Daniel Bouche

Asymptotic and hybrid techniques for electromagnetic scattering

Asymptotic and hybrid methods are widely used to compute the Radar Cross Section (RCS) of objects that are large compared to the wavelength of the incident wave, and the objective of this paper is to present an overview of a number of these methods. The cornerstone of the asymptotic methods is the Geometrical Theory of Diffraction (GTD), which was originally introduced by J. B. Keller, and which represents a generalization of the classical Geometrical Optics (GO) by virtue of the inclusion of diffraction phenomena. After a presentation of the physical principles of GTD, we provide a description of its mathematical foundations. In the process of doing this we point out that GTD gives inaccurate results at caustics and light-shadow boundaries, and subsequently present a number of alternate approaches to dealing with these problems, viz., Uniform theories; Methods for caustics curves; Physical Theory of Diffraction; and Spectral Theory of Diffraction. The effect of coating perfectly conducting bodies with dielectric materials is discussed and hybrid methods, that combine the Method of Moments (MoM) with asymptotic techniques, are briefly reviewed. Finally, the application of GTD and related techniques is illustrated by considering some representative radar targets of practical interest. >

The Boundary Layer Method

We have seen in Chap. 2, how a formal series representation, which is more general than the Luneberg-Kline series, can be used to describe the propagation of the diffracted rays. We have also seen that the formal series provides a description of the field only in regions where it is a ray field, and in the present chapter we will concern ourselves with the calculation of the field in the boundary layers.

High-Frequency Diffraction of a Plane Electromagnetic Wave by an Elongated Spheroid

An asymptotic formula for the problem of diffraction by a strongly elongated body of revolution is constructed. Its uniform nature with respect to the parameter that characterizes the rate of elongation is demonstrated. The results are in good agreement with numerical simulations.

Asymptotic of creeping waves on a strongly prolate body

We study the influence of a small transverse radius of curvature ρt on the acoustic or electromagnetic waves propagating on the surface of a convex body by the boundary- layer method. For ρt of orderk–1/3, the dependency of the field near the surface along the normal is shown to be, as when ρt is of order 1, described by the Fock- Airy function ω1. However, ρt modifies the attenuation and velocity of the waves, by introducing an exponential term. For ρt or order k–2/3, the normal dependency of the field is described by a new special function, depending on ρt The propagation constant of the wave can be obtained by solving an equation involving this function.RésuméLes auteurs étudient l’effet d’un faible rayon de courbure transverse sur les ondes, acoustiques ou électromagnétiques, se propageant à la surface d’un objet convexe, par la méthode de la couche-limite. Quand ρt est d’ordrek−1/3, la dépendance normale du champ de l’onde de surface est décrite, comme pour ρt d’ordre 1, par la fonction de Fock-Airy ω1 Toutefois, ρt modifie l’atténuation et la vitesse des ondes, en introduisant un terme exponentiel dans l’amplitude. Quand ρt est d’ordrek–1/3, la dépendance suivant la normale est décrite par une nouvelle fonction spéciale. L’exposant linéique de propagation de l’onde peut être calculé en résolvant une équation faisant intervenir cette fonction.

FORWARD AND BACKWARD WAVES IN HIGH- FREQUENCY DIFFRACTION BY AN ELONGATED SPHEROID

The asymptotics of induced current of forward and backward waves on a strongly elongated spheroid is constructed by matching the asymptotic representations to the exact solution valid in the vicinity of the rear tip of the spheroid. These asymptotic results are compared with numerical computations.

Asymptotic computation of the RCS of low observable axisymmetric objects at high frequency

A method based on high-frequency asymptotic techniques is described for rapid radar cross section (RCS) computation for arbitrary convex axisymmetric objects whose geometry is described in a computer-aided design (CAD) format. A modified version of the physical theory of diffraction (PTD), which is free from divergence problems at caustics and shadow boundaries and yields good accuracy even for low-RCS objects, is employed. The spurious contributions due to sudden truncation of the physical optics (PO) currents on the shadow boundary, which yield nonphysical results, are removed, and the accuracy of the PTD is enhanced by adding the contributions due to the creeping waves and the fringe-wave currents for discontinuities in the curvature. This modified PTD yields results that are consistent with the geometrical theory of diffraction (GTD) when the stationary phase evaluation of the fields from the induced currents is valid, and also allows the RCS to be computed for the entire range of incidence angles. The results agree well with those computed with an integral equation code. >

High-frequency currents on a strongly elongated spheroid

The problem of high-frequency diffraction by a strongly elongated spheroid is considered. The field in the boundary layer near the surface is represented as a sum of Fourier harmonics with respect to the angle of revolution. Every harmonics is approximated by the sum of forward and backward waves. The forward waves are represented asymptotically by rapidly converging integrals involving Whittaker functions. The amplitudes of forward waves are determined by matching with the incident plane wave. To find the amplitudes of backward waves, the surface of strongly elongated spheroid near its rear tip is approximated by a paraboloid, and the solution proposed by Fock is used. At large distances from the tip, this solution transforms to the sum of incoming and outgoing waves, which are matched to the forward and backward waves, respectively. This defines the amplitudes of backward waves. Finally, the field of backward waves, initially given in the form of a series with respect to the solutions of a complicated dispersion equation, is represented in the form of an integral similar to that for the forward waves. The asymptotic results are compared with a number of numerical tests, which confirm good approximating properties of the derived asymptotic representations.

Etude des ondes rampantes sur un corps convexe vérifiant une condition ďimpédance par une méthode de développement asymptotique

RésuméOn étudie les ondes rampantes se propageant à la surface ďun objet convexe ďimpédance de surface égale à Z. Pour cela, on recherche, par une technique de développement asymptotique, une solution des équations de Maxwell, se propageant suivant une géodésique, vérifiant la condition de radiation de Silver-Müller à ľinfini et la condition ďimpédance à la surface de ľobjet. En utilisant un système de coordonées géodésiques adapté au problème, on obtient une solution explicite. Toutes les composantes des champs électriques et magnétiques s’expriment en fonction des seules composantes des champs suivant la binormale è la géodésique. On montre qu’il existe deux types ďondes rampantes: ľonde rampante électrique (resp magnétique), avec une composante binormale non nulle du champ électrique (resp magnétique). Ces deux ondes rampantes sont écouplées, sauf au voisinage de Z = 1, où on met en évidence un effet de rotation de polarisation analogue à la loi de Rytov.AbstractWe study the creeping waves propagating on a convex object, whose surface impedance is Z. To this end, we seek, by using an asymptotic expansion method, a solution of Maxwell equations, propagating along a geode sic, and satisfying Silver-Müller radiation condition at infinity, and the impedance boundary condition at the surface of the body. By using a geodesic coordinate system suited to the problem, we obtain a closed form solution. The electric and magnetic fields are given in term of the components of these fields along the binormal to the geodesic. We show that two types of creeping waves exist: the electric (resp. magnetic) type, with a non zero binormal component of the electric (resp magnetic) field. They are uncoupled, except in the vicinity of Z = 1, where a rotation of the polarization, similar to Rytov’s law, is evidenced.

Calcul du second terme de l’exposant linéique de propagation des ondes rampantes par une méthode de couche limite

RésuméLes ondes rampantes se propagent en s’atténuant à la surface des obstacles convexes. L’exposant linéique de propagation de ces ondes admet un développement asymptotique en puissances fractionnaires du nombre d’onde k. Le premier terme de ce développement est connu dans le cas général, mais le second terme n’a été déterminé que pour un conducteur parfait, et pour des géométries particulières : obstacle de révolution, problèmes canoniques. Dans cet article, les auteurs calculent ce second terme pour un objet convexe quelconque décrit par une condition d’impédance, par une méthode de couche-limite. Le résultat fait apparaître l’effet des paramètres géométriques de la géodésique suivi par le rayon rampant, et des variations de l’impédance de surface.AbstractCreeping waves propagate and decay on the boundary of convex obstacles. The propagation constant of these waves has an asymptotic expansion in fractional powers of the wavenumber k. The first term of this expansion is well-known, but the second term has only been determined for perfectly conducting objects, and specific shapes : bodies of revolution, canonical problems. In this paper, we compute this second term in the case of a general convex object satisfying an impedance boundary condition, by using a boundarylayer method. The result shows the effects of the geometrical parameters of the geodesic along which the creeping wave propagates, and of the variations of the surface impedance.

Ondes rampantes sur un objet convexe Décrit par une condition D’impédance anisotrope

RésuméLes auteurs étudient les ondes rampantes sur un corps convexe décrit par une condition d’impédance anisotrope. On obtient, comme dans le cas d’une impédance isotrope, deux modes indépendants. Toutefois, la présence d’un terme extradiagonal non nul dans la matrice d’impédance introduit des effets physiques nouveaux : les deux modes ont des composantes non nulles des champs électrique et magnétique suivant la binormale au rayon rampant; de plus, les exposants linéiques de propagation dépendent de ce terme extradiagonal ; enfin, l’équation donnant l’amplitude de l’onde est modifiée par l’apparition de nouveaux termes, un terme algébrique et un terme exponentiel. Ces effets sont spécifiques au cas où le terme extradiagonal est non nul. Quand il est nul, on retrouve des résultats similaires au cas de l’impédance isotrope.AbstractWe study the propagation of creeping waves on a convex object satisfying an anisotropic impedance boundary condition. We obtain, as in the case of an Isotropie impedance, two independant creeping wave modes. However, the presence of a non zero extra-diagonal term in the impedance matrice is responsible for new physical phenomena : both modes have non zero components of electric and magnetic fields along the creeping ray binormal; moreover, the propagation constants are modified by this extradiagonal term ; finally, the equation giving the amplitude of the creeping wave is modified by new terms, one algebric and one exponential. These effects only appear when the extradiagonal term is different from zero. When this term is equal to zero, the results are similar to the Isotropie impedance case.