Three-dimensional subsurface analysis of electromagnetic scatteringfrom penetrable/PEC objects buried under rough surfaces: use of thesteepest descent fast multipole method

The electromagnetic scattering from a three-dimensional (3D) shallow object buried under a two-dimensional (2D) random rough dielectric surface is analyzed. The buried object can be a perfect electric conductor (PEC) or can be a penetrable dielectric with size and burial depth comparable to the free-space wavelength. The random rough ground surface is characterized with Gaussian statistics for surface height and for surface autocorrelation function. The Poggio, Miller, Chang, Harrington, and Wu (PMCHW) integral equations are implemented and extended. The integral equation-based steepest descent fast multipole method (SDFMM), that was originally developed at UIUC, has been used and the computer code based on this algorithm has been successfully modified to handle the current application. The significant potential of the SDFMM code is that it calculates the unknown moment method surface electric and magnetic currents on the scatterer in a dramatically fast, efficient, and accurate manner. Interactions between the rough surface interface and the buried object are fully taken into account with this new formulation. Ten incident Gaussian beams with the same elevation angle and different azimuth angles are generated for excitation as one possible way of having multiple views of a given target. The scattered electric fields due to these ten incident beams are calculated in the near zone and their complex vector average over the multiple views is computed. The target signature is obtained by subtracting the electric fields scattered from the rough ground only from those scattered from the ground with the hurled anti-personnel mine     

Inversion of roughness profile of heterogeneous fractal surface using Gaussian beam incidence at low grazing angle

As a Gaussian beam is incident upon a rough surface at low grazing angle, the Helmholts scalar wave equation may be replaced by the parabolic approximate equation. As the incident field is known, the scattered field and surface current give the Volterra integral equation. Surface roughness profile can be formulated by the integral equation of the surface currents. These two coupled equations are applied to invert the roughness profile of heterogeneous fractal surface. Using Monte Carlo method, the fractal rough surfaces with a band-limited Weistrass-Manderbrot function are numerically simulated and the scattered fields along a line parallel to the mean surface are solved. The Gaussian beam incidence and scattered fields are used to progressively invert the surface roughness profile. Reconstructed profile and its inverted fractal dimension, roughness variance and correlation length are well matched with the simulated surfaces.     

Depolarization and scattering of electromagnetic waves by irregular boundaries for arbitrary incident and scatter angles full-wave solutions

Explicit expressions are presented for the radiation fields scattered by rough surfaces. Both electric and magnetic dipole sources are assumed, thus excitations of both vertically and horizontally polarized waves are considered. The solutions are based on a full-wave approach which employs complete field expansions and exact boundary conditions at the irregular boundary. The scattering and depolarization coefficients axe derived for arbitrary incident and scatter angles. When the observation point is at the source these scattering coefficients are related to the backscatter cross section per unit area. Solutions based on the approximate impedance boundary condition are also given, and the suitability of these approximations are examined. The solutions are presented in a form that is suitable for use by engineers who may not be familiar with the analytical techniques and they may be readily compared with earlier solutions to the problem. The full-wave solutions are shown to satisfy the reciprocity relationships in electromagnetic theory, and they can be applied directly to problems of scattering and depolarization by periodic and random rough surfaces.     

Numerical simulation of scattering from three-dimensional randomlyrough surfaces

Randomly rough surface patches in three dimensions are generated on the computer. The FD-TD method is used to compute scattering from surface patches by converting the Maxwell's equations into difference equations using a central difference approximation for the space and time derivatives. The volume of grids above the rough surface is divided into the total field and the scattered field regions. In between these two regions, obliquely incident waves are generated. To reduce computation, the volume of grids is chosen to be small, and a transformation is used to convert the scattered field into far zone fields for bistatic scattering coefficient calculations. Possible errors near the edge of the surface due to the use of a relatively small volume are suppressed by introducing a windowing function. Very good agreements are obtained between the results obtained by this method and those calculated by an integral equation method (IEM) for scattering from randomly rough perfectly conducting and dielectric surfaces     

New results on coherent scattering from randomly rough conducting surfaces

An approximate solution for the average field scattered by a perfectly conducting randomly rough surface having a correlation length much smaller than the electromagnetic wavelength is presented. The analysis is based on the use of a substitute surface which gives rise to the same describing equations as the true surface relative to the average scattered field. The substitute surface comprises large nonoverlapping fiat areas having random elevations with respect to the mean planar surface and arbitrary correlation between adjacent areas. The average scattered field is shown to depend upon the number of interacting areas and the surface roughness. For a given range of surface roughness there is a specific number of interacting areas which dominate the average scattered field. It is demonstrated how this number can be computed and how a continuous curve of average scattered field as a function of surface roughness is obtained. Of particular importance is the quantitative correspondence established in this paper between the surface roughness and the degree of multiple interaction on the rough surface.     

Coupling Between Guided Surface Waves, Lateral Waves, and the Radiation Fields by Rough Surfaces--Full-Wave Solutions

In this paper explicit expressions are presented for the guided surface waves and lateral waves that are excited when radiation fields are incident upon rough surfaces. Similarly, expressions are presented for the radiation fields scattered by rough surfaces that are excited by surface waves and lateral waves. In addition, coupling between the surface waves and the lateral waves due to surface irregularities is considered in detail. The solutions, which are based on a full-wave approach to the problem, are subject to the exact boundary conditions at the irregular interface. These are shown to be consistent with the reciprocity relationship in electromagnetic theory. The validity of the approximate impedance boundary condition is examined and consideration is given to excitation at the grazing incidence, the Brewster angle, and to waves incident at the critical angle for total internal reflection. Optimum conditions are determined for coupling between the radiation fields, the surface waves, and the Iateral waves incident upon irregular boundaries. Thus this work is applicable to problems of radio wave propagation near an irregular interface between two media and excitation of guided waves by irregular dielectric structures.     

A Two-Step Procedure for Characterizing Obstacles Under a Rough Surface From Bistatic Measurements

A two-step electromagnetic detection procedure is proposed to characterize a dielectric obstacle buried at low depth under a rough surface from single-frequency and multistatic data. First, we have developed, in the framework of the small perturbation theory, a correlation procedure of the scattered field, which enables us to recover an estimation of the roughness profile. This method is tested for various cases with synthetic data provided by a rigorous boundary integral solver. Second, the obtained surface profile is introduced into the numerical simulation due to a finite-element code. An iterative process is then used, based on a level-set formulation, to obtain the shape of the buried target. The influence of the prior step on the accuracy of the reconstruction of the target is studied via various criteria and for different configurations.     

An integrated quasi-planar leaky-wave antenna

A new quasi-planar leaky-wave antenna is presented. It consists of microstrip line on one side of the substrate and uniplanar circuit on the other side placed in a partially opened waveguide. The leakage is produced by the excitation of the first higher order (odd) microstrip mode coupled electromagnetically through a slotline on the opposite side of the substrate. Theoretic results based on rigorous Green's impedance integral equation method show that the new microstrip-slotline-coupled leaky-wave antenna has a broadband tuning range via structure parameters and is insensitive to the microstrip line width variation. Measured relative power absorbed (RPA) results indicate that the useful frequency bandwidth agrees with that predicted by rigorous field theory. The measured antenna radiation patterns also agree very well with the approximate theoretic computations. The theory and experiments show that the proposed leaky-wave antenna can interface to feeding structure easily and directly. The new antenna may become a good candidate for microwave and millimeter-wave integrated antenna design     

Full-wave solutions for the scattered radiation fields from rough surfaces with arbitrary slope and frequency

Full-wave solutions are derived for the scattered radiation fields from rough surfaces with arbitrary slope and electromagnetic parameters. These solutions bridge the wide gap that exists between the perturbational solutions for rough surfaces with small slopes and the quasi-optics solutions. Thus it is shown, for example, that for good conducting boundaries the backscattered fields, which are dependent on the polarization of the incident and scattered fields at low frequencies, become independent of polarization at optical frequencies. These solutions are consistent with reciprocity, energy conservation, and duality relations in electromagnetic theory. Since the full-wave solutions account for upward and downward scattering, shadowing and multiple scatter are considered. Applications to periodic structures and random rough surfaces are also presented.     

Full-wave solutions for the depolarization of the scattered radiation fields by rough surfaces of arbitrary slope

Employing a variable coordinate system associated with the local features of two-dimensionally rough surfaces with arbitrary slope, full-wave solutions are derived for the depolarization of the scattered radiation fields. An outline of the analytical procedures used in the derivations of the solutions are presented. Furthermore, the engineer who is not familiar with them can also use the final result which is expressed as a definite integral whose integrand is given explicitly and in closed form. These full-wave solutions are compared with the quasi-optics solution and the iterative or perturbational solutions for slightly rough surfaces, and they are shown to bridge the wide gap that exists between them. The full-wave solutions are consistent with energy conservation, duality, and reciprocity relationships in electromagnetic theory. These solutions account for upward and downward scattering of the incident waves with respect to the horizontal reference plane, thus shadowing and multiple scattering are considered. Applications to two-dimensionally periodic structures and random rough surfaces are also presented. The fullwave solutions are examined for Brewster, grazing, and specular angles and backscatter. Special consideration is also given to good conducting boundaries.     

On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface

A method for the scattering of electromagnetic waves from cylindrical bodies of arbitrary materials and cross sections buried beneath a rough interface is presented. The problem is first reduced to the solution of a Fredholm integral equation of the second kind through the Green's function of the background medium. The integral equation is treated here by an application of the method of moments (MoM). The Green's function of the two-part space with rough interface is obtained by a novel approach which is based on the assumption that the perturbations of the rough surface from a planar one are objects located at both sides of the planar boundary. Such an approach allows one to formulate the problem as a scattering of cylindrical waves from buried cylindrical bodies which is solved by means of MoM. The method is effective for surfaces having a localized and arbitrary roughness. Numerical simulations are carried out to validate the results and to show the effects of some parameters on the total field. The present formulation permits one to get the near and far field expression of the scattered wave.     

Calculation of Radiation from a Helically Cut Waveguide for a Gyrotron Mode Converter in the Quasi-Optical Approximation

We present results of calculations of the radiation from a helically cut waveguide launcher, a so-called Vlasov launcher, which is commonly used either internal or external to a gyrotron for purposes of mode conversion. A gyrotron internal mode converter consists of such a launcher that radiates the waveguide mode as a nearly Gaussian beam in free space followed by a set of mirrors to focus and direct the radiation. The radiation from the launcher is first calculated using a geometric optics representation of the waveguide mode. Then the radiation is calculated in the quasi-optical limit, including diffraction. These analytic results are compared to a rigorous calculation using the computer code SURF3D, which uses an electric field integral equation (EFIE) approach. Good agreement is obtained between the quasi-optical theory and the SURF3D calculation. The present results provide new insights into the accuracy of the quasi-optical theory and may be useful for the design and improvement of Vlasov-type mode converters.     

New Full Wave Theory for Plane Wave Scattering From Rough Dielectric Surfaces—The Correction Current Method: TE Polarization

A new full-wave theory for scattering from rough dielectric surfaces—called the correction current (CC) method—is presented. An iterative solution is developed leading to a first-order scatter pattern in the form of a single integral, making it computationally efficient and capable of showing surface parameter dependencies explicitly. The first-order CC-scatter solution is shown to satisfy reciprocity, to comply with a pattern symmetry relation, and to be accurate over a wide range of surface parameters. The theory is also shown to be capable of quantifying its field errors resulting in an error criterion that is derived from the theory itself (which is not generally available in other theories). Radar cross sections for random rough surfaces with Gaussian statistics are compared to data generated by solving the electric field integral equation using the method of moments. Good agreement was shown to result for a wide range of surface parameters.      

Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface

An analytical solution is presented for the electromagnetic scattering from a dielectric circular cylinder embedded in a dielectric half-space with a slightly rough interface. The solution utilizes the spectral (plane-wave) representation of the fields and accounts for all the multiple interactions between the rough interface and the. buried cylinder. First-order coefficients from the small perturbation method are used for computation of the scattered fields from the rough surface. The derivation includes both TM and TE polarizations and can be easily extended for other cylindrical buried objects (e.g., cylindrical shell, metallic cylinder). Several scattering scenarios are examined utilizing the new solution for a dielectric cylinder beneath a flat, sinusoidal, and arbitrary rough surface profile. Results indicate that the scattering pattern of a buried object below a slightly rough surface differs from the flat surface case only when the surface roughness spectrum contains a limited range of spatial frequencies. Furthermore, the illuminated area of the incident wave is seen to be a critical factor in the visibility of a buried object below a rough surface.     

An exact line integral representation of the physical opticsscattered field: the case of a perfectly conducting polyhedral structureilluminated by electric Hertzian dipoles

An exact line integral representation of the electric physical optics scattered field is presented. This representation applies to scattering configurations with perfectly electrically conducting polyhedral structures illuminated by a finite number of electric Hertzian dipoles. The positions of the source and observation points can be almost arbitrary. The line integral representation yields the exact same result as the conventional surface radiation integral; however, it is potentially less time consuming and particularly useful when the physical optics field can be augmented by a fringe wave contribution as calculated from physical theory of diffraction equivalent edge currents. The final expression for the line integral representation is lengthy but involves only simple functions and is thus suited for numerical calculation. To illustrate the exactness of the line integral representation, comparisons of numerical results obtained from the surface and the line integral representations are performed     

Forward-backward: a new method for computing low-grazing anglescattering

A new technique called “forward-backward” has been developed for solving the magnetic field integral equation (MFIE) for perfectly conducting azimuthally homogeneous surfaces, including cases where the incident radiation has a low-grazing angle. The technique involves splitting both the surface current and the MFIE into two pieces; one representing primarily forwardly scattered energy and one representing primarily backwardly scattered energy. Each of the new integral equations can be solved by an iterative stepping procedure. The technique is applied to two sample problems: the classic Sommerfeld wedge and a peaked surface consisting of two filtered exponentials. The obtained solutions are shown to be accurate for each of these problems     

Scattering from dielectric structures above impedance surfaces andresistive sheets

Interest in understanding of electromagnetic interaction with rough surfaces has prompted the study of scattering from typical dielectric humps over impedance surfaces. It is shown that the Green's function of the problem for a resistive sheet resembles that of the impedance surface. Hence both problems are considered here. A numerical solution for the scattered field of a two-dimensional dielectric object, possibly inhomogeneous, with arbitrary cross section above the impedance surface or resistive sheet is sought. First the Green's function of the problem is derived based on the exact image theory. This form of the Green's function is amenable to numerical computation. Then the induced polarization currents are calculated by casting the integral equations into a matrix equation via the method of moments. Numerical problems in calculation of the Green's function when both source and observation points are close to the surface are discussed. Comparison of numerical results with a perturbation solution shows excellent agreement between the two methods     

Scattering of TE-polarized EM wave by discontinuity in groundeddielectric layer

The two-dimensional problem of EM wave interaction with a dielectric discontinuity in an infinite grounded dielectric layer is studied. An electric field integral equation (EFIE) for TE illumination has been derived based on the Green's function for the electric field produced by induced polarization currents in the discontinuity region. Impressed electric fields consist of either plane waves incident from space above the dielectric layer or surface waves supported by that layer. Method of Moments (MoM) numerical solutions for the induced electric field in the discontinuity region are implemented. The amplitudes of surface waves excited by excess discontinuity-region polarization currents are calculated, as well as the pattern of the scattered field and the associated scattering width. It is observed that the excitation of a surface-wave mode reduces the back scattered radiation for TE-polarized plane wave incidence. The accuracy of the theory is verified by comparison of numerical results with those of existing studies     

Scattering by an arbitrary cylinder at a plane interface: broadsideincidence

The problem of the determination of the fields scattered by an infinite dielectric cylinder of arbitrary cross section located at the interface between two semi-finite dielectric media is reduced to the solution of integral equations for unknown functions defined on the boundaries. These boundary functions are chosen so as to minimize their number. The incident field is that of a plane monochromatic wave. The derivation of the integral equations is given for the transverse electric (TE) mode for a dielectric cylinder and for a perfectly conducting cylinder. The exact electromagnetic fields are obtained from the solutions of the integral equations by integration, and the radar cross section can be computed from the far-field approximation. Sample outputs of the computer programs that implement this solution are shown     

On the theory of Δk radar observations of ocean surface waves

We consider the theory of waves scattered from a moving, rough, and dispersive surface in the small perturbation limit. The first-order scattered field for a time-dependent surface is obtained in the far zone of scattering in terms of the two-dimensional spectral amplitude of the surface and its dispersion relation. We develop a rigorous Δk radar theory and show that the nonzero output of a Δk radar occurs only when the Bragg condition for each signal component is satisfied separately. The frequency correlation function of the scattered field is then proportional to the mean value of the product of the spectral amplitudes of the surface at the corresponding Bragg wavenumbers. The mean value of this product is nonzero only for surfaces that have a locally varying spectrum and is proportional to the Fourier transform (with the argument Δk) of the variation of the local spectrum with respect to the pattern position. Such variations may be caused by either amplitude or phase modulation of the surface structure. In the former case, our results are similar to the results of existing theory. The latter case of phase modulation of the surface (for example, internal waves interacting with capillary waves) cannot be explained by previous theory