimensional waveguide

Abstract

Fields due to an electromagnetic wave propagating in a long irregular two-dimensional waveguide are calculated efficiently, using the method of left-right splitting to solve the coupled integral equations. Results are compared with those obtained from independent ray-theoretic calculations and give very close agreement. The approach has previously been applied to rough surfaces at low angles of incidence; here it is found to converge rapidly for surface slopes of 30°, and after a few iterations for incident angles up to 60° from grazing.     

Curvature effects in the composite model for low-grazing-angle rough-surface scatter

Abstract

The conventional composite surface model used for calculations of radar backscatter treats a large-scale surface component as an ensemble of tilted flat facets. This approach has limitations for low grazing angles. In the paper we develop a new composite model that takes into account the diffraction of an incident field on a curved, large-scale undulating surface. It is shown that, whereas corrections related to curvature effects are small for a steep incidence, those corrections for low-grazing-angle backscatter can be quite essential. Results of numerical simulations of radar backscattering cross sections for two polarizations are presented. Scattering from a surface with one-dimensionally varying roughness is considered. We have studied the role of undulating surface slopes and curvatures in the behaviour of the polarization ratio and found regimes in which the curvature influence on this ratio is more pronounced.     

The SSA+ for scattering from dielectric surfaces at very low grazing angles

Previously we developed a practical model for scattering from randomly-rough surfaces at very low grazing angles for the Dirichlet problem which was found to give good numerical results. In this paper, we derive the expression for the bistatic scattering cross-section for the non-local small slope approximation for dielectric interfaces. We then extend our practical model to dielectric surfaces based on this result. We discuss numerical results for scattering at low forward grazing angles for a Gaussian roughness spectrum with an angle of incidence of 80^○.     

Electromagnetic propagation in a curved two-dimensional waveguide

Fields due to an electromagnetic wave propagating in a long irregular two-dimensional waveguide are calculated efficiently, using the method of left-right splitting to solve the coupled integral equations. Results are compared with those obtained from independent ray-theoretic calculations and give very close agreement. The approach has previously been applied to rough surfaces at low angles of incidence; here it is found to converge rapidly for surface slopes of 30°, and after a few iterations for incident angles up to 60° from grazing.     

“Anomalous” pseudodielectric function of GaN: Experiment, modelling and application to the study of surface properties

Studying GaN films exposed to Ar plasma by spectroscopic ellipsometry and reflectance, we found an “anomalous” pseudodielectric function (PDF) for which the imaginary part is significantly higher as compared to GaN, while the real part of the PDF remains close to the value for GaN. In addition, a higher reflectance at low angles of incidence was observed. The data are explained in terms of a thin highly absorbing surface layer arising due to non-stoichiometry in the near-surface region. Comparison to samples grown by molecular beam epitaxy shows that similar mechanisms are responsible for optical properties of the surfaces of films obtained under Ga-rich conditions.     

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.     

Specular propagation over rough surfaces: numerical assessment of Uscinski and Stanek's mean Green's function technique

The purpose of this paper is to numerically evaluate the effectiveness and accuracy of Uscinski and Stanek's mean Green's function technique for computing the mean field of a wave scattered by a rough surface. We present here a direct comparison of this technique with a rigorous numerical method, the forward scattering integral equation method, and another analytical method, the first-order smoothing approximation. Furthermore, we compare the roughness generated equivalent admittance using the three methods. Numerical computations reveal that the scattered field calculated by this technique is not accurate particularly for the equivalent admittance at low grazing angles, even though the mean surface current density is recovered when the wave has traversed several correlation lengths on the surface.     

A practical cross-section for scattering from rough surfaces at very low grazing angles in both the forward and backward directions

In this paper we develop an extension of the small slope approximation (SSA) for scattering from randomly rough Dirichlet surfaces, which includes some multiple scattering. This extension is designated by SSA+. We focus on scattering at very low grazing angles where multiple scattering of both the incident and scattered fields is of importance. Numerical results for the SSA+ bistatic scattering cross-section for very low forward grazing angles are presented using the Gaussian roughness spectrum and for both very low forward and very low backward grazing angles using the Pierson–Moskowitz and modified power law spectra. The results are restricted to an angle of incidence of 80°. It is shown that when the lowest-order SSA gives reasonably accurate results, the SSA+ increases the accuracy up to at least the final 0.2° of grazing in the forward direction. In the backward direction, the SSA+ gives good results for the Pierson–Moskowitz spectrum, but the results are less dramatic.     

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

Abstract

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.     

Penetration depth measurement in high quality thin films

The parallel plate resonator method has been used for measuring high quality (YBCO) thin films, which have low temperature residual losses comparable to those previously obtained in single crystals. The surface resistance and the real part of the conductivity show a non-monotonic behaviour with a broad peak around 45 K. The penetration depth and the real part of the conductivity vary linearly at low temperatures. The lowest penetration depth linear fitting has a slope value of to up to 20 K which is lower than previous measurements on YBCO single crystals. An interpretation of this smaller slope in terms of the generally accepted d-wave order parameter symmetry presents difficulties. Received: 22 July 1997 / Revised: 11 March 1998 / Accepted: 23 June 1998     

Theoretical model for scattering of radar signals in K u - and C-bands from a rough sea surface with breaking waves

Abstract

A small-slope approximation (SSA) is used for numerical calculations of a radar backscattering cross section of the ocean surface for both K _u - and C-bands for various wind speeds and incident angles. Both the lowest order of the SSA and the one that includes the next-order correction to it are considered. The calculations were made by assuming the surface-height spectrum of Elfouhaily et al for fully developed seas. Empirical scattering models CMOD2-I3 and SASS-II are used for comparison. Theoretical calculations are in good overall agreement with the experimental data represented by the empirical models, with the exception of HH-polarization in the upwind direction. It was assumed that steep breaking waves are responsible for this effect, and the probability density function of large slopes was calculated based on this assumption. The logarithm of this function in the upwind direction can be approximated by a linear combination of wind speed and the appropriate slope. The resulting backscattering cross section for upwind, downwind and cross-wind directions, for winds ranging between 5 and 15 m s^?1, and for both polarizations in both wave bands corresponds to experimental results within 1–2 dB accuracy.     

Propagation of centimeter and millimeter radio waves at small grazing angles: The model of multiple screen diffraction

Specific features of the microwave propagation over a disturbed sea surface at small grazing angles are studied numerically in the approximation of multiple Fresnel diffraction by semiplanes. We consider two cases: 1) the case of very small grazing angles and low altitudes of the observation points over the sea surface and 2) the case of moderately small grazing angles where the asymptotic approximation of the Fresnel integral is applicable. It is shown that such a model permits one to explain, in particular, the high field-intensity levels and fluctuation characteristics observed at observation-point altitudes comparable with the sea-wave height. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 8, pp. 700–710, August 2008.     

Theoretical and computational aspects of scattering from periodic surfaces: two-dimensional perfectly reflecting surfaces using the spectral-coordinate method

We consider the scattering from a two-dimensional periodic surface. From our previous work on scattering from one-dimensional surfaces (1998 Waves Random Media 8 385) we have learned that the spectral-coordinate (SC) method was the fastest method we have available. Most computational studies of scattering from two-dimensional surfaces require a large memory and a long calculation time unless some approximations are used in the theoretical development. By using the SC method here we are able to solve exact theoretical equations with a minimum of calculation time.

We first derive in detail (part I) the SC equations for scattering from two-dimensional infinite surfaces. Equations for the boundary unknowns (surface field and/or its normal derivative) result as well as an equation to evaluate the scattered field once we have solved for the boundary unknowns. Special cases for the perfectly reflecting Dirichlet and Neumann boundary value problems are presented as is the flux-conservation relation.

The equations are reduced to those for a two-dimensional periodic surface in part II and we discuss the numerical methods for their solution. The two-dimensional coordinate and spectral samples are arranged in one-dimensional strings in order to define the matrix system to be solved.

The SC equations for the two-dimensional periodic surfaces are solved in part III. Computations are performed for both Dirichlet and Neumann problems for various periodic sinusoidal surface examples. The surfaces vary in roughness as well as period and are investigated when the incident field is far from grazing incidence ('no grazing') and when it is near-grazing. Extensive computations are included in terms of the maximum roughness slope which can be computed using the method with a fixed maximum error as a function of the azimuthal angle of incidence, the polar angle of incidence and the wavelength-to-period ratio.

The results show that the SC method is highly robust. This is demonstrated with extensive computations. Furthermore, the SC method is found to be computationally efficient and accurate for near-grazing incidence. Computations are presented for grazing angles as low as 0.01°. In general, we conclude that the SC method is a very fast, reliable and robust computational method to describe scattering from two-dimensional periodic surfaces. Its major limiting factor is high slopes and we quantify this limitation.     

Contribution of steep irregularities to the radio-brightness temperature of the ocean

We propose a new mechanism of a change in the brightness temperature of the ocean due to the contribution from steep mesoscale waves and estimate the contribution of such waves to the brightness temperature of the ocean. A steep wave is simulated by an inclined surface. The estimates show that variations in the radio-brightness temperature due to steep irregularities can reach several kelvins at low grazing angles. For short observation distances and low grazing angles, the brightness temperature has bursts similar to those observed in the case of backscattering. These bursts occur when breaking waves hit the observation area. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 3, pp. 217–223, March, 2000     

Pressure and temperature-dependent raman study of ZnWO4

Abstract

The Raman spectra of monoclinic ZnWO₄ (c⁴ _2h-space group) are reported over wide ranges of pressure (0-24GPa) and temperature (13-970 K). All 18 Raman active pho-nons are observed throughout these ranges. Combining the pressure and temperature Raman data, an identification scheme is suggested which makes possible to distinguish the internal modes (vibrations in the octahedral WO₆ units) from the external ones (pure lattice modes). All phonons harden with pressure, thus showing normal positive dω/dP slopes, and soften with temperature in the region 250 - 970 K (normal negative dω/dT slopes). However, two phonons show a change of slope sign below 300 K, resulting in small positive slopes at low temperatures. Since the pressure dependence of these two phonons is normal and linear, it is concluded that their anomalous, non-linear dependence with temperature is due to anharmonic (temperature induced) phonon interactions rather that volume expansion. It is found that ZnWO₄ remains stable throughout the pressure and temperature ranges.     

Acoustic scattering from a rough sea surface: the mean field by the integral equation method

Abstract

The scattering of an acoustic signal incident from below at low angles on a rough sea surface is treated by the integral equation method in the parabolic approximation. Equations are obtained allowing the mean scattered field to be calculated even when the surface causes a large phase modulation in the incident wave. Solutions are found using the method of Laplace transforms and some results are presented for a specific type of rough surface.     

Angular-dependent XPS valence-band spectra from single-crystal copper

High-resolution angular-dependent X-ray photoemission (XPS) valence-band spectra from a copper single crystal with (001) orientation are presented and analyzed in terms of a direct-transition model of photoemission. This model includes the effects of photon momentum, spectrometer geometry, and, for very low angles of emission, refraction of electrons in leaving the surface, but neglects matrix-element variation. Theoretical calculations are found to be in good agreement with experiment for six distinct emission directions. For low (grazing) angles of emission, a sharpening of the d-band peak is observed that may be associated with surface-specific band-structure changes.     

Multiple scattering in the high-frequency limit with second-order shadowing function from 2D anisotropic rough dielectric surfaces: I. Theoretical study

Abstract

In this paper the first- and second-order Kirchhoff approximation is applied to study the backscattering enhancement phenomenon, which appears when the surface rms slope is greater than 0.5. The formulation is reduced to the geometric optics approximation in which the second-order illumination function is taken into account. This study is developed for a two-dimensional (2D) anisotropic stationary rough dielectric surface and for any surface slope and height distributions assumed to be statistically even. Using the Weyl representation of the Green function (which introduces an absolute value over the surface elevation in the phase term), the incoherent scattering coefficient under the stationary phase assumption is expressed as the sum of three terms. The incoherent scattering coefficient then requires the numerical computation of a ten- dimensional integral. To reduce the number of numerical integrations, the geometric optics approximation is applied, which assumes that the correlation between two adjacent points is very strong. The model is then proportional to two surface slope probabilities, for which the slopes would specularly reflect the beams in the double scattering process. In addition, the slope distributions are related with each other by a propagating function, which accounts for the second-order illumination function. The companion paper is devoted to the simulation of this model and comparisons with an ‘exact’ numerical method.     

Universal behaviour of scattering amplitudes for scattering from a plane in an average rough surface for small grazing angles

Abstract

It is shown that for scattering from a plane in an average rough surface, the scattering cross section of the range of small grazing angles of the scattered wave demonstrates a universal behaviour. If the angle of incidence is fixed (in general it should not be small), the diffusive component of the scattering cross section for the Dirichlet problem is proportional to θ² where θ is the (small) angle of elevation, and for the Neumann problem it does not depend on θ. For the backscattering case these dependences correspondingly become θ⁴ and θ°. The result is obtained from the structure of the equations that determine the scattering problem rather than by use of an approximation.