Frequency-domain Green's function for a planar periodic semi-infinite phased array .I. Truncated floquet wave formulation

The array Green's function (AGF) represents the basic constituent for the full wave description of electromagnetic radiation from rectangular periodic arrays and scattering from periodic surfaces. In modeling the performance of such structures: one of the main objectives is the reduction of the often prohibitive numerical effort that accompanies the full-wave analysis based on an integral equation, which is structured around the ordinary individual element Green's function. For a periodic array, this array Green's function is composed of the sum of the individual dipole radiations. Under certain assumptions the integral equation can be restructured around the active Green's function, which is the field collectively radiated by an array of elementary dipoles. A series of papers have investigated the description of the AGF in terms of propagating and evanescent Floquet waves (FWs) together with corresponding FW-modulated diffracted fields which arise from FW scattering at the array edges and vertices. The above mentioned formulation, being based on an asymptotic evaluation of the constituent radiation integrals, fails when observing too close to the edge and vertices. The present work is intended to illustrate a complementary strategy for treating the observation point also close to the array contour, as implicated by a full-wave scheme.

For pt.I see ibid., vol.48, no.1, p.67-74, 2000. This second part of a two-paper sequence deals with the physical interpretation of the rigorously derived high-frequency asymptotic wave-field solution in part I, pertaining to a semi-infinite phased array of parallel dipole radiators. The asymptotic solution contains two parts that represent contributions due to truncated Floquet waves (FWs) and to the corresponding edge diffractions, respectively. The phenomenology of the FW-excited diffracted fields is discussed in detail. All possible combinations of propagating (radiating) and evanescent (nonradiating) FW and diffracted contributions are considered. The format is a generalization of the conventional geometrical theory of diffraction (GTD) for smooth truncated aperture distributions to the truncated periodicity-induced FW distributions with their corresponding FW-modulated edge diffractions. Ray paths for propagating diffracted waves are defined according to a generalized Fermat principle, which is also valid by analytic continuation for evanescent diffracted ray fields. The mechanism of uniform compensation for the FW-field discontinuities (across their truncation shadow boundaries) by the diffracted waves is explored for propagating ad evanescent FWs, including the cutoff transition from the propagating to the evanescent regime for both the FW and diffracted constituents. Illustrative examples demonstrate: (1) the accuracy and efficiency of the high-frequency algorithm under conditions that involve the various wave processes outlined above and (2) the cogent interpretation of the results in terms of the uniform FW-modulated GTD.

This second part of a three-paper sequence deals with the spatial domain parametrization and physical interpretation of the relevant asymptotic high-frequency Green's function for a semi-infinite phased array of parallel dipoles on an infinite stratified grounded dielectric slab. This array Green's function (AGF) has been previously derived using a spectral domain formulation; the relevant asymptotic solution contains contributions associated with Floquet waves (FWs), and corresponding surface, leaky and diffracted waves excited at the array edge. Both the truncated-FW series and the series of corresponding diffracted field contributions exhibit excellent convergence properties. In the present paper, through application of the Poisson summation, the AGF for a plane-stratified grounded dielectric slab is developed in terms of space domain FW-dependent Kirchhoff radiation integrals which are synthesized by superposition of periodicity-modulated phased line sources oriented parallel to the edge. The asymptotic evaluation of each Kirchhoff radiation integral leads to a grouping of various asymptotic terms, which provide physically appealing interpretations of a variety of wave processes, encompassing slab-modulated propagating (radiating) and evanescent (nonradiating) FWs, slab-guided surface waves (SWs) or leaky waves (LWs), and their edge-coupled phenomenologies. The present space domain parametrization leads to the same asymptotics as that from the spectral domain parametrization, but allows a clear description of the spatial wave interaction processes.

A uniform high-frequency solution is presented for the field radiated at finite distance by a semi-infinite beam-scanning array of magnetic line sources located on a perfectly conducting half-plane. The field is represented in terms of Floquet waves plus their relevant singly and doubly diffracted rays, which arise from both the end of the array and the edge of the half-plane. This representation is uniformly valid also when transition conditions from propagating to evanescent Floquet waves occur. Furthermore, it provides a simple and attractive physical interpretation and is found numerically very effective, due to the fast convergence of the Floquet wave expansion for the field.

Floquet wave and element-by-element summation representations are implemented to investigate the radiation from a periodic, rectangular array of linearly phased/sequentially pulsed dipoles in the frequency- and time-domains. A synthetic aperture type formulation for Floquet wave representation and an asymptotic formulation for the element-by-element summation representation are developed. Numerical results are presented to demonstrate the efficiency of the Floquet wave representation in terms of radiating, non-radiating and diffracted wave constituents and then validated via comparison with direct summation over the contributions of individual dipole radiations.

A uniform, high-frequency solution is presented for the electromagnetic field radiated at finite distance by a sectoral array of elementary dipoles placed on an infinite grounded dielectric slab. Our attention is focused to give an efficient asymptotic representation for observation point laying on the array plane, where boundary conditions are applied. This solution is useful for the efficient analysis of large finite printed arrays. The field is represented in terms of a series encompassing propagating and evanescent truncated Floquet waves together with their corresponding diffracted rays, which arise from the edges and the vertex of the array. The diffracted waves contain discontinuities which compensate the disappearance of surface wave and truncated Floquet waves at their pertinent shadow boundaries.

A new hybrid technique has been recently proposed for the analysis of large periodic arrays. This method is based on a high-frequency representation of the active Green's function of a finite array, involving a Floquet wave (FW) expansion with the relevant edge and vertex diffracted ray fields originating at the array border. Using this representation, the unknown current of an appropriate fringe integral equation is expanded in terms of a few basis functions shaped as FW-induced diffracted rays. This representation drastically reduces the numerical effort and provides a physical insight into the mechanism of the array truncation. In this paper, this method is applied to the analysis of large arrays of rectangular waveguides opened on an infinite ground plane. Numerical results are shown to demonstrate the effectiveness and the numerical efficiency of the method.

Several efficient methods are proposed in this paper to evaluate and accelerate the Sommerfeld integrals for stratified media Green's functions, with the emphasis on the case of half-space. For reference, the slowly convergent method of integration along the Sommerfeld integration path (SIP) is first summarily discussed. To speed up the convergence of the SIs, the deformation is made from the SIP to the steepest descent path (SDP). If branch point is surrounded by the contour, the constant phase path passing through the branch point (BSDP) is added to calculate the contribution of the branch point singularity. Along the deformed path, the integrands have a characteristic of constant phase and fast decaying, which results in the contribution of the path integration mainly from the vicinity of the saddle point and the computation time reduced. Further reducing computation time is realized through the asymptotic approximation based stationary phase method or saddle point method (SPA) along the SDP or the BSDP and physical explanation is enlightened. Numerical results show the validity of the proposed methods.

Transmission of a directional antenna in near proximity of a building wall modelled by a semi-finite dielectric slab above a perfectly conducting ground is investigated and an accurate and fast method to predict wall penetration loss (PL) once the source/observation points are in proximity of the wall is proposed. First, the antenna is replaced by an array of electric Hertzian dipoles with approximately the same half power beamwidth. Second, the dipole array radiation is calculated both numerically and asymptotically. They are based on computation of Sommerfeld integrals using Filon quadrature (FQ) technique and the steepest descent method, respectively. Third, PL of a typical x -polarised and y -polarised horn and a z -polarised wire antenna, simulated in FEKO using the method of moments (MoM) is compared with that of, respectively, the x -, y - and z -directed dipole arrays calculated using FQ technique. A good agreement between MoM and the dipole model results are obtained for relatively short stand-off distances for which estimation based on slab transmission coefficient is not an accurate approximation. The proposed dipole array model with FQ technique makes calculation at least 7.5 times faster than FEKO. Fourth, PL for different wall permittivity and conductivity is studied.

Transform inversions, in which density and survival functions are computed from their associated moment generating function $\mathcal{M}$, have largely been based on methods which use values of $\mathcal{M}$ in its convergence region. Prominent among such methods are saddlepoint approximations and Fourier-series inversion methods, including the fast Fourier transform. In this paper we propose inversion methods which make use of values for $\mathcal{M}$ which lie outside of its convergence region and in its analytic continuation. We focus on the simplest and perhaps richest setting for applications in which $\mathcal{M}$ is either a meromorphic function in its analytic continuation, so that all of its singularities are poles, or else the singularities are isolated essential. Asymptotic expansions of finite- and infinite-orders are developed for density and survival functions using the poles of $\mathcal{M}$ in its analytic continuation. For finite-order expansions, the expansion error is a contour integral in the analytic continuation, which we approximate using the saddlepoint method based on following the path of steepest descent. Such saddlepoint error approximations accurately determine expansion errors and, thus, provide the means for determining the order of the expansion needed to achieve some preset accuracy. They also provide an additive correction term which increases accuracy of the expansion. Further accuracy is achieved by computing the expansion errors numerically using a contour path which ultimately tracks the steepest descent direction. Important applications include Wilks’ likelihood ratio test in MANOVA, compound distributions, and the Sparre Andersen and Cramér–Lundberg ruin models.

The theoretical analysis for predicting the acoustic radiation from an infinite fluid-loaded, elastic plate excited by a line force is presented. The acoustic pressure radiated by this coupled plate-fluid system is given in an integral representation. A uniformly asymptotic series solution, obtained by the use of the modified saddle point and subtraction of poles method, is valid in the farfield and the nearfield, and is especially accurate at the coincidence angle where the first-order approximation by the saddle point method overpredicts the acoustic pressure. The uniformly asymptotic series solution also predicts a decaying but nonvanishing surface acoustic pressure while the approximate solution from the saddle point method vanishes at the grazing angle. The solution was also obtained by numerical integration along the steepest descent path. Results obtained from the numerical integration are compared with the uniformly asymptotic series solution and excellent agreement has been observed.

The coverage dependence of the activation energy of desorption for a planar array of electrical dipoles has been calculated in order to understand thermal desorption traces of alkali metal atoms from transition metal surfaces. The successful comparison between the computed spectra and available experimental data for K/Fe(110), K/Fe(100), K/Fe(polycrystalline), and K/Pt(111) allows us to demonstrate that the mutual dipolar repulsion within the adlayer is responsible for the coverage dependence of the thermal desorption spectra (TDS) characteristic of the above mentioned systems. Shifts of the TDS peaks up to about 450 K due to the lowering of ∼50% in the energy of desorption are well described by the proposed model. Based on these results a simple and accurate method to assess the coverage dependence of the desorption energy is presented. The coadsorption of electropositive and electronegative atoms has been modeled assuming a planar array of electrical dipoles of opposite directions. It is demonstrated that oxygen atoms substantially reduce the repulsion energy on the alkali adlayer which becomes thermally stabilized in agreement with published experimental results. In fact, for the systems K+O/Fe(polycrystalline) and K+O/Pt(111) the onset of K desorption is found to be shifted by about 200 K due to the presence of coadsorbed oxygen atoms.

It is demonstrated that, for a planar array of discrete elements, the accuracy of the reconstruction is more dependent upon the location of the reconstruction points than on the actual resolution (i.e., spacing) of the reconstruction points. The surface field distribution of a planar array of infinitesimal dipoles is reconstructed from complex far-field and near-field data with the use of microwave holographic metrology. A comparison is made between the discrete Fourier transform (DFT) and the more traditional fast Fourier transform (FFT) to reconstruct the surface distribution. The former technique gives the user more freedom in choosing the location of the reconstruction points. A 22 by 22 element planar array of infinitesimal dipoles was simulated. The FFT required less computer time to perform the reconstruction than the DFT, but the increased accuracy provided by the DFT warrants its use in cases where the element anomalies are small or a more accurate determination of the amplitude and phase needs to be made.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

: Traveling waves on 2D and 3D infinite periodic arrays of small lossless acoustic monopoles, electric dipoles, and magnetodielectric spheres are investigated. The waves are assumed to propagate along the axis of the arrays. The focus is on obtaining the kd-BETAd equations (diagrams) characterizing the traveling waves. Initial forms of the kd-BETAd equations are obtained by summing the fields scattered from all the array elements incident on a reference element of the array located at the origin of the coordinate system. These initial forms of the kd-BETAd equations are, however, useless for computational purposes since the summations converge extremely slowly. The core of the investigation is, therefore, to convert the slowly convergent summations to forms that can be used for efficient calculation of the kd-BETAd curves. These conversions are performed by first using either the Poisson summation formula or a method based on the use of Floquet mode expansions. Expressions for the efficient summation of Schloemilch series are then used to obtain the final forms of the kd-BETAd equations. Exact computable expressions for the fields of 3D acoustic monopole, electric dipole, and magnetodielectric sphere arrays that are finite in the direction of the array axis, illuminated by a plane wave parallel to the array axis, are obtained from the analyses performed to obtain the kd-BETAd curves for the corresponding infinite arrays.

This paper presents a novel method for synthesizing sparse dipole arrays with shaped patterns by joint optimization of element positions, rotations and phases. Such joint optimization exploits maximum available degrees-of-freedom under the assumption of uniform amplitude excitations to improve the shaped power pattern synthesis performance, and it leads to a reduced number of elements and simpler feeding network without using unequal power dividers for the array. The copolarization (COP) and cross-polarization patterns for a rotated dipole array are derived in a 2-D case, and then the vectorial shaped pattern synthesis problem is formulated as finding the optimal solution by maximally approaching the desired COP pattern shape under multiple constraints on the sidelobe level (SLL) and cross-polarization level (XPL). In addition, an asymmetric mapping method is incorporated into the proposed joint optimization, so that the minimum element spacing constraint can be also used for the synthesis of planar sparse dipole arrays. The comprehensive learning particle swarm optimization (CLPSO) algorithm is adopted to optimize the element positions, rotation angles and phases. A set of synthesis experiments including linear and planar rotated dipole arrays with a different pattern shape requirements are conducted to validate the effectiveness and robustness of the proposed approach. Synthesis results show that the proposed approach can save about 17.24% – 55.86% elements as well as many unequal power dividers for test cases.