## 1. Introduction

[2] Because of its generality and rigorousness the incorporation of the spatial domain method of moments in the mixed-potential integral equation formulation has been one of the most popular methods for analyzing planar microwave integrated circuits [*Tsai et al.*, 1997; *Chang and Zheng*, 1992; *Mosig*, 1988]. With the planar circuits being more complicated, multilayer media are widely employed to make room for more versatile designs [*Tsai et al.*, 1997]. So, many efforts have been made to evaluate the Green's function of the multilayer medium [*Rahmat-Samii et al.*, 1981; *Kathei and Alexopoulos*, 1983; *Chow et al.*, 1991; *Yang et al.*, 1992; *Aksun*, 1996; *Hsieh and Kuo*, 1998], which is crucial for the success of the integral equation methods. All of these efforts fall into roughly two categories: the direct integration approaches and the complex image methods (CIM). The former numerically integrate the well-known Sommerfeld integral along either the real axis [*Kathei and Alexopoulos*, 1983] or a deformed path on the complex plane [*Rahmat-Samii et al.*, 1981]. When the integration is performed along the real axis, singularities of the integrand should be found out and removed in advance. As for the deformed path integration, the associated Bessel function with complex arguments may cause difficulty in obtaining accurate results. Recently, the fast Hankel transformation (FHT) algorithm has been proposed to accelerate the real-axis integration scheme for the calculation of the spatial domain Green's function [*Hsieh and Kuo*, 1998].

[3] By applying approximation techniques to the integrand, CIM can get the closed-form Green's function in the spatial domain finally via the Sommerfeld identity (SI) [*Chow et al.*, 1991; *Yang et al.*, 1992; *Aksun*, 1996]. With spectrum estimation techniques like the generalized pencil of function (GPOF) [*Sarkar and Pereira*, 1995], only a few closed-form complex images (images with complex amplitudes and locations) are needed to approximate the SI over a moderate distance range [*Aksun*, 1996]. The two-level scheme [*Aksun*, 1996] can make the choice of the numbers of complex images and sampling points and the endpoints of the sampling regions robust, which provides accurate representation of the Green's function and is much faster compared with the original one-level approximation.

[4] Microwave circuits are often enclosed in metal boxes to avoid radiation and coupling. The commonly used model for the analysis of the boxed circuits ignores the sidewalls; that is, only a layered structure between the two ground planes is considered. The features of the electromagnetic field by a horizontal electric dipole in such a structure have been studied [*Yang et al.*, 1992]. There exist not only the surface waves of both longitudinal section electric (LSE) and longitudinal section magnetic (LSM) types trapped by layered dielectrics but also the waveguide modes of both types trapped by the two ground planes. For the lossless cases all the propagating modes of these two kinds correspond to the rational parts with the real poles in the spectral domain. The only difference is as follows: All the poles of the propagating surface waves fall in the interval [*k*_{0}, ε_{r,max}*k*_{0}], while all the poles of propagating waveguide modes fall in [0, *k*_{0}]. Since both of them attenuate slowly, they dominate the far field together.

[5] The accurate extraction of surface waves and waveguide modes play an important role on the evaluation of the Green's function in the spatial domain. With these modes extracted analytically, the remainder of the spectral domain Green's function is smooth on the real axis. The smoothed integrand can facilitate the numerical integration along the real axis and is also ready for acceleration techniques such as FHT [*Hsieh and Kuo*, 1998]. The preextraction of the surface waves and waveguide modes is also helpful for CIM, because it can improve the approximation efficiency and provide a more precise characterization for the far field. A recent piece of research has even indicated that the extraction of surface waves is a prerequisite for the CIM when applied to the two-dimensional (2-D) cases [*Zhou et al.*, 1999].

[6] Because of the existence of the layered medium it is hard to accurately determine all the poles of the spectral domain Green's function on the real axis and their residues. Raising the operation frequency will make it even harder since the counts of poles will increase accordingly. To address the difficulty, a new extraction algorithm for the lossless layered isotropic medium was presented in our previous work [*Song and Hong*, 2002]. Different from the commonly used root-search-like algorithms, such as the bisection method, Muller's method, and Davidenko's method [*Muller*, 1956; *Johnson*, 1982; *Talisa*, 1985], there is no need to solve any transcendental equations, and all the real poles and their residues can be found out simultaneously with high accuracy and efficiency.

[7] In the practice of microwave circuit design and production, certain materials used as substrates exhibit dielectric anisotropy (occurring naturally or introduced during the manufacturing process). Furthermore, the anisotropy can also be used to improve the design [*Alexoplous and Krowne*, 1978]. So, in this paper, we will extend the algorithm to the more complex case of a layered uniaxially anisotropic/isotropic medium between the two ground planes. First, the spectral domain Green's function is formulated in terms of the Green's function of the cascading equivalent transmission lines model. Then a technique initialized for the simulation of transmission lines [*Celik and Cangellaris*, 1996] is applied to the model. Thus we can get the rational approximation of the spectral domain Green's function. The parts with real poles can be drawn and transformed into the spatial domain via Cauchy's residue theorem, and thus the propagating surface waves and waveguide modes of both LSE and LSM types are extracted.

[8] This paper is organized as follows: Section 2 presents the principle of the recommended algorithm. In section 3 some numerical examples are given for the validation of the method. In section 4 the conclusion is drawn. Some details about the presented method are described in Appendix A.