# Showing papers in "IEEE Transactions on Antennas and Propagation in 1990"

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TL;DR: In this paper, an accurate and general procedure for the analysis of electromagnetic radiation and scattering by perfectly conducting objects of arbitrary shape embedded in a medium consisting of an arbitrary number of planar dielectric layers is developed.

Abstract: An accurate and general procedure for the analysis of electromagnetic radiation and scattering by perfectly conducting objects of arbitrary shape embedded in a medium consisting of an arbitrary number of planar dielectric layers is developed. The key step in this procedure is a formulation of the so-called mixed-potential electric field integral equation (MPIE) that is amenable to an existing advanced solution technique developed for objects in free space and that employs the method of moments in conjunction with a triangular-patch model of the arbitrary surface. Hence, the goal is to immediately increase analysis capabilities in electromagnetics, yet remain compatible with the large existing base of knowledge concerning the solution of surface integral equations. Three alternative forms of the MPIE in plane-stratified media are developed, and their properties are discussed. One of the developed MPIEs is used to analyze scatterers and antennas of arbitrary shape that penetrate the interface between contiguous dielectric half-spaces. >

773Â citations

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TL;DR: In this paper, the concept of soft and hard surfaces is treated in detail, considering different geometries, and it is shown that both the hard and soft boundaries have the advantage of a polarizationindependent reflection coefficient for geometrical optics ray fields, so that a circularly polarized wave is circularly polarization in the same sense after reflection.

Abstract: A transversely corrugated surface as used in corrugated horn antennas represents a soft boundary. A hard boundary is made by using longitudinal corrugations filled with dielectric material. The concept of soft and hard surfaces is treated in detail, considering different geometries. It is shown that both the hard and soft boundaries have the advantage of a polarization-independent reflection coefficient for geometrical optics ray fields, so that a circularly polarized wave is circularly polarized in the same sense after reflection. The hard boundary can be used to obtain strong radiation fields along a surface for any polarization, whereas the soft boundary makes the fields radiated along the surface zero. >

677Â citations

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TL;DR: In this paper, a technique to compensate for mutual coupling in a small array is developed and experimentally verified, which consists of a matrix multiplication performed on the received-signal vector.

Abstract: A technique to compensate for mutual coupling in a small array is developed and experimentally verified. Mathematically, the compensation consists of a matrix multiplication performed on the received-signal vector. This, in effect, restores the signals as received by the isolated elements in the absence of mutual coupling. This technique is most practical for digital beamforming antennas where the matrix operation can be readily implemented. >

393Â citations

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TL;DR: In this paper, the theory of chirowaveguides is discussed and their salient features are analyzed, and the corresponding dispersion relations, cutoff frequencies, and propagating and evanescent modes are obtained.

Abstract: The theory of chirowaveguides is discussed, and their salient features are analyzed. It is shown that the Helmholtz equations for the longitudinal components of electric and magnetic fields in chirowaveguides are always coupled and that, consequently, in these waveguides individual transverse electric (TE), transverse magnetic (TM), or transverse electromagnetic (TEM) modes cannot be supported. As an illustrative example, the parallel-plate chirowaveguide is analyzed in detail and the corresponding dispersion relations, cutoff frequencies, and propagating and evanescent modes are obtained. In the dispersion (Brillouin) diagram for a chirowaveguide, three regions are identified: the fast-fast-wave region, the fast-slow-wave region and the slow-slow-wave region. For each of these regions, the electromagnetic field components in a parallel-plate chirowaveguide are analyzed and the electric field components are plotted. Potential applications of chirowaveguides in integrated optical devices, communications systems, and printed circuit antennas are mentioned. >

247Â citations

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TL;DR: In this paper, two antennas are considered, a cylindrical monopole and a conical monopole, which are driven through an image plane from a coaxial transmission line and analyzed by a straightforward application of the finite-difference-time-domain (FD-TD) method.

Abstract: Two antennas are considered, a cylindrical monopole and a conical monopole. Both are driven through an image plane from a coaxial transmission line. Each of these antennas corresponds to a well-posed theoretical electromagnetic boundary value problem and a realizable experimental model. These antennas are analyzed by a straightforward application of the finite-difference-time-domain (FD-TD) method. The computed results for these antennas are shown to be in excellent agreement with accurate experimental measurements for both the time domain and the frequency domain. The graphical displays presented for the transient near-zone and far-zone radiation from these antennas provide physical insight into the radiation process. >

242Â citations

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TL;DR: In this article, three mixedpotential electric field integral equations (MPIEs) for conducting surfaces of arbitrary shape residing in plane-stratified dielectric media with an arbitrary number of layers were formulated.

Abstract: For pt.I see ibid., vol.38, no.3, p.335-44 (1990). In pt.I, three mixed-potential electric field integral equations (MPIEs) for conducting surfaces of arbitrary shape residing in plane-stratified dielectric media with an arbitrary number of layers were formulated. One of the MPIEs (formulation C) was found to be particularly well suited for the application of the method of moments (MM). In pt.II, formulation C is specialized to the important case of a scatterer or antenna of arbitrary shape residing in contiguous half-spaces. This MPIE is solved by the MM employing a triangle-patch model of the surface of the object. Sample numerical results for several cases of interest are presented. >

209Â citations

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TL;DR: In this paper, the FDTD algorithm for the solution of electromagnetic scattering problems is formulated in generalized coordinates in two dimensions and implemented in a code with the lowest-order Bayliss-Turkel radiation boundary condition expressed in cylindrical coordinates.

Abstract: The finite-difference-time-domain (FDTD) algorithm for the solution of electromagnetic scattering problems is formulated in generalized coordinates in two dimensions and implemented in a code with the lowest-order Bayliss-Turkel radiation boundary condition expressed in cylindrical coordinates. It is shown that, for a perfect conductor, such a formulation leads to a stable, well-posed algorithm and that, in regions where the curvature of the coordinate lines is not great, the dispersion and anisotropy effects are negligible. Such effects become more pronounced in regions of high curvature, leading to unphysical phase shifts. The magnitude of such shifts and the amount of wavefront distortion is studied via numerical experiments using a cylindrical mesh. Near-field results are given for two canonical shapes for each polarization: the circular cylinder and cylinders of square and rectangular cross sections. These results are compared with those obtained by exact eigenfunction expansion techniques, with method-of-moments (MM) solutions, and with solutions obtained from an alternate FDTD approach. In each case, agreement is excellent. The propagation of a plane wave through a polar space in the absence of a scatterer is also examined, and it is shown that the FDTD algorithm is capable of tracking the incident wave closely. >

200Â citations

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TL;DR: In this paper, the characteristics of two-layer electromagnetically coupled rectangular patch antennas were investigated and the relative sizes of the parasitic and fed patches were found to have significant effects on the resonant input resistance and bandwidth.

Abstract: Experimental results on the characteristics of two-layer electromagnetically coupled rectangular patch antennas are presented. In addition to the relatively large bandwidth region that occurs when the separation between the two layers is less than 0.15 wavelength, a high-gain region is found when the separation exceeds 0.3 wavelength. The relative sizes of the parasitic and fed patches are found to have significant effects on the resonant input resistance and bandwidth. >

197Â citations

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TL;DR: In this paper, a method for improving the convergence of the series representing the doubly periodic free-space Green's function is presented, which consists of successively applying three different transformations to the green's function spectral representation.

Abstract: A method for improving the convergence of the series representing the doubly periodic free-space Green's function is presented. The method consists of successively applying three different transformations to the Green's function spectral representation. Kummer's transformation is first applied to convert the slowly converging spectral representation into the sum of a rapidly converging series and a slowly converging series. The latter series is recognized as the spectral representation of the original periodic source distribution radiating in a medium with an imaginary wavenumber. Application of the Poisson transformation to this series renders it exponentially convergent since it effectively represents propagation of point source contributions through a medium with imaginary wavenumber. Finally, Shanks' transform is plotted versus the number of terms taken in the series. Numerical results confirm that an improvement in the convergence rate of the series is achieved for a particular convergence criterion. >

177Â citations

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TL;DR: A numerical technique for pattern synthesis in arrays is presented, which allows one to find a set of array coefficients that steer the main beam in a given direction and yield sidelobes meeting a specified criterion, if such a setof array coefficients exists.

Abstract: A numerical technique for pattern synthesis in arrays is presented. For a given set of elements, the technique allows one to find a set of array coefficients that steer the main beam in a given direction and yield sidelobes meeting a specified criterion, if such a set of array coefficients exists. If the pattern specifications cannot be met with the given elements, the algorithm finds the best attainable pattern. The advantage of this technique is that it can be used with an arbitrary set of array elements. Different elements in the array can have different element patterns, and the array can have arbitrary nonuniform spacing between elements. The synthesis technique is based on adaptive array theory. The given array elements are assumed to be used as the elements of an adaptive array. The main beam is pointed in the proper direction by choosing the steering vector for that direction, and the sidelobes are controlled by introducing a large number of interfering signals at many angles throughout the sidelobe region. The algorithm iterates on the interference powers until a suitable pattern is obtained. >

164Â citations

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TL;DR: An analytic model for computing the radiation properties of the quadrifilar helix volute antenna is discussed and various design considerations for GPS applications are presented.

Abstract: An analytic model for computing the radiation properties of the quadrifilar helix volute antenna is discussed and various design considerations for GPS applications are presented. The effects of modifying the antenna length and diameter on the antenna amplitude and phase performance are presented, and using the antenna for dual-frequency operation is discussed. The effects of phase imbalances are presented and compared with measured pattern anomalies. >

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TL;DR: In this article, a monolithic two-dimensional horn imaging array has been fabricated for millimeter wavelengths, where a dipole is suspended in an etched pyramidal cavity on a 1- mu m silicon-oxynitride membrane.

Abstract: A monolithic two-dimensional horn imaging array has been fabricated for millimeter wavelengths. In this configuration, a dipole is suspended in an etched pyramidal cavity on a 1- mu m silicon-oxynitride membrane. This approach leaves room for low-frequency connections and processing electronics. The theoretical pattern is calculated by approximating the horn structure by a cascade of rectangular-waveguide sections. The boundary conditions are matched at each of the waveguide sections and at the aperture of the horn. Patterns at 93 and 242 GHz agree well with theory. Horn aperture efficiencies of 44+or-4%, including mismatch and resistive losses, have been measured. A detailed breakdown of the losses is presented. The coupling efficiency to various f-number imaging systems is investigated, and a coupling efficiency of 24% for an f0.7 imaging system (including spillover, taper, mismatch and resistive losses) has been measured. Possible application areas include imaging arrays for remote sensing, plasma diagnostics, radiometry and superconducting tunnel-junction receivers for radio astronomy. >

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TL;DR: In this article, the equivalence principle is used to predict the radiation pattern of a general class of leaky-wave antennas, consisting of a finite-size source which excites a radially propagating leaky wave on some planar surface.

Abstract: Formulas are derived for the far-infrared radiation pattern of cylindrical leaky waves propagating on a planar surface. The formulas can be used to predict the radiation pattern of a general class of leaky-wave antennas, consisting of a finite-size source which excites a radially propagating leaky wave on some planar surface. Leaky-wave antennas consisting of antenna elements embedded in dielectric layers (microstrip elements) fall into this category. Using the equivalence principle, formulas are derived for both transverse electric (TE) and transverse magnetic (TM) leaky waves with arbitrary propagation constants. The formulas allow for radiation from cylindrical apertures of arbitrary size, so that the effect of truncating the supporting planar surface with an absorbing material can be determined. Particular attention is devoted to the case of a leaky wave for which the real and imaginary parts of the complex propagation constant are equal, since this type of wave has been shown to be responsible for broadside radiation in certain leaky-wave antennas comprised of dielectric layers. >

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TL;DR: In this paper, the current distribution on the patch is rigorously formulated using a cylindrically stratified medium approach, and a set of vector integral equations is derived which governs the current distributions on a patch.

Abstract: The radiation from a cylindrical microstrip antenna excited by a probe is analyzed. Both the cylindrical-rectangular and the wraparound elements are discussed. The current distribution on the patch is rigorously formulated using a cylindrically stratified medium approach. A set of vector integral equations is derived which governs the current distribution on the patch. The set of equations is then solved using a moment method. The input impedance and the radiation pattern are derived both exactly and in the small substrate thickness limit. >

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TL;DR: In this article, a field decomposition is used to simplify the derivation of the solutions of Green's dyadics and the Cherenkov radiation description for a four-parameter biisotropic medium, and Huygens' principle is applied to the problem of a thin linear antenna.

Abstract: Solutions to Maxwell's equations for a four-parameter biisotropic medium are analyzed. A field decomposition provides remarkable simplification in the derivation of solutions and provides a qualitative picture of biisotropy. Green's dyadics and the Cherenkov radiation description are easily derived. A broad class of biisotropic materials that is not optically active is found (for instance, Cherenkov radiation and dipole radiation are linearly polarized). Green's dyadics for a more general constitutive relation, which includes diffusion terms, are solved in terms of the biisotropic solutions. Huygens' principle is obtained and applied in surface integral equation form to the problem of a thin linear antenna in order to demonstrate the feasibility of upgrading existing numerical codes to handle the case of biisotropic materials. >

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TL;DR: In this paper, the equivalence principle is applied to transform the original problem into interior and exterior problems, which are coupled on the exterior dielectric body surface through the continuities of the tangential electric field and magnetic field.

Abstract: A hybrid formulation which combines the method of moments (MM) with the finite element method (FEM) to solve electromagnetic scattering and/or absorption problems involving inhomogeneous media is discussed. The basic technique is to apply the equivalence principle and transform the original problem into interior and exterior problems, which are coupled on the exterior dielectric body surface through the continuities of the tangential electric field and magnetic field. The interior problem involving inhomogeneous medium is solved by the FEM, and the exterior problem is solved by the MM. The coupling of the interior and exterior problems on their common surface results in a matrix equation for the equivalent current sources for the interior and exterior problems. Combining advantages of both methods allows complicated inhomogeneous problems with arbitrary geometry to be treated in a straightforward manner. The validity and accuracy of the formulation are checked by two-dimensional numerical results, which are compared with the exact eigenfunction solution, the unimoment solution, and Richmond's pure moment solution. >

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TL;DR: In this article, a method is discussed to overcome the slow convergence by using the Poisson summation formula and summing in a combination of spectral and spatial domains, and a parameter study is performed to determine an optimum way to weigh the combination of domains.

Abstract: Electromagnetic scattering from periodic structures can be formulated in terms of an integral equation that has as its kernel a periodic Green's function. The periodic Green's function can be derived as a response to an array of line/point sources (spatial domain) or as a response to series of current sheets (spectral domain). These responses are a Fourier transform pair and are slowly convergent summations. The convergence problems in each domain arise from unavoidable singularities in the reciprocal domain. A method is discussed to overcome the slow convergence by using the Poisson summation formula and summing in a combination of spectral and spatial domains. A parameter study is performed to determine an optimum way to weigh the combination of domains. simple examples of scattering from a one-dimensional array of strips and two-dimensional array of plates are used to illustrate the concepts. >

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TL;DR: In this paper, the authors discussed the factors affecting the realizable sidelobe performance of microstrip arrays, including excitation amplitude and phase accuracies, mutual coupling, diffraction effects, positioning errors and errors due to imperfect element matching and feed network isolation.

Abstract: The factors affecting the realizable sidelobe performance of microstrip arrays are discussed and quantified. These include excitation amplitude and phase accuracies, mutual coupling, diffraction effects, positioning errors, and errors due to imperfect element matching and feed network isolation. It is shown that low-sidelobe microstrip arrays require a very tight tolerance on the resonant frequencies of the elements, and the elimination of spurious radiation from the feed network. Cross-polarization and surface wave effects are discussed. An experimental 16-element microstrip array prototype incorporated these considerations into the design, and achieved a -35 dB relative sidelobe level. >

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TL;DR: In this article, the resonance frequency of coax-fed circular microstrip antennas with and without air gaps has been modeled (in its Section III) by incorporating and rearranging some results previously reported by others.

Abstract: For original paper by Abboud et al. see IEEE Trans. Antennas Propagat., vol.38, p.1882-5 (1990). In the original paper, the resonance frequency of coax-fed circular microstrip antennas with and without air gaps has been modeled (in its Section III) by incorporating and rearranging some results previously reported by others. The present author, while working with the formulas of that Section III noticed a discrepancy in (8) and (9). This has been investigated thoroughly and the observations are presented. A brief reply is given by Damiano et al.

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TL;DR: In this paper, a microstrip antenna with large bandwidth was developed using a parasitic technique, which gave a less distorted radiation pattern with frequency and achieved an impedance bandwidth eight times that of a conventional patch antenna of the same size.

Abstract: A microstrip antenna with large bandwidth is developed using a parasitic technique. Compared to the available wideband antennas, the proposed antenna structure is very compact and gives a less distorted radiation pattern with frequency. An impedance bandwidth eight times that of a conventional patch antenna of the same size is achieved. The concept of coupled microstrip line model is extended for theoretical interpretation of the impedance loci. The experimental procedure and results are described, and a theoretical analysis is presented. >

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TL;DR: In this article, an E-field integral equation for the analysis of finite printed circuit antennas with multiple dielectric regions is developed, where the ground plane is considered to be finite.

Abstract: An E-field integral equation for the analysis of finite printed circuit antennas with multiple dielectric regions is developed. In this analysis, the ground plane is considered to be finite. The dielectric substrates may be either lossless or lossy, and they may be inhomogeneous but must be finite. The equivalence principle is used to replace all conducting bodies by equivalent surface electric currents and all dielectrics by equivalent volume polarization currents. The respective boundary conditions on the dielectrics and the conductors are utilized to solve for the electric current on the entire structure. Typical results are presented to illustrate the potential of this method. >

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TL;DR: In this article, ground-based zenith-viewing observations of atmospheric thermal emission were made at frequencies of 20.6, 31.65, and 90.0 GHz.

Abstract: During 1987 and 1988, ground-based zenith-viewing observations of atmospheric thermal emission were made at frequencies of 20.6, 31.65, and 90.0 GHz. At the locations of the experiments (San Nicolas Island, CA, and Denver, CO) radiosonde observations of temperature and humidity were also available. Both National Weather Service and CLASS radiosondes were used in the study. After conversion to attenuation by use of the mean radiating temperature approximation, the data were processed to derive attenuation statistics. Both clear and cloudy attenuation characteristics were examined and compared with results from recent theories. For the clear atmosphere, water-vapor models of Waters (1976) and of Liebe (1989) were compared. At 20.6, and 31.65 GHz, the model of Waters agrees better with measurements; at 90.0 GHz, the model of Liebe is far superior. A recent model of P.W. Rosenkranz (1988) was used for oxygen absorption. For the average mass absorption coefficients for liquid clouds, measurement and theory generally agreed to within 30%. The predictability and interdependence of the three separate channels were also examined. It was found that attenuation for any two channels can predict that of the third to within 25%. >

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TL;DR: Numerical techniques based on the matrix factorization method are developed for reducing the computational complexity of determining the optimal weight vector and a set of eigenvector constraints are used to approximate the effect of the quadratic constraint.

Abstract: A technique of synthesizing an antenna array pattern with prescribed broad nulls is presented. The array pattern synthesis problem is formulated as a least-square null constrained optimization problem. Numerical techniques based on the matrix factorization method are developed for reducing the computational complexity of determining the optimal weight vector. Subsequently, a set of eigenvector constraints are used to approximate the effect of the quadratic constraint. Numerical results are presented to illustrate the performance achievable. >

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TL;DR: In this article, a new near-field/far-field transformation technique is developed, based on the measurement of the near field amplitude over two surfaces surrounding the antenna under test.

Abstract: The possibility of determining the far field of radiating systems by measuring only the near-field amplitude is investigated. The main difficulties of the problem are examined in some detail and a new near-field/far-field transformation technique is developed, based on the measurement of the near-field amplitude over two surfaces surrounding the antenna under test. The accuracy of the far-field reconstruction results are related both to the distance between such surfaces and to some a priori information concerning the near-field phase and/or the radiating system. The information on the radiating system allows relaxation of the need for any information on the near-field phase provided that the distance between the measurement surfaces is high enough. Conversely, the knowledge of a more or less corrupted near-field phase allows reduction of such distances without affecting the accuracy of the far-field reconstruction. Numerical examples validating the effectiveness of the developed algorithm are provided for the planar scanning case. >

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TL;DR: In this paper, the authors explored the characteristics of space and surface-wave fields produced by an electromagnetic source in a multilayered structure using the integral transformation technique, and showed that the space and the surface wave modes are orthogonal along the longitudinal direction with respect to an appropriate weighting function.

Abstract: Characteristics of space- and surface-wave fields produced by an electromagnetic source in a multilayered structure are explored. Using the integral transformation technique it is shown that the space and surface-wave modes are orthogonal along the longitudinal direction with respect to an appropriate weighting function. It is demonstrated that these properties, together with reciprocity, can be utilized to determine the amplitudes of various surface-wave modes produced by an arbitrarily shaped source. Numerical results for the space- and surface-wave power for a circular patch antenna are presented. The study may find application for millimeter-wave printed antennas where the surface waves will play an important role in determining the radiation and impedance characteristics. >

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TL;DR: In this paper, the trends and magnitudes of sea reflectivity at low (1 degrees -10 degrees ) grazing angles were investigated. But at extremely low grazing angles ( > 0.

Abstract: Existing sea clutter models are in general agreement on the trends and magnitudes of sea reflectivity at low (1 degrees -10 degrees ) grazing angles. However, at extremely low grazing angles ( >

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TL;DR: The uniform geometrical theory of diffraction (UTD) solutions can be inversely transformed analytically to obtain a time-domain version of the UTD as discussed by the authors.

Abstract: The uniform geometrical theory of diffraction (UTD) solutions can be inversely transformed analytically to obtain a time-domain version of the UTD. The time-domain solutions are valid in the early time period where an observation time t is close to the time after the arrival of the first diffracted wavefront. Comparisons with GTD (geometrical theory of diffraction) and also with available rigorous results (J.B. Keller and A. Blank, 1951) reveal that the UTD solutions are accurate for substantial early time periods while the GTD (Keller and Blank) results are valid for very early time periods. >

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IBM

^{1}TL;DR: In this article, a full-wave approach is presented for calculating the scattered fields produced by structures that involve finite-size dielectric regions, and the matrix elements are calculated, depending upon the distance between the source and field locations.

Abstract: A full-wave approach is presented for calculating the scattered fields produced by structures that involve finite-size dielectric regions. The dielectric is first approximated by an array of interlocking thin-wall sections; the electric field boundary conditions are then applied through the use of appropriate surface impedances. Rooftop basis functions, chosen to represent the surface current, are appropriately placed on the thin-wall sections in such a way as to accurately represent the polarization current while preventing fictitious charge within the dielectric. Rooftop currents are also used to represent the current on any conductor that may be present. The matrix elements are calculated, depending upon the distance between the source and field locations, through a scheme that employs Taylor series expansions and point source approximations. The technique is applied to scattering from dielectric cubes and composite dielectric-conductor structures, and to radiation from microstrip structures. Numerical convergence and agreement with the literature are demonstrated. >

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TL;DR: In this article, a full-wave analysis for determining the resonant frequency of a rectangular microstrip patch on multiple uniaxial anisotropic layers with or without ananisotropic overlay is presented.

Abstract: A full-wave analysis for determining the resonant frequency of a rectangular microstrip patch on multiple uniaxial anisotropic layers with or without an anisotropic overlay is presented. Two independent methods are used to derive the immittance matrix for the patch, from which the resonant frequency is determined. They are the Hertz vector potentials and the modified spectral domain immittance approach. Numerical results of the resonant frequency are given for several patch configurations, including cases of a patch on a single anisotropic layer, a patch on a double layer with one layer anisotropic and one layer isotropic, a suspended patch resonator with anisotropic substrate, a patch with anisotropic overlay and a patch on two anisotropic substrates with an anisotropic overlay. Changes in the resonant frequency of up to 58% are reported as n/sub x//n/sub y/ is changed from 1.0 (for isotropic substrates) to 2.0. >

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TL;DR: In this article, a relatively simple closed-form asymptotic representation for the single-layer microstrip dyadic surface Green's function is developed, which is surprisingly accurate even for very small (a few tenths of a free-space wavelength) lateral separation of the source and field points along the air-dielectric interface.

Abstract: A relatively simple closed-form asymptotic representation for the single-layer microstrip dyadic surface Green's function is developed. The large parameter in this asymptotic development is proportional to the lateral separation between the source and field points along the air-dielectric interface. This asymptotic solution remains surprisingly accurate even for very small (a few tenths of a free-space wavelength) lateral separation of the source and field points. Thus, using the present asymptotic approximation of the Green's function can lead to a very efficient moment method (MM) solution for the currents on an array of microstrip antenna patches and feed lines. Numerical results based on the efficient MM analysis using the present closed-form asymptotic approximation to the microstrip surface Green's function are given for the mutual coupling between a pair of printed dipoles on a single-layer grounded dielectric slab. The accuracy of the latter calculation is confirmed by comparison with numerical results based on a MM analysis which employs an exact integral representation for the microstrip Green's function. >