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.     

Theoretical and computational aspects of scattering from periodic surfaces: one-dimensional transmission interface

We consider the scattering from and transmission through a one-dimensional periodic surface. For this problem, the electromagnetic cases of TE and TM polarization reduce to the scalar acoustic examples. Three different theoretical and computational methods are described, all involving the solution of integral equations and their resulting discrete matrix system of equations for the boundary unknowns. They are characterized by two sample spaces for their discrete solution, coordinate space and spectral space, and labelled by the sampling of the rows and columns of the discretized matrices. They are coordinate-coordinate (CC), the usual coordinate-space method, spectral-coordinate (SC) where the matrix rows are discretized or sampled in spectral space and spectral-spectral (SS) where both rows and columns are sampled in spectral space. The SS method uses a new topological basis expansion for the boundary unknowns.

Equations are derived for infinite surfaces, then specialized and solved for periodic surfaces. Computational results are presented for the transmission problem as a function of roughness, near-grazing incidence as well as many other angles, density and wavenumber ratios. Matrix condition numbers and different sampling methods are considered. An error criterion is used to gauge the validity of the results.

The computational results indicated that the SC method was by far the fastest (by several orders of magnitude), but that it became ill-conditioned for very rough surfaces. The CC method was most reliable, but often required very large matrices and was consequently extremely slow. It is shown that the SS method is computationally efficient and accurate at near-grazing incidence and can be used to fill a gap in the literature. Extensive computational results indicate that both SC and SS are highly robust computational methods. Spectral-based methods thus provide viable computational schemes to study periodic surface scattering.     

Theoretical and computational aspects of scattering from periodic surfaces: one-dimensional transmission interface

Abstract

We consider the scattering from and transmission through a one-dimensional periodic surface. For this problem, the electromagnetic cases of TE and TM polarization reduce to the scalar acoustic examples. Three different theoretical and computational methods are described, all involving the solution of integral equations and their resulting discrete matrix system of equations for the boundary unknowns. They are characterized by two sample spaces for their discrete solution, coordinate space and spectral space, and labelled by the sampling of the rows and columns of the discretized matrices. They are coordinate-coordinate (CC), the usual coordinate-space method, spectral-coordinate (SC) where the matrix rows are discretized or sampled in spectral space and spectral-spectral (SS) where both rows and columns are sampled in spectral space. The SS method uses a new topological basis expansion for the boundary unknowns.

Equations are derived for infinite surfaces, then specialized and solved for periodic surfaces. Computational results are presented for the transmission problem as a function of roughness, near-grazing incidence as well as many other angles, density and wavenumber ratios. Matrix condition numbers and different sampling methods are considered. An error criterion is used to gauge the validity of the results.

The computational results indicated that the SC method was by far the fastest (by several orders of magnitude), but that it became ill-conditioned for very rough surfaces. The CC method was most reliable, but often required very large matrices and was consequently extremely slow. It is shown that the SS method is computationally efficient and accurate at near-grazing incidence and can be used to fill a gap in the literature. Extensive computational results indicate that both SC and SS are highly robust computational methods. Spectral-based methods thus provide viable computational schemes to study periodic surface scattering.     

Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces

We discuss the scattering of acoustic or electromagnetic waves from one-dimensional rough surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces (TE polarization). The theoretical development is for both infinite and periodic surfaces, the latter equations being derived from the former. We include both derivations for completeness of notation. Several theoretical developments are presented. They are characterized by integral equation solutions for the surface current or normal derivative of the total field. All the equations are discretized to a matrix system and further characterized by the sampling of the rows and columns of the matrix which is accomplished in either coordinate space (C) or spectral space (S). The standard equations are referred to here as CC equations of either the first (CC1) or second kind (CC2). Mixed representation, or SC-type, equations are solved as well as SS equations fully in spectral space.

Computational results are presented for scattering from various periodic surfaces. The results include examples with grazing incidence, a very rough surface and a highly oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics or resonance regime, and a renormalization regime.

The objective of this study was to determine the best computational method for these problems. Briefly, the SC method was the fastest, but it did not converge for large slopes or very rough surfaces for reasons we explain. The SS method was slower and had the same convergence difficulties as SC. The CC methods were extremely slow but always converged. The simplest approach is to try the SC method first. Convergence, when the method works, is very fast. If convergence does not occur with SC, then SS should be used, and failing that CC.     

Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces

Abstract

We discuss the scattering of acoustic or electromagnetic waves from one-dimensional rough surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces (TE polarization). The theoretical development is for both infinite and periodic surfaces, the latter equations being derived from the former. We include both derivations for completeness of notation. Several theoretical developments are presented. They are characterized by integral equation solutions for the surface current or normal derivative of the total field. All the equations are discretized to a matrix system and further characterized by the sampling of the rows and columns of the matrix which is accomplished in either coordinate space (C) or spectral space (S). The standard equations are referred to here as CC equations of either the first (CC1) or second kind (CC2). Mixed representation, or SC-type, equations are solved as well as SS equations fully in spectral space.

Computational results are presented for scattering from various periodic surfaces. The results include examples with grazing incidence, a very rough surface and a highly oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics or resonance regime, and a renormalization regime.

The objective of this study was to determine the best computational method for these problems. Briefly, the SC method was the fastest, but it did not converge for large slopes or very rough surfaces for reasons we explain. The SS method was slower and had the same convergence difficulties as SC. The CC methods were extremely slow but always converged. The simplest approach is to try the SC method first. Convergence, when the method works, is very fast. If convergence does not occur with SC, then SS should be used, and failing that CC.     

A general method for analyzing em wave scattering from arbitrarily shaped two-dimensional periodic surfaces

In this paper the Method of Lines (MoL) is successfully extended to solve the EM wave scattering problems of periodic surfaces with arbitrary profile. As examples, the scattering coefficients of space harmonics of corrugated and sinusoidal surfaces are calculated. The results are in good agreement with available data from Wirgin and from A.K.Jordan et al. In addition, the results of comb structure are also calculated. The flexibility and less computation of this method make it eligible for analyzing various two-dimensional periodic structures.     

Light scattering from rough surfaces with superimposed periodic structures

0 ), where K is the wave vector of the periodic structure and λ₀ is the correlation length for random roughness. The surface height h of the periodic structure plays a less important role in the suppression of the diffuse scattering, but it gives an oscillating term in grating scattering intensity that can produce the “rainbowing” (or coloration) effect for such a surface. In practice, this may result in increased visual brightness of textured metallic surfaces and also in a new and interesting method of surface coloration control. The rapid development of focused beam texturing technologies leaves no doubt that patterns with a given spatial frequency and amplitude can be easily produced in experiments. Received: 16 January 1997/Revised version: 19 June 1997     

Calculations of the Mueller matrix for scattering of light from two-dimensional surfaces

Abstract

Results are presented of an analysis of the Mueller matrix parameters for the problem of scattering of light from two-dimensional rough surfaces. The Mueller matrix fully describes the polarization properties of the scattered light. It is shown, using symmetry arguments, that for normal incidence it is necessary to measure the Mueller matrix terms in only one plane, thus reducing the amount of data to be analysed. Examples of the form of the Mueller matrix terms are calculated using a simple ray-trace approach.     

Field transformational approach to three-dimensional scattering from two-dimensional rough surfaces

In 1985, Tappert and Nghiem-Phu introduced a field-transformation technique for computing rough surface scattering from a parabolic equation model utilizing a split-step Fourier marching algorithm. The approach was based on a two-dimensional parabolic equation with a standard operator approximation that was capable of computing scattering from a one-dimensional rough surface. Although this approach has been used extensively and effectively, extensions of this approach to higher order approximations or three-dimensional propagation have only recently been investigated. In this work, the expressions that incorporate higher-order approximations and three-dimensional scattering from two-dimensional rough surfaces are presented. The implications of some computationally necessary approximations are also provided.     

Implementation and validation of the curvilinear coordinate method for the scattering of electromagnetic waves from two-dimensional dielectric random rough surfaces

The curvilinear coordinate method is applied for analysing 2-D dielectric random rough surfaces. The theory is based on Maxwell's equations written in a non-orthogonal coordinate system. For each medium, this method leads to an eigenvalue system. The scattered fields within two media are expanded as linear combinations of eigensolutions satisfying the outgoing wave condition. The boundary conditions allow the scattering amplitudes to be determined. The coherent and incoherent intensities are estimated by averaging the scattering amplitudes over several realizations. The theory is verified by comparison with results obtained by other exact method. A discussion on the C-method and the Sparse-Matrix CAnonical Grid method is proposed in terms of accuracy and computation time.     

Doppler spectrum of polarimetric scattering field from two-dimensional time-varying nonlinear sea surfaces

Polarimetric scattering model of second-order small-slope approxi-mation combined with “choppy wave" model (CWM) for describing nonlinear hydrodynamic interactions between ocean waves is utilized in this paper to investigate the influence of sea surface nonlinearities on backscattering coefficient as well as Doppler spectrum signatures including Doppler shift and spectral bandwidth. Simulation results show that at moderate to large incidence angles the Doppler shift and spectral bandwidth of the CWM nonlinear sea surfaces are significantly larger than those of linear sea surfaces, in particular at low grazing angles. In addition, Doppler signatures show distinct polarization dependence, and most importantly the cross-polarized Doppler signatures significantly differ from the co-polarized ones. It is also indicated that co-polarized Doppler shift increases obviously with wind speed increasing, whereas the cross-polarized Doppler shift looks less sensitive to wind speed variations. The difference of Doppler signatures between co- and cross-polarization is potentially valuable for ocean remote sensing applications, especially for observing very high winds.     

The use of the virtual source technique in computing scattering from periodic ocean surfaces

In this paper the virtual source technique is used to compute scattering of a plane wave from a periodic ocean surface. The virtual source technique is a method of imposing boundary conditions using virtual sources, with initially unknown complex amplitudes. These amplitudes are then determined by applying the boundary conditions. The fields due to these virtual sources are given by the environment Green's function. In principle, satisfying boundary conditions on an infinite surface requires an infinite number of sources. In this paper, the periodic nature of the surface is employed to populate a single period of the surface with virtual sources and m surface periods are added to obtain scattering from the entire surface. The use of an accelerated sum formula makes it possible to obtain a convergent sum with relatively small number of terms (～40). The accuracy of the technique is verified by comparing its results with those obtained using the integral equation technique.     

Investigation of light scattering from rough periodic surfaces — numerical solutions

The Beckmann scalar scattering model based on the Kirchhoff approximation was used to investigate the scattering of light from periodic surfaces whose roughness amplitude is comparable to, and greater than, the illumination wavelength. Solutions by numerical integration were obtained for surfaces of different profiles and of different roughnesses. It was found that the scattering patterns from these surfaces were very different at large roughness amplitudes. As the incident angle was varied on a given surface, it was also observed that the intensity of any individual diffraction order oscillated and the degree of oscillation was directly related to roughness. By utilizing this property, a new procedure could be developed for surface roughness assessment.     

Theoretical aspects of incoherent nuclear resonant scattering

Hyperfine Interactions - The influence of nonrotational atomic motion on the scattering of X-rays by nuclei with sharp resonances is investigated. Two incoherent scattering channels, nuclear...     

Design of two-dimensional random surfaces with specified scattering properties

A method is proposed for designing a two-dimensional randomly rough Dirichlet surface that, when illuminated at normal incidence, scatters a scalar plane wave with a specified angular distribution of its intensity. The method is validated by computer simulation calculations.     

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.     

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

Abstract

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.     

Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interaction

Abstract

Calculations, using the method of ordered multiple interaction (MOMI), of the scattering of electromagnetic waves from a two-dimensional, randomly rough, perfectly conducting surface with a ratio of RMS height [sgrave] to correlation length a of 1.0 or smaller are presented which demonstrate the robustness of the method. Convergence is achieved in six iterations or less. Some surfaces with [sgrave] = a = 1.0λ and certain topological features exhibited slow convergence. The MOMI inherently will show slow convergence when there are multiple back and forth scatterings. Since resonant scattering is characterized by this type of scattering, this suggests the presence of surface resonances on these surfaces.