IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 20, NO. 4, APRIL 2010 181 Green’s Function Calculation for a Line Source Exciting a 2-D Periodic Printed Structure Simone Paulotto, Member, IEEE, Giampiero Lovat, Member, IEEE, Paolo Baccarelli, Member, IEEE, and Paolo Burghignoli, Senior Member, IEEE Abstract—The electromagnetic field excited by an electric line source in the presence of a printed structure periodic along two directions with arbitrary geometry of the metallization is calcu- lated. The Array Scanning Method (ASM) is adopted to express the field excited by the aperiodic source in terms of an integral su- perposition of suitable auxiliary Floquet-periodic fields. The latter are solutions of electric-field integral equations that are cast in a mixed-potential formulation in the unit cell and are numerically discretized with an efficient implementation of the method of mo- ments in the spatial domain. The accuracy of the proposed ap- proach is validated through an independent ASM code developed for metal-strip gratings and also against consolidated analytical models available for specific strip- and patch-based periodic struc- tures. Index Terms—Array scanning method (ASM), Green’s func- tions, near field, periodic structures, planar structures. I. INTRODUCTION T HE efficient description of electromagnetic interaction between aperiodic sources and periodic structures is of primary importance to develop effective integral-equation solvers for the analysis and design of microwave circuits or antennas operating in periodic environments. Recently, an efficient method, known as Array Scanning Method (ASM) [1], has been proposed for the analysis of line sources in 1-D periodic structures [2] and dipole sources in 2-D periodic artificial materials [3] through a Method of Moments (MoM) approach. In both cases, however, a free-space environment was considered. (General 1-D and 2-D periodic structures with a finite excitation have been studied in [4] and [5], respectively, but in a Finite-Difference Time-Domain framework). Since 2-D periodic structures printed on dielectric layered media (such as frequency-selective surfaces, electromagnetic band-gap structures, and leaky-wave antennas) are of great in- terest, in this letter, we present an efficient ASM-MoM pro- cedure for calculating the Green’s function for an aperiodic source in the presence of a planar structure periodic along two directions, with arbitrary metallization within the spatial period (unit cell), printed on a grounded dielectric slab (see Fig. 1). An electric line source is considered here for simplicity, but the extension to a dipole source can readily be done. A fast Manuscript received September 03, 2009; revised December 01, 2009. First published February 22, 2010; current version published April 07, 2010. S. Paulotto is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77004 USA (e-mail: paulottos@ieee.org). G. Lovat is with the Department of Electrical Engineering, Sapienza Univer- sity of Rome, Rome 00184, Italy. P. Baccarelli and P. Burghignoli are with the Department of Electronic Engi- neering, Sapienza University of Rome, Rome 00184, Italy. Digital Object Identifier 10.1109/LMWC.2010.2042542 Fig. 1. (a) Periodic structure printed on a grounded dielectric slab with arbitrary planar metallization in the unit cell, excited by an electric line source. (b) Detail of the unit cell. periodic MoM solver in the spatial domain is employed for the auxiliary Floquet-Periodic (FP) problem in layered media [6], [7]. The accuracy of the results is illustrated on structures for which well-established homogenized models exist in the low-frequency regime [8], i.e., 1-D arrays of metal strips and 2-D arrays of square patches. Moreover, in the former case, when the line source is parallel to the strips, results from a sim- pler 2D-1D ASM code (as illustrated and implemented in [9]) are also used as a further independent validation. II. ARRAY SCANNING METHOD FOR A LINE-SOURCE EXCITATION OF A 2-D PERIODIC STRUCTURE The electric line source in Fig. 1(a), located at , can be expressed as . To determine the field excited by this source, an auxiliary FP problem is considered, by periodically replicating the source along the direction with phase shift . The resulting auxiliary source is then (1) This auxiliary source can also be obtained by periodically repli- cating along and a source equal to inside the unit cell and zero outside, with a zero phase shift along the axis. The resulting 2D-FP auxiliary problem is illustrated in Fig. 1(b): the boundaries of the unit cell are phase-shift walls with phase shifts and along and , respectively, where and . The original source can be synthesized by exploiting the basic ASM relation (2) 1531-1309/$26.00 © 2010 IEEE