An Efficient Formulation for Numerical Evaluation of the Green’s Functions of Large Planar Phased Arrays

The present work illustrates a formulation to evaluate spectral integral that represent the array Green’s function (AGF) of large finite planar phased array for observation points close to the array contour. The procedure is based on the AGF representation in terms of a double spectral integral, whose numerical evaluation takes advantage of the exponential attenuation on a proper integration path. Thanks to this convergence properties, the final algorithm is numerically accurate, stable and more efficient with respect to the individual element summation. INTRODUCTION 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 integral equation, which is structured around the ordinary individual element Green's function. For a periodic array, this AGF is composed of the sum over the individual dipole contributions. 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 recent series of papers have investigated about 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 vertexes [1], [2]. The above mentioned formulation, being based on an asymptotic evaluation of the constituent radiation integrals, fails when observing too close to edge and vertexes. The present work is intended to illustrate a complementary strategy for treating observation point also close to the array contour, as implicated by a full-wave scheme. The same scheme, which imposes the solution of an integral equation, needs in most of cases the evaluation of the field at the array plane, i.e., where the boundary conditions are imposed. Consequently, the formulation will be carried out for this case, which is however the most useful and, at the same time, critical from numerical point of view. The procedure is based on the AGF representation in terms of double spectral integral, whose numerical evaluation takes advantage by the exponential attenuation on a proper integration path. Thanks to this convergence properties the final algorithm is numerically accurate and more efficient with respect to the individual element summation. Although example are presented here for the case of free-space Green's function, the same procedure can be modify to account for stratified media. A sample calculation is included to demonstrate the accuracy of this numerical method. FORMULATION A rectangular planar array linearly phased and uniformly illuminated in amplitude can be rigorously decomposed in terms of planar angular sector arrays [2]. Thus, it is not restrictive to start from a sectoral periodic array of linearly phased dipoles located in the z1, z2-plane (Fig.1 a). The inter-element spatial period along z1 and z2 is given by d1 and d2, and the inter-element phase gradient by γ1 and γ2, respectively. All dipoles are oriented along the unit vector p̂ (a bold character denotes a vector quantity, and a caret ∧ denotes a unit vector). A time dependence exp(jωt), is understood and suppressed. In [1] it is shown that the electromagnetic vector field at the observation point y y z z z z ˆ ˆ ˆ 2 2 1 1 + + = r can be represented in the spectral ) , ( 2 1 z z k k form of the free space Green's function Figura 1 a) Geometry of the planar sectoral array of parallel dipoles oriented along a direction p̂ . b) Geometry of the actual and the complementary array; this latter is used for extending the procedure for z1>0, z2>0 . 2 1 2 2 1 1 2 1 2 1 ) ( 2 1 2 ) ( ) ( ˆ ) , ( 8 1 ) ( z z y y k z k z k j z z z z dk dk k e k B k B p k k G y z z + + − ∞