A numerical strategy for evaluating the array Green's function for a class of planar problems

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.

This two-part sequence deals with the derivation and physical interpretation of a uniform high-frequency solution for the field radiated at finite distance by a planar semi-infinite phased array of parallel elementary electric dipoles. The field obtained by direct summation over the contributions from the individual radiators is restructured into a double series of wavenumber spectral integrals whose asymptotic reduction yields a series encompassing propagating and evanescent Floquet waves (FWs) together with corresponding diffracted rays, which arise from scattering of the FW at the edge of the array. The formal aspects of the solution are treated in the present paper. They involve a sequence of manipulations in the complex spectral wavenumber planes that prepare the integrands for subsequent efficient and physically incisive asymptotics based on the method of steepest descent. Different species of spectral poles define the various species of propagating and evanescent FW. Their interception by the steepest descent path (SDP) determines the variety of shadow boundaries for the edge truncated FW. The uniform asymptotic reduction of the SDP integrals, performed by the Van der Waerden (1951) procedure and yielding a variety of edge-diffracted fields, completes the formal treatment.

The development of novel techniques capable of accelerating the analysis of large finite arrays is a subject of current interest for a large variety of engineering applications. Recently, a uniform high-frequency formulation of the Green's function (GF) for semi-infinite or sectoral arrays of dipoles relying on a Floquet wave diffraction theory has been introduced. These results can be exploited for the analysis of large finite phased array; indeed, the GF of such arrays can be efficiently expressed in terms of a series of propagating and evanescent FWs augmented by the relevant diffracted waves arising from the scattering at the edges and vertices of the array. However, this formulation becomes cumbersome for arrays with irregular contour, due to the intrinsic ambiguity in defining edges and tips. This contribution proposes a more general representation, in which the array GF is obtained as the sum of spherical diffracted wave contributions arising from the actual array rim and conical wave contributions coming from the array surface. This is obtained by regarding the planar distribution of sources as the superposition of parallel suitably excited finite linear arrays, and next by asymptotically evaluating the contributions of each linear array. These latter consist of the sum of truncated single-indexed FWs relevant to the infinite linear array plus the corresponding diffracted waves at the two end-points.

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.

Periodic arrays of radiating or scattering elements play an important role in phased array antennas, frequency selective surfaces and related applications. To gain an understanding of the sparsely explored time domain (TD) behavior of such structures, we have initiated a systematic investigation of relevant canonical TD dipole-excited Green's functions (GF), which so far include those for infinite and truncated line periodic arrays, parameterized in terms of TD-Floquet waves (FW) and truncation-induced TD-FW-modulated tip diffractions. Such waves on semi-infinite and finite square arrays of dipoles have been investigated in the frequency domain, and shown to be useful in practical array applications. This article extends our TD studies to an infinite periodic sequentially pulsed planar array. Like the predecessor GFs, this canonical prototype is simple enough to admit closed form exact solutions, whose interpretation is discussed phenomenologically aided by asymptotic parameterization in terms of instantaneous frequencies. Preliminary numerical results demonstrate the efficiency of the TD-FW algorithms.

In the use of the time-domain integral equation (TDIE) method for the analysis of layered media, it is important to have the time-domain layered medium Green's function computed for many source-to-field distances /spl rho/ and time instants t. In this paper, a numerical method is used that computes the mixed potential Green's functions G/sub v/(/spl rho/,t) and G/sub A/(/spl rho/,t) for a multilayered medium for many /spl rho/'s and t's simultaneously. The method is applicable to multilayered media and for lossless or lossy dispersive media. Salient features of the method are: 1) the use of complex /spl omega/ so that the surface wave poles are lifted off the real k/sub /spl rho// axis such that pole extractions are not required; 2) the use of half-space extraction so that the integrand for the Sommerfeld integral decays exponentially along the k/sub /spl rho// axis to obtain fast convergence of the integral; and 3) the use of the fast Hankel transform so that the Green's function is calculated for many values of /spl rho/ simultaneously. For a four-layer medium, we illustrate the numerical results by a three-dimensional plot of /spl rho/G/sub v/(/spl rho/,t) versus /spl rho/ and t and demonstrate the space-time evolution of these Green's functions. For a maximum frequency range of 8 GHz, the method requires only a few CPU minutes to compute a table of 100 (points in /spl rho/) /spl times/ 168 (points in t) uniformly spaced values of G/sub v/(/spl rho/,t) on an 867-MHz Pentium PC.

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.

SECTION 12 - Green's Function

Electromagnetic fields created in a homogeneous medium by a known distribution of electric current are determined by integral equations involving free space Green's functions. In actual situations, however, current distributions are not known beforehand and must be determined, in such a way that the total fields satisfy the boundary conditions on the conductors. The integral equation technique was extended to treat also heterogeneous structures, in order to analyze and design multilayer printed circuits and antennas, such as the broadband strip-slot-foam-inverted patch printed antenna. Sommerfeld integrals provide Green's functions for stratified media, which are evaluated by means of specifically developed, efficient computer algorithms. Structures including coaxial or line feeds and coupling through slots can now be easily analyzed with commercially available software packages.

We consider the method of evaluating matrix elements of electromagnetic volume integral equations with the help of suitably constructed Laplacian-type representations of singular kernels (Green functions) appearing in electromagnetic volume and surface integral equations. The method consists of representing the singular kernels (Green functions) in terms of a generalized Laplacian operator acting on some auxiliary functions. Such a representations allow us, by using Gauss divergence theorem, to convert volumetric and surface integrals representing matrix elements to integrals always involving only non-singular integrands. The task of of finding particular Laplacian representation of different kernels amounts to solving appropriate ordinary or partial inhomogeneous differential equations. We apply the Laplacian method to analytic evaluation of matrix elements of the volume integral equation by employing two different Laplacian representations of the integral equation kernels. The first representation allows us to reduce 6-dimensional volume integrals over tetrahedra to 2-dimensional surface integrals over tetrahedra faces. The second representation is subsequently used reducing surface integrals over tetrahedra faces to pairs of of 1-dimensional line integrals over tetrahedra edges. We also derive analytic expressions for the resulting line integrals given in terms of elementary functions.

In this paper, we theoretically characterize the performance of array of plasmonic core-shell nano-radiators located over layered substrates. Engineered substrates are investigated to manipulate the radiation performance of nanoantennas. A rigorous analytical approach for the problem in hand is developed by applying Green's function analysis of dipoles located above layered materials. It is illustrated that around the electric scattering resonances of the subwavelength spherical particles, each particle can be viewed as an induced electric dipole which is related to the total electric field upon that particle by a polarizability factor. Utilizing this, we can effectively study the physical performance of such structures. The accuracy of our theoretical model is validated through using a full-wave finite difference time domain (FDTD) numerical technique. It is established that by novel arraying of nano-particl and tailoring their multilayer substrates, one can successfully engineer the radiation patterns and beam angles. Several optical nanoantennas designed on layered substrates are explored. Using the FDTD the effect of finite size substrate is also explored.

This paper presents an analytical and simulation-based treatment of the far-field radiation characteristics of a Multiple Log-Periodic Dipole Array (MLPA) antenna using Green’s function–based magnetic vector potential formulations. The study consolidates established electromagnetic theory by explicitly combining log-periodic element scaling, cumulative spatial phase delays, and array-level superposition into a single, transparent analytical workflow. The scalar free-space Green’s function is employed to derive the magnetic vector potential, from which far-field electric field expressions are obtained under standard approximations. Radiation characteristics such as half-power beamwidth, peak directivity, and sidelobe levels are extracted from MATLAB-based simulations and compared with representative theoretical LPDA performance ranges reported in the literature. The results demonstrate consistency with expected broadband LPDA behavior and serve primarily to illustrate the applicability of Green’s function methods to hierarchically structured log-periodic arrays. The work is intended as a reproducible analytical reference and pedagogical baseline for MLPA modeling, rather than as a replacement for full-wave numerical solvers or experimental validation.

The spatial frequency technique (SFT) was developed for solving the integral equation (IE) in electromagnetic scattering and radiation problems and shown to work faster than many existing solution methods. In this paper, the new formulation, which is based on the efficient FFT algorithm, is analyzed further for analytically solving the IE for certain types scattering and radiation problems. It is shown that the SFT description of the symmetric matrix which represents the Green's function may be decomposed into an orthogonal basis using Takagi factorization. Transforming the Green's function into this orthogonal basis has the potential to invert the Green's function analytically. The SFT is also compared with the Fast Multipole Method (FMM) and shown to have similar underlying properties which may also be utilized to analytically invert the Green's function using FMM.

A new approach for the scattering of electromagnetic (EM) waves from periodic dielectric rough surfaces is addressed. The method is an extension of the buried object approach (BOA), which is developed for rough surfaces of infinite extend, to the present problem. The BOA allows to model the original problem as the scattering of EM waves from cylindrical objects located in a two-half-space medium with planar interface. Then, the problem is reduced to the solution of a Fredholm integral equation of second kind through the periodic Green's function of two-half-space medium. The periodic Green's function of two-half-space medium is calculated via the Floquet mode expansion, whose numerical evaluation can be accelerated by using effective methods. The method can also be used to solve the scattering problems of rough surfaces of infinite extend and having a localized roughness. Numerical simulations show that the method yields effective and accurate results for surfaces of arbitrary variation.

In this paper, we present the multilevel Green's function interpolation method (MLGFIM) for analyses of three-dimensional doubly periodic structures consisting of dielectric media and conducting objects. The volume integral equation (VIE) and surface integral equation (SIE) are adopted, respectively, for the inhomogeneous dielectric and conducting objects in a unit cell. Conformal basis functions defined on curvilinear hexahedron and quadrilateral elements are used to solve the volume/surface integral equation (VSIE). Periodic boundary conditions are introduced at the boundaries of the unit cell. Computation of the space-domain Green's function is accelerated by means of Ewald's transformation. A periodic octary-cube-tree scheme is developed to allow adaptation of the MLGFIM for analyses of doubly periodic structures. The proposed algorithm is first validated by comparison with published data in the open literature. More complex periodic structures, such as dielectric coated conducting shells, folded dielectric structures, photonic bandgap structures, and split ring resonators (SRRs), are then simulated to illustrate that the MLGFIM has a computational complexity of O(N) when applied to periodic structures.