Closed-form Expressions for Layered Media Green's Functions That are Reliable Both in the Near Field and in the Far Field

Abstract

The authors have developed an algorithm for the determination of accurate closed- form expressions of Green's functions for multilayered media. The algorithm is based on the fltting of the spectral domain Green's functions in terms of an asymptotic term plus a ratio of two polynomials. The fltting is carried out via the method of total least squares, which is a non-iterative method demanding very few computational resources. The resulting closed-form expressions for the Green's functions consist of a quasistatic term plus a sum of cylindrical waves. The quasistatic term provides the near-fleld behavior and the cylindrical waves provide the farfleld behavior. As a consequence of this, the closed-form expressions are very accurate irrespective of the distance between the source point and the observation point. DOI: 10.2529/PIERS060807124124 The application of the method of moments (MoM) to the solution of mixed potential integral equations (MPIE) has proven to be an e-cient numerical tool for the analysis of planar circuits and antennas (1). In fact, several commercial software products that are currently used in the design of planar structures (such as \Ansoft Ensemble, \Zeland IE3D, and \Agilent Momentum) are based on the solution of MPIE by means of the MoM. In order to solve the MPIE arising from the analysis of planar structures, it is necessary to calculate the Green's functions (GF) for the scalar and vector potentials in multilayered media. These GF can be determined via numerical computation of inflnite integrals that are commonly known as Sommerfeld integrals (SI). However, the highly-oscillatory nature of the integrands involved makes the numerical computation of SI cumbersome and time consuming (1). Some researches have developed speciflc techniques that make it possible to accelerate the nu- merical computation of SI (2,3). Unfortunately, these techniques have the drawback that they have to be repeatedly used every time the distance between the source and observation points changes. One technique for the computation of multilayered media GFs that has received much attention in the last flfteen years is the discrete complex image method (DCIM) (4). This technique makes it possible to obtain closed-form expressions of the GFs without requiring numerical integration. However, the closed-form expressions have been recently found to produce unpredictable errors in the far-fleld computation of the GFs (5). Although it is possible to raise the threshold value of the distance between source and observation points that marks the onset of numerical errors, this is achieved at the expense of a considerable CPU time consumption (5). The sensitivity of the DCIM to far-fleld numerical errors seems to decrease when a surface waves term is included in the closed-form expression of the GFs (4,6). Unfortunately, the surface-waves term contains poles of the spectral domain GFs as well as residues of the spectral domain GFs at these poles, and the accurate determination of both the poles and the residues requires time-consuming algorithms (6). Very recently, the rational function fltting method (RFFM) has been proposed as an alternative technique to DCIM for the determination of closed-form expressions of multilayered media GFs (7). In the new technique the GFs are expressed in terms of a quasistatic term plus a linear com- bination of cylindrical waves. The amplitudes and propagation constants of the cylindrical waves are obtained in the spectral domain by means of a vector fltting algorithm, which is an iterative algorithm. The main drawback of the RFFM is that the closed-form expressions for the GFs in- troduce unacceptable errors in the near-fleld (7). These errors are caused by the Hankel functions representing the cylindrical waves, and in particular, by the non-physical near-fleld singularities present in these Hankel functions. In this paper, we present a revised version of the RFFM which substantially improves the original approach published in (7). In one hand, the determination of