## 1. Introduction

[2] Relatively simple and accurate closed form expressions are obtained, in the format of the asymptotic high frequency UTD ray method [*Kouyoumjian and Pathak*, 1974; *Pathak*, 1992], for describing the first-order diffraction, as well as slope diffraction, by discontinuities in thin, planar 2-D, canonical, isotropic and homogeneous DPS/DNG material configurations which are excited by a line (or line dipole) source. As is commonly classified, DPS materials are those which exhibit positive values of electrical permittivity and permeability while DNG materials are supposed to exhibit negative values for these quantities [*Engheta and Ziolkowski*, 2005]. Actually the UTD solutions obtained here also remain valid if one of these electrical parameters of the material is positive, while the other is negative. The source can be placed nearby or directly on, or even be embedded in the material; furthermore the source can also be allowed to recede far away from the surface. These UTD solutions would be useful for predicting the behavior of the fields radiated by antennas far from, near, or on metallic structures with a finite size material coating, and for predicting the coupling between such antennas. The specific canonical configurations of interest are shown in Figures 1 and 2. In particular, Figure 1a illustrates the diffraction from the junction between two thin, planar DPS/DNG grounded material slabs of different thickness and different electrical properties. Also Figure 2a illustrates the problem of wave diffraction by a thin, DPS/DNG material half plane. The configurations in Figures 1 and 2 are surrounded by free space. The material thickness is assumed to be a small fraction (e.g., one tenth) of the free space wavelength. The uniform line source or line dipole source can be either of the electric or magnetic type. Unfortunately, exact analytical solutions to the above problems of discontinuities in diffraction by material coated metallic surfaces are not currently available in a form suitable and tractable for engineering applications. Also it is noted that conventional numerical approaches for solving problems in Figures 1a and 2a become highly inefficient and lack physical insight. In contrast, the present work, based on a heuristic spectral synthesis method, provides relatively simple closed form solutions that describe, in a physically appealing manner, the fields associated with the various UTD ray contributions, namely, the geometrical optics (GO), surface wave, and diffraction effects, respectively, which contribute to an observation point as shown in Figures 1b and 2b. It is noted that while the source may be near, or on, or far from the surface, it must remain at a distance which is somewhat larger than a free space wavelength from the discontinuity at “0”. When the source is sufficiently far from the discontinuity and from the material surface, and when this source is a line dipole in which the direction of the dipole axis is oriented (or pointed) either directly toward the discontinuity, or close to that direction, then the UTD slope diffraction phenomena dominates over the first-order UTD based diffraction effects. In this case, the pattern of the field incident on the discontinuity from the line dipole is rapidly varying in angle as measured from the dipole axis on which its radiation pattern exhibits a null. One notes that the first-order UTD diffracted field is proportional to the value of the incident field at the discontinuity, while the slope diffraction is proportional to the derivative (slope) of the angular variation of the incident field at the discontinuity; hence, the slope effect alone remains dominant when the first-order diffraction either vanishes or becomes small in comparison. Furthermore, it is important to note that the first-order UTD ray field directly incident from a line source, or a line dipole source, which is placed either in or directly on a DPS/DNG material also vanishes along the material interface in accordance with the Karp-Karal lemma [*Zucker*, 1969], irrespective of the dipole orientation, but again the slope of the field incident from the source in this case does not. Thus, in the latter case, which arises for antennas placed on a DPS/DNG material, the space wave excited directly by the source also undergoes slope diffraction at the discontinuity, in addition to surface wave diffraction which also exists if that source is able to directly excite a surface wave (SW) in the material. Clearly, slope diffraction effects are therefore very important in most practical antenna and scattering problems involving material discontinuities. It is noted that DPS materials typically excite forward surface waves (FSWs), while DNG materials can support backward surface waves (BSWs) [*Engheta and Ziolkowski*, 2005]. BSWs are also included in the UTD solution developed below.

[3] Most previous works in the literature dealing with the diffraction by material discontinuities generally replace the original coated metallic surfaces or material slabs by approximate impedance or transmissive (for the geometry of Figure 2) boundary conditions, respectively, in order to arrive at a rigorous analytical solution to the resulting approximate problem configuration. These previous works primarily addresses the scattering problem in which the illumination is a uniform plane wave that is incident on the thin material discontinuity. In contrast, the present work applies not only to scattering situations but also to antenna problems which are of equally great practical importance. Moreover, slope diffraction effects were not treated in almost all previous works on the problems of scattering by material discontinuities, since the illumination used therein was typically a uniform plane wave. The present work incorporates plane, cylindrical, and surface wave illumination. Among related previous works, the one by *Tiberio et al.* [1989] provides a cylindrical or plane wave diffraction by a 2-D impedance wedge; and another paper by *Manara et al.* [1993] analyzes surface wave diffraction by the same geometry. However, although valid for the more general wedge geometry, those solutions in *Tiberio et al.* [1989] and *Manara et al.* [1993] are given in terms of rather complicated Maliuzhinets functions. Some initial, useful, related work is discussed by *Burnside and Burgener* [1983] where the earlier work by *Kouyoumjian and Pathak* [1974] for a perfectly conducting wedge is generalized heuristically for constructing a UTD solution for the diffraction by a DPS material half plane; however, the resulting diffracted field is non reciprocal and does not satisfy the Karp-Karal lemma on the material half plane; also, the solution does not contain surface wave (SW) effects. In *Luebbers* [1989] and *Nechayev and Constantinou* [2006], the work of *Burnside and Burgener* [1983] is directly extended to study the approximate UTD scattering by a wedge with impedance boundary conditions; these solutions also suffer from the same limitations as those in *Burnside and Burgener* [1983]. A W-H solution is available in *Volakis* [1988] for the plane wave diffraction by a thin DPS material half plane modeled using an approximate transmissive boundary condition. Also *Rojas and Pathak* [1989] analyzed the plane wave diffraction by a junction formed by a thin, planar two-part DPS material backed by an infinite ground plane which was modeled approximately by higher-order impedance boundary conditions on either side of the junction, respectively. Unlike the W-H and Maliuzhinets type solutions based on the impedance type approximation, the present solutions recover the proper, local plane wave Fresnel reflection and transmission coefficients (FRTCs), and surface wave constants, respectively, for the actual material. More importantly, the expressions for the first-order UTD as well as slope diffracted UTD fields obtained in this paper remain free of the complicated integral forms of the W-H split (or factorization) functions, and they also remain free of any complicated Maliuzhinets functions.

[4] In this paper, the solutions to the problems in Figures 1 and 2 are formulated initially in terms of a cylindrical wave spectral (CWS) integral for the scattered field (satisfying the wave equation). First the spectral weight function in the CWS for the problem in Figure 1 is synthesized heuristically based on an ansatz provided by the W-H solution to a special canonical problem of the plane wave diffraction by a two part impedance surface [*Rojas and Pathak*, 1989]. A bisection method, also described by *Rojas and Pathak* [1989], can be employed to directly synthesize a CWS for the problem in Figure 2 in terms of the CWS for the problem in Figure 1.

[5] This paper is organized as follows. Section 2 summarizes UTD ray solutions for the problem in Figure 1, including the slope diffraction terms. The UTD for the launching and diffraction of SWs is also discussed later in that section. The radiation and scattering from canonical problems of interest is calculated in section 3 using the UTD solutions obtained here, and are shown to compare very well with the modified W-H solutions obtained from *Volakis* [1988] and *Rojas and Pathak* [1989], which were developed originally to deal with very thin DPS materials via approximate boundary conditions. The case of a surface wave type antenna on a thin finite material strip, with a perfect electric conductor (PEC) backing is also analyzed via the UTD based on the results presented in section 2. It is noted that all the fields in this work are assumed to have an *e*^{jωt} time dependence which is suppressed throughout the paper.

[6] Although this paper analyzes 2-D configurations, the corresponding 2-D UTD solutions developed here serve as crucial building blocks for constructing the related three-dimensional (3-D) configurations involving point source (or spherical wave) excitation of the geometries in Figures 1 and 2, respectively. The latter 3-D analysis is currently being completed; furthermore, it appears that the development of such 3-D solutions will not only solve more realistic problems, but would also potentially provide a physical picture, based on the UTD ray physics, for describing focusing effects in DNG flat lens geometries. The 3-D solutions pertaining to configurations in Figures 1 and 2, respectively, will be reported separately.