Impedance Wedge Research Articles

Polarimetric characterization of debris and faults in the highway environment at millimeter-wave frequencies

In this paper, measurements and models for the polarimetric backscatter response of various paint targets on roads and road surface faults are presented. Of particular interest are debris and faults that could lead to fatal accidents and damage of property. A desired safety feature for automotive radar sensors is the capability of detecting such debris and faults. The detectability of a point target is evaluated by comparing its RCS value with the RCS threshold value defined by the backscatter response of the road surface. Extensive backscatter measurements at W-band were conducted to obtain the backscatter response of typical debris and faults on asphalt surfaces at near grazing incidence angles (76/spl deg/-86/spl deg/). On the other hand, theoretical models, based on diffraction from impedance wedges and scattering from impedance cylinders, respectively, as well as physical optics approximation, were developed to predict the backscatter response of road surface faults and targets with planar facets on road surfaces. Experimental results indicate that detectability in all cases is a function of target size, its azimuthal angle with respect to radar boresight, and the polarization state of the system. The measured backscatter response is used to verify the validity of the theoretical models. Angular polarimetric backscatter measurements of targets defining roadside boundaries such as a concrete curb, a guardrail, and a pebble surface are also presented. The results of these measurements could be used to alert fatigued drivers should their vehicles be heading sideward.

Diffraction of a skewly incident plane wave by an anisotropic impedance wedge – a class of exactly solvable cases

The Sommerfeld–Malyuzhinets’ technique and the special function χ Φ, which is originally introduced in the study of wave diffraction by a wedge located in a gyroelectric medium, have been used to find the exact solution for diffraction of a skewly incident and arbitrarily polarized plane wave by wedges with an arbitrary opening angle and with a class of specific, but in general non-axial anisotropic face impedances. Just for these impedance faces suitable linear combinations of the field components parallel to the edge of the wedge are no longer completely related to each other on the wedge surfaces; an application of the Sommerfeld–Malyuzhinets’ technique to these boundary conditions then leads to inhomogeneous difference equations for the spectral functions; in terms of the χ Φ function these functional equations are transformed to such simple forms that their closed-form exact solutions are given immediately. The uniform asymptotic expansion is then obtained via the method of saddle point. This solution coincides with exact solutions for tensor impedance wedges illuminated by a normally incident plane wave and agrees very well with both analytical perturbation solution as well as numerical results of the method of parabolic equation for a skewly incident plane wave. Typical diffraction behavior dependent on the skewness of the incident wave is also shown.

Far-field analysis of the Malyuzhinets solution for plane and surface waves diffraction by an impedance wedge

The basic problem of determining the far-field scattered from the edge of a wedge of exterior angle 2 Φ with arbitrary impedance conditions on either face is considered. An accurate solution in the form of a Sommerfeld integral obtained by Malyuzhinets is evaluated for kr≫1. A fairly complete discussion of the far-field response is provided, including uniform and non-uniform asymptotic approximations. The far-field is split into edge-diffracted, surface, and geometrical optics waves, including multiply reflected components. The edge-diffracted field is defined by the diffraction coefficient, which we show has a simple factorisation: D= u 0( φ) u 0( φ 0) F Φ ( φ, φ 0), where φ and φ 0 are the source and observation directions, u 0( φ) is the value of the wave function at the edge for a plane wave of unit amplitude incident from the direction φ, and F Φ ( φ, φ 0) involves only trigonometric functions. We demonstrate that the monostatic tip diffraction from a wedge of arbitrary angle can be made to vanish by appropriate choice of the surface impedance. The unique value of impedance is always real, and an explicit formula is given for its evaluation. New results are presented for the reflection and transmission of surface waves on an impedance wedge, including simple approximations for an internal wedge with small Φ. Finally, a complete uniform description of the far-field is given in the format of the Uniform Asymptotic Theory of Diffraction.

The Malyuzhinets theory for scattering from wedge boundaries: a review

The Malyuzhinets technique is reviewed based on his fundamental papers of the 1950s. Subsequent developments are surveyed and recent advances are presented. The review is focused around the basic problem of determining the wave field scattered from the edge of a wedge of exterior angle 2Φ with arbitrary impedance conditions on either face. We begin by establishing a direct relationship between the Sommerfeld integral representation and the Laplace transform. This provides fresh insight into Malyuzhinets’ inferences about functions representable via the Sommerfeld integral and, simultaneously, allows us to prove both the inversion formula for the Sommerfeld integral and the crucial nullification theorem. The special functions η Φ( z) and ψ Φ( z) occurring in Malyuzhinets’ theory of diffraction from a wedge-shaped region are described. Based on this theoretical background we present a detailed derivation of the well-known Malyuzhinets expressions for the wave field diffracted by an impedance wedge. An alternative representation of the Malyuzhinets solution as a series of Bessel functions is also presented that is completely equivalent to the integral form of the Malyuzhinets solution. This permits a description of the wave field in the vicinity of the edge of an impedance wedge, when kr≤1, and simple expressions are given for the tip values of the field and its first derivatives. The edge value u 0 can be expressed in terms of Malyuzhinets functions, and its magnitude is easily evaluated if the impedances of the wedge faces are purely imaginary. Thus, ∣ u 0∣≤ π/Φ with equality only for a wedge with Neumann boundary conditions.

Elliptic Cylinder Modeling of Obstacles for Low Altitude Radar Systems

In this work obstacle models for wave propagation over irregular terrain are investigated. Obstacles can be modeled both as perfectly conducting wedges or ellipses and impedance wedges and ellipses depending on the terrain structure and electrical properties. The irregularities of the terrain can be taken into account by surface roughness attenuation factor. The effects of different obstacle models, topographical and electrical properties of the terrain are calculated using Uniform Geometrical Theory of Diffraction for perfectly conducting and impedance surfaces. Results are compared to each other. It is shown that there is a significant difference between impedance wedge-wedge and impedance ellipse-ellipse models for low target altitudes. To calculate propagation factor precisely ellipse-ellipse model must be used for some types of terrain geometries.

New results for diffraction from an impedance wedge

Acoustic diffraction from the tip of a wedge with impedance boundary conditions is a well covered topic, with extensive work reported in the 1950s. One of the main quantities of interest is the diffraction coefficient, D, for plane-wave incidence. Some new, simple, and useful results for the diffraction coefficient are reported. It will be demonstrated that D factors into the product of three terms: (1) the amplitude at the tip for a unit plane-wave incident from the direction of interest, (2) the same amplitude for a plane wave coming from the observation direction, and (3) an expression involving only trigonometric functions. Furthermore, the magnitudes of the tip, or edge, amplitudes (1) and (2) are very simple if the faces of the wedge are purely reactive (imaginary impedances). The simplicity of the formula for D allows us to answer questions such as: Can the diffraction vanish if the impedances of the wedge faces are properly chosen? The answer is generally in the affirmative, at least for selected incidence and observation directions. These findings will be illustrated by several examples and applications will be discussed.

A UTD solution for the scattering by a wedge with anisotropic impedance faces: skew incidence case

Asymptotic expressions for the fields scattered by an anisotropic impedance wedge at oblique incidence are derived in the context of the uniform geometrical theory of diffraction (UTD). They are obtained by resorting to a perturbative approach, considering the normal incidence case as the imperturbed configuration. We observe that the limits of applicability of this approximate analytical solution extend far beyond those of standard perturbative approaches, allowing us to account for deviations from the normal incidence case of 20/spl deg/ to 30/spl deg/.

Green Function of an Impedance Wedge

ABSTRACT The Green function of an angular sector with impedance boundary conditions is considered on the basis of Malyuzhinets’ solution for diffraction of a plane wave by an impedance wedge of external angle 2Φ. From alternative representations of the Malyuzhinets solution, by integrating them over the plane wave spectrum of a line source, two exact representations, integral and series, of the Green function are constructed so as to provide a complete set of analytic expressions for the Green function, covering all possible positions of the source and observation point with respect to the vertex. Specifically, the integral representation can be asymptotically evaluated in the high-frequency limit, while the series form is more preferential when the source or the observer or both, are located in an immediate vicinity of the edge of the wedge. Assuming that the wedge is acute (Φ>π/2), a complete description of the Green function in the far zone is given including geometrical optic, surface wave, and edge diffracted components. A uniform formula for the Green function is obtained and applied to the analysis of attenuation of an incident cylindrical wave by a wedge-like impedance barrier. Near-field approximations of the Green function are also presented and radiation from a current/ dipole linear source placed near the edge of an impedance wedge is analysed as a function of the wedge angle and face impedances.

EM Scattering from Anisotropic Impedance Wedges at Oblique Incidence: Application to Artificially Hard and Soft Surfaces

ABSTRACT Three-dimensional electromagnetic scattering of plane waves from a class of anisotropic impedance wedges is analyzed. All the configurations considered are characterized by anisotropic impedance faces with their principal anisotropy axes parallel and perpendicular to the edge, with one of the two elements characterizing the impedance tensors vanishing or diverging. Rigorous integral representations for the field are determined by resorting to the Maliuzhinets’ method; they are defined along the Sommerfeld's integration path, so that are suitable for their asymptotic evaluation. Although valid only for specific configurations, we note that the new rigorous solutions can be used for testing the accuracy of numerical or approximate analytical methods, applicable to more general cases. The particular category of impedance tensors considered for the wedge faces can be usefully applied for analyzing the specific case of the artificially hard and soft surfaces.

Diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory: a new class of canonical cases

An original mathematical approach for the diffraction of an electromagnetic skew-incident wave by a wedge allows us to reduce such a problem to new elementary functional equations. Using this method, we present an exact analytical solution with closed form expressions for a class of wedge of any angle with a certain type of anisotropic boundary condition which does not require any field component to vanish on the faces. We thereby consider the important case where the principal axes of anisotropy are along and normal to the edge, the relative impedance matrix attached to each face being diagonal with its determinant equal to unity.

The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium

The diffraction of an inhomogeneous plane wave by an impedance wedge embedded in a lossy medium is analyzed. The rigorous integral representation for the field is asymptotically evaluated in the context of the uniform geometrical theory of diffraction (UTD) so that the asymptotic expressions obtained can be employed in a ray analysis of the scattering from more complex edge geometries located in a dissipative medium. Surface wave excitations at the edge and their propagation along the wedge faces are discussed with particular emphasis on the effects of losses.

Electromagnetic scattering by a wedge with anisotropic impedance faces

Electromagnetic scattering from the edge of an anisotropic impedance wedge, illuminated at oblique incidence, is addressed. In particular, the paper provides a review of existing solutions for this important topic in diffraction theory. Both numerical and analytical techniques, suitable to properly account for the scattering properties of the wedge's anisotropic impedance faces, are considered.

Resistive treatment to reduce edge diffraction from large wedge‐shaped objects and planar antennas

The application of a tapered resistive card to reduce the EM scattering from the edge of an impedance wedge is presented. The problem is two‐dimensional, where the incident field is polarized transverse magnetic or electric to the axis of the wedge, and the resistive card is infinitesimally thin and is attached to the vertex of the wedge. An integral equation is formulated to solve for the scattering from the structure which utilizes a specialized Green's function for the impedance wedge. The integral equation is then solved using a method of moments technique. Since the presence of the wedge is taken into account by the Green's function, unknowns need to be placed only over the region of the resistive card and not along the boundaries of the wedge, thus reducing the size of the moment method matrix. Numerical optimization techniques are then employed to determine the resistance profiles for the cards which yield designs that minimize the scattered field over broad spatial regions and over broad frequency bands as well. The resulting designs for the resistive cards have therefore been optimized to yield maximum performance for scattering reduction. The application of this work to the reduction of edge diffraction from the edges of the ground plane of a microstrip antenna is also discussed, and measured as well as calculated results are presented.

Numerical study of diffraction and slope-diffraction at anisotropic impedance wedges by the method of parabolic equation: space waves

The method of parabolic equation (PE) has been successfully applied to the numerical determination of diffraction, slope-diffraction, and multiple-diffraction coefficients of scalar impedance wedges illuminated by a line source. As a continuation, this paper studies-for the first time to the authors' knowledge-another important canonical problem for the uniform geometrical theory of diffraction (UTD), namely, diffraction and slope-diffraction of an incident cylindrical wave at wedges with anisotropic impedance surfaces, by using the same method. For the diffracted fields, the exact Helmholtz equation is asymptotically approximated by the corresponding parabolic one. It is proved that the sufficient conditions for the unique solution of the Helmholtz equation also guarantee the uniqueness of the solution of the parabolic one. The latter is then efficiently solved by using Crank-Nicholson finite-difference (FD) scheme. Due to the lack of exact solutions, the PE results were compared to uniform asymptotic theory of diffraction (UAT) ones for weak anisotropy and, in this case, very good agreement has been achieved. The diffraction and slope-diffraction behavior dependent upon the measure of the weakness of the anisotropy has been demonstrated by several examples.

Numerical analysis of the diffraction at an anisotropic impedance wedge

A numerical approach to the electromagnetic scattering of a plane wave impinging at skew incidence on the edge of a wedge with two different anisotropic face impedances is presented. The diffracted field is obtained by applying the finite-difference (FD) method to solve two coupled parabolic equations. The solution obtained is valid in the high-frequency region.

FD solution for the diffraction of an inhomogeneous plane wave at an impedance wedge

FD solution for the diffraction of an inhomogeneous plane wave at an impedance wedge

Virtual-ray method and its application in the plane wave scattering by an impedance wedge

The virtual-ray method for treating HF electromagnetic scattering problems is derived from the plane wave of free space, and using this the plane wave scattering by an impedance wedge is studied. In the solution process a novel concept of generalized circle is introduced so that the complete amplitude function is obtained. And a reasonable physical interpretation for the term w2, which was neglected previously, is given. The calculated results agree well with those of the analytical solution obtained by G.D. Maliuzhinets(1958).

Two-dimensional Green's function for a wedge with impedance faces

The solution for the two-dimensional (2-D) Green's function for a wedge with impedance faces is presented. The important feature of this Green's function is that there are no restrictions on the locations for the source and observation points-they can be anywhere. Its development proceeds along two separate lines: one for when the source or observation point is far from the wedge vertex and another one for when it is close. Much of the effort that has been expended in these formulations has been in obtaining forms for the Green's function which are efficient to evaluate numerically. This involved deforming the various contours of integration so that they are rapidly convergent and separating the contributions from the numerous singularities that occur in the integrands and evaluating them in closed form. The formulations that are employed here allow for the individual field components such as the diffracted, geometrical optics, and surface wave components to be identified and studied individually so that a physical understanding for the various scattering mechanisms for the impedance wedge can be appreciated.

Diffraction by a wedge with higher‐order boundary conditions

This paper presents a powerful analytical procedure that combines the higherorder boundary condition approximation and the modified Malyuzhinets technique to treat electromagnetic scattering from arbitrarily angled wedge‐like composite configurations with penetrable faces. The configuration may be either a perfectly conducting wedge covered with a dielectric/ferrite material or a composite wedge with strongly absorbing faces, so that the field transmitted directly through the configuration can be neglected. With special emphasis on retaining the passivity of the media interfaces, boundary conditions on the wedge faces are approximated using impedance conditions with higher‐order field derivatives. To close the formulation of the diffraction problem, a class of contact conditions is described that ensures the uniqueness and reciprocity. A general solution which is valid for passive boundary conditions of arbitrary odd orders is derived in an explicit form as the Sommerfeld integral involving arbitrary constants. By choosing these constants in such a way as to fit its analytical behavior for kr ≪ 1 to the exact one obtained by directly integrating the Maxwell equations, a specific form of the contact conditions can be deduced that reflects the design features of the configurations near their edges and forces the solution to be an asymptotic one for vanishing thicknesses of the coatings or skin layers. As a result, the unique and reciprocal closed‐form solution associated with third‐order boundary conditions is presented which uniformly describes the diffraction of an H‐ polarized plane electromagnetic wave by a variety of the composite wedge‐like structures with penetrable faces, extending thereby Malyuzhinets' solution for an impedance wedge to more sophisticated boundary conditions.

PTD analysis of impedance structures

Based on an approximate dyadic diffraction coefficient, equivalent currents (ECs) are derived for computing the scattering by a finite-length impedance wedge of arbitrary angle. The derived equivalent currents are implemented in a standard general purpose physical theory of diffraction (PTD) code and results are presented demonstrating the accuracy of the formulation for a number of impedance and (dielectrically) coated structures. These include typical shapes such as plates, finite-length cones, and cylinders which have been partially or fully coated. The PTD implementation requires a dyadic physical optics diffraction coefficient which is presented in the appendix.